Cross helicity and related dynamo

The turbulent cross helicity is directly related to the coupling coefficients for the mean vorticity in the electromotive force and for the mean magnetic-field strain in the Reynolds stress tensor. This suggests that the cross-helicity effects are important in the cases where global inhomogeneous flow and magnetic-field structures are present. Since such large-scale structures are ubiquitous in geo/astrophysical phenomena, the cross-helicity effect is expected to play an important role in geo/astrophysical flows. In the presence of turbulent cross helicity, the mean vortical motion contributes to the turbulent electromotive force. Magnetic-field generation due to this effect is called the cross-helicity dynamo. Several features of the cross-helicity dynamo are introduced. Alignment of the mean electric-current density J with the mean vorticity Ω , as well as the alignment between the mean magnetic field B and velocity U , is supposed to be one of the characteristic features of the dynamo. Unlike the case in the helicity or α effect, where J is aligned with B in the turbulent electromotive force, we in general have a finite mean-field Lorentz force J ?×? B in the cross-helicity dynamo. This gives a distinguished feature of the cross-helicity effect. By considering the effects of cross helicity in the momentum equation, we see several interesting consequences of the effect. Turbulent cross helicity coupled with the mean magnetic shear reduces the effect of turbulent or eddy viscosity. Flow induction is an important consequence of this effect. One key issue in the cross-helicity dynamo is to examine how and how much cross helicity can be present in turbulence. On the basis of the cross-helicity transport equation, its production mechanisms are discussed. Some recent developments in numerical validation of the basic notion of the cross-helicity dynamo are also presented.     

Topological invariants of field lines rooted to planes

Abstract

Two open curves with fixed endpoints on a boundary surface can be topologically linked. However, the Gauss linkage integral applies only to closed curves and cannot measure their linkage. Here we employ the concept of relative helicity in order to define a linkage for open curves. For a magnetic field consisting of closed field lines, the magnetic helicity integral can be expressed as the sum of Gauss linkage integrals over pairs of lines. Relative helicity extends the helicity integral to volumes where field lines may cross the boundary surface. By analogy, linkages can be defined for open lines by requiring that their sum equal the relative helicity.

With this definition, the linkage of two lines which extend between two parallel planes simply equals the number of turns the lines take about each other. We obtain this result by first defining a gauge-invariant, one-dimensional helicity density, i.e. the relative helicity of an infinitesimally thin plane slab. This quantity has a physical interpretation in terms of the rate at which field lines lines wind about each other in the direction normal to the plane. A different method is employed for lines with both endpoints on one plane; this method expresses linkages in terms of a certain Gauss linkage integral plus a correction term. In general, the linkage number of two curves can be put in the form L=r + n, |r|≦1J2, where r depends only on the positions of the endpoints, and n is an integer which reflects the order of braiding of the curves.

Given fixed endpoints, the linkage numbers of a magnetic field are ideal magneto-hydrodynamic invariants. These numbers may be useful in the analysis of magnetic structures not bounded by magnetic surfaces, for example solar coronal fields rooted in the photosphere. Unfortunately, the set of linkage numbers for a field does not uniquely determine the field line topology. We briefly discuss the problem of providing a complete and economical classification of field topologies, using concepts from the theory of braid equivalence classes.     

Broken ergodicity,magnetic helicity,and the MHD dynamo

We consider an unforced, incompressible, turbulent magnetofluid constrained by concentric inner and outer spherical surfaces. We define a model system in which normal components of the velocity, magnetic field, vorticity, and electric current are zero on the boundaries. This choice allows us to find a set of Galerkin expansion functions that are common to both velocity and magnetic field, as well as vorticity and current. The model dynamical system represents magnetohydrodynamic (MHD) turbulence in a spherical domain and is analyzed by the methods similar to those applied to homogeneous MHD turbulence. We find a statistical theory of ideal (i.e. no dissipation) MHD turbulence analogous to that found in the homogeneous case, including the prediction of coherent structure in the form of a large-scale quasistationary magnetic field. This MHD dynamo depends on broken ergodicity, an effect that is enhanced when total magnetic helicity is increased relative to total energy. When dissipation is added and large scales are only weakly damped, quasiequilibrium may occur for long periods of time, so that the ideal theory is still pertinent on a global scale. Over longer periods of time, the selective decay of energy over magnetic helicity further enhances the effects of broken ergodicity. Thus, broken ergodicity is an essential mechanism and relative magnetic helicity is a critical parameter in this model MHD dynamo theory.     

Non-linear hydrodynamic stability of oceanic flows

Abstract

In this paper an analytical method to study the hydrodynamic stability of simple barotropic, non-divergent flows is discussed. The method is based on the variational approach introduced by Arnold and derived from the Lyapunov stability criteria. In this context, the sufficient condition for the stability of a steady barotropic flow ψ(x,y) is obtained when dP(ψ)/dP_{ψ = ψ}, the derivative of the absolute vorticity P(ψ), is positive definite. In this case, we discuss the effect of higher derivatives dⁿP(ψ)/dψⁿψ_{ψ = ψ} on the non-linear stability. Then we show that some classical examples of oceanic non-divergent flows (i.e. lee waves downstream an Island, steady flows through a Strait, the Fofonoff gyre) are stable to finite-amplitude perturbations.     

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.     

The instability of barotropic circular vortices

Abstract

The linear, normal mode instability of barotropic circular vortices with zero circulation is examined in the f-plane quasigeostrophic equations. Equivalents of Rayleigh's and Fjortoft's criteria and the semicircle theorem for parallel shear flow are given, and the energy equation shows the instability to be barotropic. A new result is that the fastest growing perturbation is often an internal instability, having a finite vertical scale, but may also be an external instability, having no vertical structure. For parallel shear flow the fastest growing perturbation is always an external instability; this is Squire's theorem. Whether the fastest growing perturbation is internal or external depends upon the profile: for mean flow streamfunction profiles which monotonically decrease with radius, the instability is internal for less steep profiles with a broad velocity extremum and external for steep profiles with a narrow velocity extremum. Finite amplitude, numerical model calculations show that this linear instability analysis is not valid very far into the finite amplitude range, and that a barotropic vortex, whose fastest growing perturbation is internal, is vertically fragmented by the instability.     

Generation and evolution of interplanetary slow shocks

It is well known that most MHD shocks observed within 1 AU are MHD fast shocks. Only a very limited number of MHD slow shocks are observed within 1 AU. In order to understand why there are only a few MHD slow shocks observed within 1 AU, we use a one-dimensional, time-dependent MHD code with an adaptive grid to study the generation and evolution of interplanetary slow shocks (ISS) in the solar wind. Results show that a negative, nearly square-wave perturbation will generate a pair of slow shocks (a forward and a reverse slow shock). In addition, the forward and the reverse slow shocks can pass through each other without destroying their characteristics, but the propagating speeds for both shocks are decreased. A positive, square-wave perturbation will generate both slow and fast shocks. When a forward slow shock (FSS) propagates behind a forward fast shock (FFS), the former experiences a decreasing Mach number. In addition, the FSS always disappears within a distance of 150R_⊙ (where R_⊙ is one solar radius) from the Sun when there is a forward fast shock (with Mach number \geq1.7) propagating in front of the FSS. In all tests that we have performed, we have not discovered that the FSS (or reverse slow shock) evolves into a FFS (or reverse fast shock). Thus, we do not confirm the FSS-FFS evolution as suggested by Whang (1987).     

Barotropic local instability and severe storm process

By means of barotropic model, the characteristic and initial value problems are investigated to reveal the local two-dimensional barotropic instability of the nonuniform current to the dynamic mechanism of the formation of the Yangtze-Huaihe River severe storm in July 1991. Analytical theory and numerical experiment show that (i) the unstable developing modes are chiefly the two periods of about 44 d and 10 d, which are fundamentally consistent with that of the precipitation change of the Yangtze-Huaihe River. (ii) The growth rate of the local perturbation is dominated by the meridional wave numbern = 1–5 and zonal wave numberk = 1–12, i.e. the severe storm over the Yangtze-Huaihe River results from the interaction of the systems at different latitudes and waves of different scales, (iii) The perturbation over the Yangtze-Huaihe River possesses the property of local intensification, which slowly migrates westward over the lower and middle reaches of the Yangtze-Huaihe River. (iv) The growth rate of the instability, especially the propagation velocity of the perturbation, is sensitive to the external parameters ū and α. Project supported by the National Natural Science Foundation of China.     

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.     

Controlling boundary conditions with a four-dimensional variational data-assimilation method in a non-stratified open coastal model

An innovative way to take the large-scale circulation influence into account in coastal primitive-equation models is explored by an inverse modelling approach. Restricted to barotropic external forcing, this work is a first step in the development of a four-dimensional variational (4DVAR) data-assimilation approach to estimate the best initial and open-boundary conditions that force a coastal model according to interior observations. This development is founded on the OPA modelling system which representation of barotropic coastal dynamics is restricted to motions of long time scales ( a day) due to its rigid lid approximation. Twin experiments are performed in an academic configuration of the Gulf of Lions (located in the northwestern Mediterranean Sea) to study the sensitivity of a remote barotropic forcing to different observational networks measuring surface currents deployed in this area. Three monitoring designs are tested for a large-scale barotropic perturbation in the hindcast mode. It is shown that the space and time distribution of observations acts on the efficiency of the 4DVAR method and then allows coarser datasets.Responsible Editor: Phil Dyke     

地球磁尾中不同类型磁结构的磁螺度演化特征

在二维三分量MHD数值模拟的基础上 ,对地球磁尾不同类型磁结构的形成作磁螺度分析 .研究表明 ,对于由晨昏电场产生的磁尾驱动重联过程 ,通过系统边界输运的磁螺度通量是引起系统总磁螺度变化的直接原因 .不同的初始磁螺度密度分布和磁螺度通量输运 ,可以引起中性片区域磁螺度密度分布的不同演化 ,从而导致具有不同拓扑位形磁结构的形成 .     

Buoyancy-driven perturbations in a rapidly rotating,electrically conducting fluid: part II – dynamo action

The helicity, electromotive force and α-effect produced in a homogeneous, rapidly rotating, electrically conducting fluid by an isolated source of buoyancy at small Elsasser number are calculated, visualized and analyzed. Due to physical symmetries of the system, the integrals of helicity and electromotive force over all space are zero. However, each has a significant non-zero value when integrated over the cross section of the Taylor column. The local α-effect is found to be significantly anisotropic; it is strongest when the applied magnetic field is toroidal and the resulting EMF is parallel to the applied field.     

Zonal shear flows with a free surface: Hamiltonian formulation and linear and nonlinear stability

General theoretical results via a Hamiltonian formulation are developed for zonal shear flows with the inclusion of the vortex stretching effect of the deformed free surface (equivalent barotropic model). These results include a generalization of the Flierl–Stern–Whitehead zero angular momentum theorem for localized nonlinear structures (whether or not on a β-plane), and sufficient conditions for linear and nonlinear stability in the Liapunov sense–the latter are given as estimates in terms of an L ²-type perturbation norm which are global in time and are derived via bounds on the equilibrium potential vorticity gradient.     

Kinematic dynamos associated with large scale fluid motions

Abstract

A standard approach to the kinematic dynamo problem is that pioneered by Bullard and Gellman (1954), which utilizes the toroidal-poloidal separation and spherical harmonic expansion of the magnetic and velocity fields. In these studies, the velocity field is given as a combination of small number of toroidal and poloidal harmonics, with their radial dependences prescribed by some physical considerations. Starting from the original paper of Bullard and Gellman (1954), a number of authors repeated such analyses on different combination of velocity fields, including the most recent and comprehensive effort by Dudley and James (1989). In this paper, we re-examine the previous kinematic dynamo models, using the computer algebra approach initiated by Kono (1990). This method is particularly suited to this kind of research since different velocity fields can be treated by a single program. We used the distribution of magnetic energies in various harmonics to infer the convergence of the results.

The numerical results obtained in this study for the models of Bullard and Gellman (1954), Lilley (1970), Gubbins (1973), Pekeris et al. (1973), Kumar and Roberts (1975), and Dudley and James (1989) are consistent with the previously reported results, in particular, with the extensive calculation of Dudley and James. In addition, we found that the combination of velocities used by Lilley can support the dynamo action if the radial dependence of the velocity is modified.

We also examined the helicity distributions in these dynamo models, to see if there is any correlation between the helicity and the efficiency of dynamo action. A successful dynamo can result both from the cases in which the helicity distributions are symmetric or antisymmetric with respect to the equator. In both cases, it appears that the dynamo action is efficient if the volume integral of helicity over a hemisphere is large.     

Quasigeostrophic planetary waves in a two-layer ocean with one-dimensional periodic bottom topography

Although the study of topographic effects on the Rossby waves in a stratified ocean has a long history, the wave property over a periodic bottom topography whose lateral scale is comparable to the wavelength is still not clear. The present paper treats this problem in a two-layer ocean with one-dimensional periodic bottom topography by a simple numerical method, in which no restriction on the wavelength and/or the horizontal scale of the topography is required. The dispersion diagram is obtained for a wavenumber range of [?π/L _b , π/L _b ], where L _b is the periodic length of the topography. When the topographic?β?is not negligible compared to the planetary β, the Rossby wave solutions around the wavenumbers which satisfy the resonant condition among the waves and topography disappear and separate into an infinite number of discrete modes. For convenience, each mode is numbered in order of frequency. As topographic height is increased, the high frequency barotropic Rossby wave (mode 1) becomes a topographic mode which can exist even on the f plane, and the highfrequency baroclinic mode (mode 2) becomes a surface intensified mode. Behaviors of low frequency modes are somewhat complicated. When the topographic amplitude is small, the low frequency baroclinic modes tend to be bottom trapped and the low frequency barotropic modes tend to be surface intensified. As topographic amplitude further increases, the relation between the mode number and vertical structure changes. This change can be attributed to the increase of the frequency of the topographic mode with the topographic amplitude.     

Simulation of the formation of Ushant thermal front

The formation of the Ushant thermal front off northwestern France is simulated numerically with a one-dimensional model of the vertical thermal structure of the sea. This simulation is derived from a Niiler and Kraus (1977, Modelling and prediction of the upper layers of the ocean, pp. 143–172) mixed layer model in which bottom friction is introduced. The model inputs are the meteorological parameters measured at Ushant and the barotropic tidal currents computed with a numerical model (Mariette et al., 1982, Oceanologica Acta, 5, 149–159). Both very fine time steps (Δt = 1h) and spatial steps (horizontal mesh size: Δx = Δy = 2nmi; vertical step: Δz = 1m) are used. This study is intended to be mainly concerned with the processes acting during the formation of the Ushant front. Comparison of the model results with satellite pictures and in situ measurements shows good agreement. We point out that biological blooms could occur according to a spring-neap cycle, during the thermal front formation. The prevailing effect of local processes such as tidally induced bottom friction and energy budget at the sea surface on the thermal evolution of the Mer d'Iroise is then emphasized. During the formation of the front, these effects seem to be more important than the advective processes which are not considered in this work.     

Tides off‐shore: Transition from California coastal to deep‐sea waters‡

Abstract

Tidal pressures and currents were measured with self‐contained capsules dropped to the sea floor for one month at distances of 175, 190, and 500 nautical miles from San Diego. These observations, together with a one‐week bottom pressure record by Filloux at 750 n miles, and three half‐week bottom current records by Isaacs et al, at intermediary distances, were analyzed for tidal components by cross‐correlation with a noise‐free reference time series. (For short records this method has some merit over classical tide analysis.) It was found that the tide decays seaward to e^‐1 times the coastal amplitude over a distance of order 1000 km for the semidiurnal species, slower for the diurnal species. Tidal currents turn counterclockwise, and are polarized with maximum flow parrallel to shore in the direction of tidal propagation (320°T) at local high tide. The current amplitude is roughly 2 cm/sec for the semidiurnal component, 1 cm/sec for the diurnal component. Superimposed baroclinic tidal currents lead to poor signal: noise ratios (between 1:1 and 10:1) for the barotropic currents. In contrast, the ratio is typically 1000:1 for the bottom pressures and generally exceeds that for coastal tide stations of comparable duration. Published I.H.B. tidal constants for exposed California coastal stations indicate “upshore” (towards 320°T) propagation at 140 m/sec for semidiurnal tides. 214 m/sec for diurnal tides.

To interpret these diverse observations, we have computed the dispersion laws for all possible rotationally‐gravitationally trapped waves against a straight coast with shelf. Trapped solutions are conveniently portrayed in terms of a parameter μ such that ? = sin μ = iu/v and f = ‐ cos μ = η/v define the ellipticity and impedance of the wave motion, η, u and v being off‐shelf dimensionless elevation, normal‐to‐shore and longshore components of velocity, respectively. We then attempt to fit the observations by a superposition of the possible wave classes, all of the same tidal frequency: (a) a free Kelvin‐like edge wave with small μ (mostly trapped by rotation, but somewhat slowed by the shelf); (6) a free Poincare‐like leaky wave; and (c) a forced wave (the distortion of the sea bottom by the tide producing forces plays a significant role). The mod el can account for the main features of the observed tidal heights, and gives relative amplitudes at the coast of 54:16:4 cm for components a:b:c in the case of the semidiurnal tides, 21:24:9 cm for the diurnal tides. The results place a semidiurnal amphidrome about midway between San Diego and Hawaii. Tidal currents are not well fitted by the model, and there are problems associated with the separation of barotropic and baroclinic modes, and with the benthic boundary layer. Coastal energy dissipation is small in the sea under investigation, but a “ capacitive “ phase delay appears to be associated with Northern California harbors and inland waters.     

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).     

Variational principles for topological barotropic fluid dynamics

Barotropic fluid flows with the same circulation structure as steady flows generically have comoving physical surfaces on which the vortex lines lie. These become Bernoullian surfaces when the flow is steady. When these surfaces are nested (vortex line foliation) with the topology of cylinders, toroids or a combination of both, we show how a Clebsch representation of the flow velocity can be introduced. This is then used to reduce the number of functions to be varied in the variational principles for such flows. We introduce a three function variational formalism for steady and non-steady barotropic flows.     

On the derivation of the energy flux of linear magnetohydrodynamic waves

Abstract

New light is shed on the derivation of the energy flux of the linear MHD waves. It is shown that, according to a suggestion of Lighthill, the usual perturbation procedure, which starts from the general expression for the energy flux, need not be supplemented by an averaging procedure. As a result, it is shown that to second order in the wave amplitude, a quantity identifiable as the wave energy flux is conserved. Some of the subtleties inherent in the derivation of the pertubation energy equation are discussed.