Spontaneous tangential discontinuities and the optical analogy for static magnetic fields. VI. Topology of current sheets

Abstract

The electric surface current in a tangential discontinuity in a force-free magnetic field is conserved. The direction of the current is halfway between the direction of the continuous fields on either side of the surface of discontinuity. Hence the current sheets, i.e. the surface of tangential discontinuity, have a topology that is distinct from the lines of force of the field. The precise nature of the topology of the current sheet depends upon the form of the winding patterns in the field. Hence, invariant winding patterns and random winding patterns are treated separately. Current sheets may have edges, at the junction of two or more topological separatrices. The current lines may, in special cases, be closed on themselves. The lines of force that lie on either side of a current sheet somewhere pass off the sheet across a junction onto another sheet. In most cases the current sheets extending along a field make an irregular honeycomb.

The honeycomb pattern varies along the field if the winding pattern of the field varies. The surface current density in a tangential discontinuity declines inversely, or faster, with distance from its region of origin. The edges of weaker tangential discontinuities (originating in more distant regions) are bounded by the stronger tangential discontinuities (of nearby origin).

An examination of the force-free field equations in a small neighborhood of the line of intersection of two tangential discontinuities shows that the lines of force twist around to cross the line of intersection at right angles. If the angle between the tangential discontinuities exceeds π/2, there is also the possibilitity that the lines twist around so as to come tangent to the line of intersection as they cross it.     

Tangential discontinuities and the optical analogy for stationary fields. VII. Effects of resistivity

Abstract

The relatively large resistivity in the solar photosphere and chromosphere softens the ideal tangential discontinuities of magnetostatic equilibrium into continuous transitions in field direction over scales of 0.1–10 km. This softening is communicated upward at the Alfvén speed into the active solar corona. The degree of softening is a vital part of the theory of magnetic heat input to the active X-ray corona, because the very low resistivity of the coronal gas provides effective dissipation only if the current sheets are reduced to a thickness of 10^?2km.

A close examination of the problem shows that the Alfvén transit time up into the corona is large compared to the characteristic time of 1 sec in which the coronal tangential discontinutities are formed. It also shows that the principal effect of the resistivity is to create a thin surface layer of fluid on adjacent flux bundles, which causes a general drift of the flux but does not directly broaden the current sheets higher up in the field. In fact the motions of the surface layers do not extend upwards beyond the first winding pattern at each end of a coronal loop.

It appears that the photospheric and chromospheric resistivity is without striking consequences for magnetic heating in the corona.     

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.     

Laboratory experiments on fronts

Abstract

Supercritically unstable density fronts near a vertical wall in a rotating, two-layer fluid were created on a laboratory turntable by withdrawing the outer wall of an annulus with a narrow gap, and allowing buoyant fluid from within the annulus to collapse toward a state of quasi-geostrophic balance. The resulting “coastal” current has a nearly uniform potential vorticity and is bounded by a front on which ageostrophic, wave-like disturbances grow. If the current width is comparable to the Rossby radius of deformation, the dominant length scale of disturbances is proportional to the width of the current. On the other hand, if the upper layer is much wider than the Rossby radius, then the observed length scale is a constant multiple of the Rossby radius. If the vertical boundary is omitted in the experiments, so that we are left with a circular anticyclonic vortex, the observed length scales and large-amplitude behaviour of disturbances are identical to those for the boundary currents, indicating that the wall has no significant influence on the flow.

At very large amplitude the growing waves lead to the formation of cyclone-anticyclone vortex pairs. For very wide currents, both the mean flow and the disturbances are first confined to a region within a few Rossby radii of the front. However, both the mean flow and the turbulent eddy motions slowly propagate into the previously stationary upper layer until, eventually, the whole of the upper layer is turbulent.     

The vertical structure of tidal currents

Abstract

The accuracy to which the vertical structure of tidal currents can be predicted is examined. Theoretical models for current structure are developed employing (a) a constant eddy viscosity E = ε and (b) an eddy viscosity varying linearly with height above the sea bed z; E(z)=βz. By requiring these models to satisfy the commonly accepted quadratic friction law, the condition ε>½k is deduced where k is the bed friction coefficient, W a representative velocity and D the depth.

The sense of rotation of a current ellipse is shown to be related to the configuration of co-tidal charts. The vertical structure of the current ellipse is illustrated from the theoretical models and the sensitivity of this structure is examined for the following variables: (a) eddy viscosity ε or βz, (b) the bed friction parameter kW, (c) rotation of the prescribed pressure gradients and (d) tidal period. While reasonable agreement between observed and calculated current profiles may often be reported, precise agreement is shown to depend upon accurate specification of both eddy viscosity and the bed stress condition.     

Incoherent internal tidal currents in the deep ocean

Eleven months current meter observations from the deep Bay of Biscay were examined for the residual (incoherent internal tidal; icIT) signal, left after harmonic analysis using eight tidal constituents (large-scale barotropic or coherent baroclinic signal) within the semidiurnal band. This residual signal comprised 30% of the total tidal kinetic energy and, due to its flat spectral appearance, it was responsible for typically 5–7 days intermittency. Although icIT was part of the red noise internal wave band continuum, it was not attributable to instrumental noise. It consisted of quasi-harmonics at non-tidal harmonic frequencies having amplitudes larger than N₂, the third largest semidiurnal tidal constituent. It is suggested that the kinetic energy at these non-tidal frequencies reflects interaction between semidiurnal tidal motions and the slowly varying background conditions.Responsible Editor: Roger Proctor     

A 3-D coastal tidal rectification process: observations, theory and experiments

In homogeneous rotating fluid, when there is an oscillating forcing in the interior fluid with a period long enough for an Ekman layer to develop, there is an interaction between the oscillatory Ekman layer and the vertical wall, since the latter imposes an alternating adjustment flow confined near the wall. As a result, this coastal rectification process leads to a Lagrangian transport along the coast. The Ekman number, the Rossby number and the temporal Rossby number of the forcing flow are the governing parameters of that mechanism which can be described by a simplified analytical model taking into account both the vertical time-dependent structure of the current and the presence of the wall. The model shows that the residual (rectified) current flowing with the coast to its right results from the strong nonlinear interaction between along- and cross-shore tidal currents leading to asymmetrical momentum exchanges between the Ekman bottom layer and the coastal boundary layer. The model provides simple scaling laws for the maximum intensity and width of the residual current. The latter is significantly larger than the friction (Stokes) lateral boundary layer of the forcing flow. A comprehensive set of experiments is performed in the 13 m diameter rotating tank by oscillating an 8 m×2 m horizontal plate and vertical wall in a homogeneous fluid at rest in solid-body rotation and measuring the two horizontal components of the current at several locations and depths above the central part of the plate. The predicted and experimentally measured maximum intensity and width of the residual current are in very good agreement, within the range of validity of the model, i.e. when the Ekman number is sufficiently small. However experiments also show that the residual current still occurs when the Ekman layer thickness is of the same order as the fluid depth, but it is then confined to a narrower band along the vertical wall. The flow structure found experimentally is also correctly described by a numerical model developed by Zhang et al. (1994). Current measurements in the Eastern part of the English Channel near the French coast reveal a significant coastal residual current flowing Northward and the coastal rectification process described here may account for part of it.     

Threshold of erosion of submarine bedrock landscapes by tidal currents

Since sea level stabilized 7000 yr bp , shelf seas experiencing semi‐diurnal tides will have been affected by streaming four times per day. If tidal erosion of bedrock were even only marginally efficient, the ~10 million streamings since then should have left geomorphological imprints. We examine high‐resolution multibeam sonar data from three areas with extreme tidal currents. The Minas Passage (Bay of Fundy) experiencing 8‐knot surface tidal currents was surveyed in 2007 with a multibeam sonar. In an area near to transverse dunes, which are evidence for bedload transport, the data show local overhanging surfaces near to the sediment‐rock contact, potentially created by abrasion by saltating particles. However, they are uncommon. In the Straits of Messina, where surface currents reach 10 knots, surveying revealed ridges lying oblique to the flow that are not obviously broken into separate outcrops by erosion. In the Bristol Channel, UK, sonar data collected where currents reach 3·4 knots at 1·5 m above the bed reveal outcrops of limestone with superimposed sand dunes, but only minor rounding of blocks. Holocene tidal currents have apparently been generally ineffective at eroding bedrock. We examine this issue further by compiling extreme tidal streams around the UK and from them estimate shear stresses, representing a macro‐tidal environment where peak surface currents reach 9·7 knots. Those data are compared with shear stresses in mountainous rivers where long‐term rates of erosion are comparable with tectonic uplift rates and are thus geomorphologically significant. Whereas river stresses reach 10²–10³ Pa, the largest tidal stresses are generally 10¹ and only rarely approach 10² Pa, too small for quarrying to operate generally. However, the vertical faces in the Minas Passage may represent the onset of abrasion. Given this limited evidence for abrasion, we explore conditions in the geological past for tides that may have locally eroded bedrock. Copyright © 2012 John Wiley & Sons, Ltd.     

UNIT HYDROGRAPH DERIVED FROM A SPATIALLY DISTRIBUTED VELOCITY FIELD

A unit hydrograph model is proposed in which the watershed is decomposed into subareas which are individual cells or zones of neighbouring cells. The unit hydrograph is found for each subarea and the response at the outlet to excess rainfall on each subarea is summed to produce the watershed runoff hydrograph. The cell to cell flow path to the watershed outlet is determined from a digital elevation model. A constant flow velocity is assigned to each cell and the time lag between subarea input and response at the watershed outlet is found by integrating the flow time along the path from the subarea to the outlet. The response function for a subarea is modelled as a lagged linear reservoir in which the flow time is equal to the sum of a time of translation and an average residence time in the reservoir. It is shown that the assumption of a spatially varying, but time-invariant, velocity field underlying this model produces a linear system model for all subareas whose outputs can be summed in the manner indicated. An example application is presented for the 8.70 km² Severn watershed at Plynlimon in Wales using a 50 m digital elevation model in which the cell velocity is calculated by modifying an average watershed velocity according to the terrain slope and the drainage area of each cell. The resulting model reasonably reproduces the observed unit hydrograph.     

The relationship between electric fields, conductances and currents in the high-latitude ionosphere: a statistical study using EISCAT data

The relationship between electric fields, height-integrated conductivities and electric currents in the high-latitude nightside electrojet region is known to be complex. The tristatic nature of the EISCAT UHF radar facility provides an excellent means of exploring this interrelationship as it enables simultaneous estimates to be made of the full electric field vector and the ionospheric Hall and Pedersen conductances, further allowing the determination of both field-perpendicular electric current components. Over 1300 h of common programme observations by the UHF radar system provide the basis of a statistical study of electric fields, conductances and currents in the high-latitude ionosphere, from which preliminary results are presented. Times at which there is significant solar contribution to the ionospheric conductances have been excluded by limiting the observations according to solar zenith angle. Initial results indicate that, in general, the times of peak conductance, identified from the entire set of EISCAT observations, do not correspond to the times of the largest electric field values; the relative contribution of ionospheric conductance and electric field to the electrojet currents therefore depends critically on local time, a conclusion which corroborates work by previous authors. Simultaneous measurements confirm a tendency for a decrease in both Hall and Pedersen conductances to be accompanied by an increase in the electric field, at least for moderate and large electric field value, a tendency which is also identified to some extent in the ratio of the conductances, which acts as an indicator of the energy of precipitating particles.     

Variability of baroclinic tidal currents on the Mackenzie Shelf,the Southeastern Beaufort Sea

Semidiurnal tidal currents on the outer shelf of the Mackenzie Shelf in the Beaufort Sea were found to be strongly influenced by the locally generated baroclinic tide. Two primary factors are involved in this process: (1) the sharp shelf break along the northeastern Mackenzie Shelf, promoting the generation of vigorous internal tidal waves; and (2) the proximity to critical latitudes for M₂ and N₂ motions locking these waves and preventing them from leaving the source region. As a result, internal tides are resonantly trapped between the shelf and critical latitudes. The physical properties and temporal variations of tidal motions were examined using current meter measurements obtained from 1987–1988 at four sites (SS1, SS2, SS3, and SS4) offshore of the shelf break at depths of ∼200 m. Each mooring had Aanderaa RCM4s positioned at ∼35 m below the surface and ∼50 m above the bottom. Complex demodulation was used to compute the envelopes (amplitude modulation) of these components. A striking difference in the variability of clockwise (CW) and counterclockwise (CCW) tidal currents was found. The CW tides are highly variable, have greater amplitude, exhibit a burst-like character associated with wind events and contain about 80% of the total energy of the semidiurnal tidal currents. In contrast, the CCW components have a more regular temporal regime with distinct monthly, fortnightly and 10-day modulation at astronomical periodicities associated with frequency differences M₂–N₂ (0.03629 cpd), S₂–M₂ (0.06773 cpd), and S₂–N₂ (0.10402 cpd). Significant horizontal correlation of the CW current envelopes was found only between stations near the northeast Mackenzie Shelf, indicating this to be the main area of baroclinic internal wave generation.     

Induction of auroral zone electric currents within the Alaska pipeline

The Alaskar pipeline is a highly conducting anomaly extending 800 miles (1300 km) from about 62° to 69° geomagnetic latitude beneath the most active regions of the ionospheric electrojet current. The spectral behavior of the magnetic field from this current was analyzed using data from standard geomagnetic observatories to establish the predictable patterns of temporal and spatial changes for field pulsation periods between 5 min and 4 hr. Such behavior is presented in a series of tables, graphs and formulae. Using 2- and 3-layer models of the conducting earth, the induced electric fields associated with the geomagnetic changes were established. From the direct relationship of the current to the geomagnetic field variation patterns one can infer counterpart temporal and spatial characteristics of the pipeline current. The relationship of the field amplitudes to geomagnetic activity indices,A _p, and the established occurrence of various levels ofA _p over several solar cycles were employed to show that about half of the time the induced currents in the pipe would be under 1 A for the maximum response oscillatory periods near 1 hr. Such currents should be of minimal consequence in corrosion effects for even a section of the pipeline unprotected by sacrificial electrodes. Of greater interest was the result that the extreme surges of current should reach over one-hundred amperes in the pipeline during high activity.     

Analytical description of electric currents in the magnetosheath region

Volume currents in the magnetosheath region are calculated within the framework of a new analytical model. Magnetic field structure in the region is found, satisfying boundary conditions on the bow shock and the magnetopause, and then volume currents are calculated using the Maxwell equation. Surface bow shock and magnetopause currents are calculated, too. Free parameters of the model are interplanetary magnetic field, Mach number of the solar wind flow, distances to the bow shock and to the magnetopause, and field compression at the magnetopause.     

An enhanced depth-averaged tidal model for morphological studies in the presence of rotary currents

A simple and efficient method to improve morphological predictions using depth-averaged tidal models is presented. The method includes the contribution of secondary flows in sediment transport using the computed flow field from a depth-averaged model. The method has been validated for a case study using the 3D POLCOMS model and ADCP data. The enhanced depth-averaged tidal model along with the SWAN wave model are applied to morphological prediction around the Lleyn Peninsula and Bardsey Island as a case study in the Irish Sea. Due to the presence of a headland in this area two asymmetrical tidal eddies are developed in which the cyclonic eddy is stronger as a result of Coriolis effects. The results show that the enhanced model can effectively predict formation of sand banks at the centre of cyclonic eddies, while the depth-averaged model, due to its inability to accommodate secondary flow, is inadequate in this respect.     

潮汐地电场特征及机理研究

地电场观测系统的观测对象是地电场强度矢量,其目的是通过观测地表的地电场强度,探索地震前地电场的变化,获得地下电性结构及其变化规律,开展孕震过程中大地电场的变化特征研究 。本文对翁田台地电场自建台以来的数据进行了详细的梳理,通过计算差值和相关系数来分析数据自运行以来的质量情况,总结数据质量较好的原因,并对下一步如何提高数据质量给出一些建议。翁田台地电场观测数据曲线存在明显1峰1谷的日变形态,对其进行频谱分析,优势周期为24、12、6 h,以24 h为主。运用优势方位角的方法计算发现,2017—2018年优势方位角存在收窄和偏转现象,对该异常进行了分析与探讨。

     

潮汐地电场特征及机理研究

以中国大陆一百个地电场台站的数据为分析基础,将潮汐地电场分类为近正弦形的TGF-A型和近梯形的TGF-B型.TGF-A型地电场与固体潮汐密切关联,基本分布在大面积水域附近,并与附近水域面积和距离、岩性结构、构造活动等因素有关.TGF-B型地电场与气潮作用产生的空间Sq电流关系密切,并与岩石饱和度、渗透率等有关.TGF-A型波形畸变揭示了岩石所受应力可能出现变异,导致岩石裂隙水周期性渗流突变;TGF-B型背景值跃变可能是岩石微破裂加剧导致地下水向破裂区渗流.两类潮汐地电场变异的机理可能是地电场数据应用于强震短临分析的理论基础.