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  1. Stationary plasma thrusters operate in space | SpringerLink
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In , Harry Petschek proposed a mechanism where the inflow and outflow regions are separated by stationary slow mode shocks. Simulations of resistive MHD reconnection with uniform resistivity showed the development of elongated current sheets in agreement with the Sweet-Parker model rather than the Petschek model. When a localized anomalously large resistivity is used, however, Petschek reconnection can be realized in resistive MHD simulations. Because the use of an anomalous resistivity is only appropriate when the particle mean free path is large compared to the reconnection layer, it is likely that other collisionless effects become important before Petschek reconnection can be realized.

In the Sweet-Parker model, the common assumption is that the magnetic diffusivity is constant. Nevertheless, if the drift velocity of electrons exceeds the thermal velocity of plasma, a steady state cannot be achieved and magnetic diffusivity should be much larger than what is given in the above. Another proposed mechanism is known as the Bohm diffusion across the magnetic field. Lazarian and Vishniac considered the magnetic reconnection in the presence of a random component of magnetic field in a totally ionized and inviscid plasma assuming that the resistive effects could be described with an Ohmic resistivity.

Lazarian and Vishniac showed that, in general, this cannot affect the final result. In fact, their model is independent of small scale physics which determines the local reconnection rate. This model has been successfully tested by numerical simulations. On these scales, the Hall effect becomes important.

Two-fluid simulations show the formation of an X-point geometry rather than the double Y-point geometry characteristic of resistive reconnection. The electrons are then accelerated to very high speeds by Whistler waves. Because the ions can move through a wider "bottleneck" near the current layer and because the electrons are moving much faster in Hall MHD than in standard MHD , reconnection may proceed more quickly.

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Magnetic reconnection occurs during solar flares , coronal mass ejections , and many other events in the solar atmosphere. In the past, observations of the solar atmosphere were done using remote imaging; consequently, the magnetic fields were inferred or extrapolated rather than observed directly.

However, the first direct observations of solar magnetic reconnection were gathered in and released in by the High Resolution Coronal Imager. Magnetic reconnection events that occur in the Earth's magnetosphere in the dayside magnetopause and in the magnetotail were observed by spacecrafts such as Cluster II [18] and the Magnetospheric Multiscale Mission.

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It has observed numerous reconnection events in which the Earth's magnetic field reconnects with that of the Sun i. These include 'reverse reconnection' that causes sunward convection in the Earth's ionosphere near the polar cusps; 'dayside reconnection', which allows the transmission of particles and energy into the Earth's vicinity and 'tail reconnection', which causes auroral substorms by injecting particles deep into the magnetosphere and releasing the energy stored in the Earth's magnetotail. The Magnetospheric Multiscale Mission , launched on 13 March , improved the spatial and temporal resolution of the Cluster II results by having a tighter constellation of spacecraft.

This led to a better understanding of the behavior of the electrical currents in the electron diffusion region. Vassilis Angelopoulos of the University of California, Los Angeles, who is the principal investigator for the THEMIS mission, claimed, "Our data show clearly and for the first time that magnetic reconnection is the trigger.

Magnetic reconnection has also been observed in numerous laboratory experiments. The confinement of plasma in devices such as tokamaks , spherical tokamaks , and reversed field pinches requires the presence of closed magnetic flux surfaces. By changing the magnetic topology, magnetic reconnection degrades confinement by disrupting these closed flux surfaces, allowing the hot central plasma to mix with cooler plasma closer to the wall.

From Wikipedia, the free encyclopedia. Play media. Basic theory of magnetic flipping". Journal of Geophysical Research. Bibcode : JGR Journal of Geophysical Research: Space Physics. Bibcode : JGRA.. September Solar Physics. Bibcode : SoPh.. November Astronomy and Astrophysics. December Physics of Fluids. Bibcode : PhFl Physics of Plasmas. Bibcode : PhPl Physical Review Letters. Bibcode : PhRvL.. Cosmical Magnetic Fields. Oxford: Oxford University Press. The Astrophysical Journal. Additionally, we examine evolution of the HPS thickness with heliocentric distance.

Obtained solutions are accurate within arbitrary functions; therefore, the analytical model allows the wide area of application, depending on chosen boundary conditions. The components of electric field are defined in the same way. Separatrices, which divide the areas of opened and closed magnetic lines, are indicated by blue dotted lines.

The shown coordinate system is related to the HCS position. Since it lies close to the ecliptic plane, the coordinate system may be considered as an analogue of the geocentric solar ecliptic coordinate system if it is necessary for comparisons with observations. The configuration of fields, currents, and separatrices, being consequences of the modeling, will be discussed in detail after the building the model and choosing the boundary conditions. Let us list the main assumptions. As we develop the axisymmetric model, equations for the magnetic field and currents can be written with the help of stream functions [see, for example, Landau and Lifshitz , , ].

At the axial symmetry, the magnetic field defined by means of 1 becomes divergence free. The unipolar induction due to the difference of rotation between the corona and the photosphere [ Owens et al. The initial equations of the model are listed below. Let us make one more assumption about the isothermal plasma outflow. Below, we will use designations n 0 and n 1 for concentration boundary conditions in the neutral plane and in the solar corona, respectively.

The exact type of this tie remains a subject of discussions [see, for example, Parker , ; Sittler and Scudder , ; Totten et al. Our calculations showed that the difference between the adiabatic and isothermic cases is negligibly small. This effect was discussed several times in observational papers [ Elliott et al. Corresponding approximation curves are shown with black and green. Its variation determines the variation of the electric potential with latitude and results in the electric current generation. Equations 11 and 12 that reflect plasma flowing along magnetic lines are important consequences of the model.

In other words, magnetic field lines are equipotential. Under the same assumptions, 14 it follows from the azimuthal projection of plasma balance 4. Under assumptions of parity, stationarity, and small nonzero thickness, it follows from the cross projection of 8 that 15 where 16 is a function of the flux having a dimension of the angular speed. All the three assumptions will be used further if other is not stipulated. Taking into account the assumption about a small azimuthal speed, one can get the following expression from the longitudinal projection of the force balance 4 : Thus, the radial component of the speed is a function of the magnetic flux and depends only on the boundary conditions.

Considering 18 , we can simplify 15 The equations 3 , 10 , 11 — 14 - 11 — 14 , 16 , 17 , 19 , and 20 are the main equations of our model. As we mentioned above, we use additional assumptions concerning plasma properties in the solar corona and in the photosphere to determine the boundary conditions. We set the surface at the distance of 1. Considering the unipolar induction in the corona, we develop a simplified model, which is aimed to obtain a distribution of potential in the corona as a function of the magnetic flux.

Some difficulty in finding a relationship between the plasma density in the solar corona and in the solar wind should be noted. According to Gibson et al. At the same time, we accept the view of the angular dependence of n on the heliolatitude described in Gibson et al. According to Hundhausen [ ], the full mass flux from the Sun is. We assume that the mass flux density does not depend on heliolatitude.

If this is so, then If to accept the assumption about the radial evolution of the density, the radial component of the speed is given by The stream function U is determined from 24 by its substitution in to The negative flux corresponds to the open magnetic field lines.

The corona and the photosphere rotate with different speeds. As a result, they possess different distributions of the electric potential. The difference of potentials between the corona and the photosphere results in the electric current induction. Let us imagine a surface in the corona, at which the angular speed reaches a constant. In our model we consider the photosphere and this surface as two ideally conducting thin layers which are spaced apart at a distance much lesser than their radii. The plasma between these layers is finitely conducting.

After Alfven [ , ], we assume that the longitudinal currents get closed at the HCS. From 15 and 29 , we obtain A solution of 31 is as follows: 32 where. The distribution of the potential in the corona is calculated from The above described mechanism of the appearance of potential difference and the current is known as a unipolar induction [ Landau and Lifshitz , ; Alfven , , ]. The existence of the considerable total current flowing in the radial direction was supposed by Alfven [ Alfven , , ], who considered the Sun as a unipolar inductor.

It is essential that the speed distribution 27 assumes that the photosphere rotates faster than the corona at low latitudes, and, on the contrary, the photosphere rotates slower than the corona at high latitudes. As follows from 29 and 32 , a decrease or an increase of the potential with increasing latitude or with the reduction of the flux strongly depends on the sign of difference of speeds between the rotating corona and the photosphere.

The right panel shows the latitudinal variation of the relative angular speed calculated through the difference between angular speeds of the corona and the photosphere. This dependence considerably determines the variation of the electric potential with increasing latitude. Similar variations of the azimuthal current and the electric field take place as well. The projection of this critical point along the magnetic field lines to the HPS area corresponds to the place where both the IMF and the global current change their directions at the neutral plane.

Other values of the parameter will be considered also. Results are grouped as follows: at first, we show solutions for the speed and the total current which are mainly determined by the boundary conditions and the unipolar induction. Then some features of the separatrix spatial distribution are discussed, and comparisons with Parker's model are provided. Let us note that both the real HCS and the plasma sheet may have more complicated shape as presented in the model, and, at any comparisons with observations, the coordinate Z should be considered as the instant normal to the neutral plane.

In most complicated cases, only qualitative comparisons are possible. The provided solution is consistent with observational data [ Sheeley et al. The modeled radial variation of V is determined by its distribution over the surface of the Sun 24 and following the magnetic field lines It follows from 21 and 24 that the speed is higher at higher latitudes, which perfectly corresponds to observations too [see Khabarova , , and references therein]. Since the magnetic field lines coming from higher latitudes cross the neutral plane farther than ones coming from low latitudes, the solar wind speed in low heliolatitudes increases with distance.

Therefore, the solar wind acceleration can be explained without implementations of mechanism of plasma acceleration in the solar corona. Let us estimate the total current within the HCS. As we mentioned above, the coronal current 28 appears in the region of a finite conductivity.

In its turn, it leads to formation of azimuthal magnetic field 30 of the HCS, which is situated in the ideally conductive plane. As one can see from 33 , the strength of the total current within the surface depends on distance from the Sun and is not zero even in the case of the absence of currents between the corona and the photosphere. The cause of their appearance is rotation of the ideally conducting solar corona in the magnetic field with the subsequent generation of the electrostatic potential.

According to very rough estimates, we have chosen three possible values of the parameter:. The choice of the first one gives very interesting effect, when the current changes its direction at the HCS at some heliocentric distance. At , the azimuthal magnetic field at the Earth's orbit is in accordance with observational data.

The increase of the current with a distance occurs because magnetic field lines from higher latitudes cross the neutral plane at larger distances, and, correspondingly, the volume of the current carrying flux tubes entering the HCS increases too. One can see that, in this case, the sign of the current is positive everywhere and 2 orders larger than in the previous case.

The value of the threshold depends on other parameters of the model, according to the approximate formula following from 31 : where n 1 0 is the concentration of plasma in the polar corona. To understand why the sign of the current changes at some distance, let us apply formulas 28 and 32 to the Sun's poles, where the magnetic flux is small in comparison with its value at equator. After the linear decomposition of equations 28 and 32 on the small parameter, one can estimate the surface current as According to 30 , the azimuthal magnetic field strength depends on J.

This is consistent with Parker's model [ Parker , ]. At the same time, the azimuthal magnetic field reaches zero in the turning point simultaneously with the current. Therefore, the magnetic field behavior near the HCS can be considerably different from Parker's views. The twisted inward spiral turns into the twisted outward at this distance.

Very far from the Sun, the direction of the spiral corresponds to Parker's one. The turning point exists due to the differential rotation between the corona and the photosphere and the consequent unipolar induction. The sharp form of the spiral is a consequence of high conductivity of the solar corona, as the azimuthal current is proportional to it.

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The equation 12 allows estimating the position of separatrices if to use previously obtained density and speed distributions: Formula 35 must not be used very close to the Sun at distances closer than 10 solar radii , because the main assumption of the model about a thin current sheet is violated. As follows from 12 , the radial magnetic field component varies along the magnetic field line according to the same law as the plasma density, because the speed quickly reaches its maximum value. Points represent the square law for the H r r dependence, characteristic for Parker's solar wind model [ Parker , ].

The same picture is observed along other magnetic lines between separatrices. In this regard, it is important to note that our solution for H r is different from Parker's one [ Parker , ] due to the presence of separatrices and closed magnetic lines that are not radially directed. Arrows indicate the HPS thickness L as the half of distance between separatrices. The current density quickly falls down above the separatrix, which is in accordance with the above proposed definition of the separatrix as a boarder of the HPS.

The growth of the current density is determined by the plasma speed increase at high heliolatitudes in comparison with low heliolatitudes, as the separatrices proceed from high latitudes and the solar wind flows faster closer to the pole. According to the model, the separatrices' path begins precisely from the Sun's poles, though it is not essential. There are three discontinuities in the HPS system, which are determined by violence of monotonicity. The central one corresponds to the HCS, and two others are located at separatrices. In reality, additional current sheets are formed in these areas see below.

Taking into account 14 , 15 , 18 , 32 , and 33 , we can get an expression for the mass density of the momentum of impulse: Let us estimate the angular speed values at large distances r. Using 33 and expanding 36 into series for the flux to the first order of smallness, we get an asymptotic estimation for the angular speed: The order of its magnitude is as follows: where v A is the Alfven speed.

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The obtained result is very important, because most models consider the HCS as a system that has an angular speed close to the speed of the rotating Sun. However, this contradicts to observations. The only spacecraft that could provide data for that was Ulysses. The HCS occurred not exactly in the ecliptic plane but rather close to it see corresponding heliolatitudes shown in the horizontal axis. The spacecraft crossed the HCS plane approximately in the perpendicular direction, along Z.

Such a behavior of the radial IMF component is typical for multilayer current structures. In our model, we can obtain such a solution employing nonmonotonic boundary conditions for the magnetic flux, for example, taking into account the quadrupole component of the solar magnetic field. Both mechanisms will be studied in the future in connection with observations. The obtained results let us explain some puzzling observations made close to the ecliptic plane. Perhaps, a model alternative to Parker's one must be employed [ Eyink , ].

The increase in the magnetic flux is impossible in these directions because we use the stream functions 1 and 2. However, the way of the IMF measurements supposes the crossing of the HPS under some angle, sometimes in the nearly perpendicular direction. This means that measurements performed in areas both above the HPS and inside it are involved in calculations of the IMF strength.

This is exactly what one can find in the literature see references in Khabarova and Obridko [ ] and Khabarova [ ]. Let us discuss the possibility of the second approach. Here A 1 and A 2 are functions of the flux determined by the tie of the pressure A 1 and the speed A 2 with the flux. This situation may take place if the rotation speed of the corona is independent of the heliolatitude.

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Meanwhile, such structures as bifurcated current sheets are sometimes observed in the solar wind [ Gosling and Szabo , ]. These are believed to be produced by the magnetic reconnection. The latter is known to occur recurrently at current sheets with strong guide field, which is the case for the HCS [ Zharkova and Khabarova , ].

Observations show that, indeed, the farther from the Sun, the closer to Parker's solution, at least near the ecliptic plane [ Khabarova and Obridko , ; Khabarova , ]. In the frame of our model, this is due to the decreasing influence of the HCS on the surrounding solar wind with distance. Far from the Sun, the solar wind becomes purely Parker like. As we mentioned above, Alfven discussed a possibility of unipolar induction resulting in generation of a system of longitudinal currents flowing from high latitudes and closing at the HCS neutral plane [ Alfven , , ].

Meanwhile, Alfven considered only general questions and did not pay the attention to such details as a the latitudinal dependence of the current in the corona, b the nonmonotonic change of the difference of angular speeds between the corona and the photosphere with heliolatitude, and c the influence of the limited conductivity of the solar corona on the system of currents. In the absence of the turning point, the estimated strength of radial currents in our model and in Alfven [ , ] is well corresponding each other, but if such a point exits, there is no consistency between the two models.

Interestingly, there is possible evidence for the effect found out from the behavior of Fe and O in large solar energetic particle events, which cannot be explained without attraction of the idea of the occurrence of a magnetic boundary, reflecting Fe ions beyond the Earth [ Reames and Ng , ].

This question demands further investigations. An interesting fact is that a similar picture of currents and magnetic fields with changing direction is observed in galaxies and is in agreement with some models.

Such models, called the magnetic tower models, employ the fact of differential rotation of the galactic disk, which reminds in some kind our idea to use the differential rotation of photosphere in the model [ Kharb et al.