## Dz2

This equation is solved by the magnetic flux function Fig. 5.5 Harris sheet Flux function A(z), magnetic field Bx(z) and electric current density jy (z) in non-dimensional form. Fig. 5.5 Harris sheet Flux function A(z), magnetic field Bx(z) and electric current density jy (z) in non-dimensional form. It is instructive to find out what function f (Z), introduced in Section 5.2.2, generates the solution (5.122). The answer is as is readily verified by inserting (5.125) into (5.92), identifying Z...

## Geospace

Magnetospheric activity comprises the major global dynamical phenomena of the Earth's magnetosphere including ionospheric processes. It results from the interaction of the solar wind with the Earth's magnetosphere (Fig. 2.1). The solar wind is characterized by a fast (supersonic) plasma flow from the Sun into interplanetary space. The magnetosphere is the region above the ionosphere that is dominated by the geomagnetic field. The solar wind compresses the Earth's magnetic field on the day-side...

## Observation of magnetopause reconnection

Reconnection signatures have been observed in situ at the magnetopause for negative IMF Bz since the pioneering work of Paschmann et al. (1979), confirmed and extended by Sonnerup et al. (1981), Cowley (1984) and others. Typically, observed reconnection sites are stretched configurations with strong variation only perpendicular to the current sheet (Fig. 13.2). Plasma velocities are of the order of 1 10 of the Alfven velocity, as expected for standard reconnection processes (Section 11.2). As...

## Aspects of bifurcation and nonlinear dynamics

Important aspects of activity of space plasmas can be described in terms of transitions from stable to unstable states. Therefore, it was necessary to deal with the stability properties of selected equilibria, which has been a major aspect in this part of the book. However, to obtain a deeper physical understanding of the dynamic properties of a given system it is desirable to have available a complete overview of all equilibrium states and their stability properties for every choice of a...

## Flux linkage

During the nonlinear evolution of 3D fields new reconnection configurations may form that do not have direct two-dimensional counterparts. A typical example is linkage of magnetic flux tubes. Figure 11.21 gives an example from a resistive MHD simulation (Otto, 1995). The figure shows a snapshot of selected flux tubes. Reconnection releases the tension in the linked Fig. 11.18 The magnetic field structure in the central plane y 0 of a 3D resistive MHD simulation (Hesse and Birn, 1991b). As By...

## Physics of Space Plasma Activity

Space plasma is so hot that the atoms break up into charged particles which then become trapped and stored in magnetic fields. When critical conditions are reached the magnetic field breaks up, releasing a large amount of energy and causing dramatic phenomena. A prominent example is the magnetospheric substorm occurring in the Earth's magnetosphere. It involves plasma and magnetic field structures extending from 100 km to tens of Earth radii, and can be seen as strong intensifications of the...

## Models of solar activity

In the Solar System the most spectacular manifestations of space plasma activity are the large-scale solar eruptions, such as coronal mass ejections (CMEs), solar flares and prominence eruptions, as briefly described in Section 2.2. In this chapter we attempt to address the underlying physical processes. The approach leaves aside many details, although they would be exciting from a more morphological point of view. Instead, we are interested in the basic physical mechanisms and concentrate on...

## Discussion

Having considered two major applications of space plasma activity, it seems appropriate to close this part with a discussion of their differences and similarities. We first address the role of magnetic reconnection and then suggest a general eruption scheme that covers both magnetospheric and solar activity. We begin by considering reconnection in solar activity and then bring in magnetospheric reconnection for comparison. As we have seen, models of solar activity involve magnetic reconnection...

## Info

Both vd and k are directed perpendicular to the magnetic field. Small 3p ensures that the LHD modes are approximately longitudinal. For 3 > 1 the LHD instability does not play a significant role. For turbulence based on this mode with k j one finds an expression for the turbulent resistivity from (9.49) with (9.46) where e is inserted from (9.53). The approximation (9.50) is obtained with k0 w h v and corresponding values of ur and 7 from (9.54) and (9.55). One finds In cases where e0( 1 2) 2...

## Extended simulations

There are MHD simulations that cover both growth and expansion phases. Fig. 13.7 illustrates corresponding results from a 3D resistive MHD simulation by Birn and Hesse (1996) with S 500 (Section 10.3) of the magneto-tail including the transition to a dipolar field. Slow external driving leads to a growth phase which lasts until about t 80 (in non-dimensional units (Section 10.3.1)). The accumulation of magnetic flux and the formation of a thin current sheet is clearly seen. At t 80 the current...

## Kinetic models

Consider a plasma consisting of several particle species, such as electrons and a number of different ions. To introduce kinetic models we first look at the simple case where the particles interact only through an electromagnetic field of the form E(r, t), B(r, t). In other words, the particles do not interact directly with each other so that the trajectory of each particle is governed by the equations of single particle motion. Then, Hamilton's equations of motion (3.15) and (3.16) imply a...

## Open versus closed magnetosphere

Let us, for a moment, consider the consequences of a hypothetical model, which describes the solar wind magnetosphere interaction by ideal MHD, including MHD discontinuities (Section 3.9). As the Earth's dipole field is an obstacle in the supersonic solar wind, a fast MHD shock wave, the bow shock, stands in front of the magnetosphere. The magnetopause is a tangential discontinuity, confining the geomagnetic field. It is a necessary element of the initial value problem (an analogous case was...

## DdA dAv y

Obviously, (6.24) agrees with the Grad-Shafranov equation (5.78) for constant Bz. Thus, the quasi-neutral limit of present Vlasov theory for systems with translational invariance and constant Bz reduces to a corresponding magnetohydrostatic description. This fact is easily explained by the pressure isotropy in the poloidal plane and by the fact that setting a to zero eliminates the aE term in the momentum equation (6.21) such that this equation reduces to the corresponding MHD version. We...

## Bifurcation properties of Grad Shafranov theory

Let us consider the Grad-Shafranov equation for one- or two-dimensional Cartesian coordinate spaces. For concreteness, let us assume that the domain of interest Q is bounded with a smooth boundary dQ. Equilibrium sequences with a sequence parameter A can be generated by choosing the pressure of the form p(x, A), where x stands for the coordinate(s). A simple prototype case corresponds to p A exp(2A) 2 so that the Grad-Shafranov equation becomes This case is equivalent to the choice (5.88), the...

## Plasma models

By plasma model we denote a set of equations governing the temporal evolution of a plasma under a given set of boundary and initial conditions. Ideally, plasma models should be based on first principles. Unfortunately, these are not yet available, at least not from a strict point of view. In any event, simplifications are necessary to keep applications feasible. For describing space plasma dynamics, it is largely appropriate to ignore quantum effects. The condition for this assumption to be...

## Fz dz

Where z0 is the value of z at t 0 and the sign in the integrand is fixed by the initial condition and continuity. Here we see explicitly that our system is integrable. (In fact it is a general result that every autonomous system with one degree of freedom is integrable.) Integrability does by no means imply that the orbits have a simple structure. Depending on the initial conditions, the orbits show different qualitative properties (e.g., Sonnerup, 1971). Orbits of particles with sufficiently...

## Bifurcations of the Harris sheet

Let us apply this approach to the Harris sheet equilibrium. For a rectangular domain 0 < x < a, -b 2 < z < b 2 the corresponding solution of (12.27) is where v > 0 (the inverse thickness of the sheet) acts as a parameter so that A can be understood as a function of x, z, A and AH defines the (Dirichlet) boundary condition. As in the case of fixed current density of Fig. 12.7, but unlike the case of the catastrophe of Fig. 12.6, the solution exists for all positive A. So, the next...

## Jdjd

This property holds for arbitrary smooth, real displacements 1 and 2 satisfying the boundary condition. For a derivation of (10.23) see, for instance, Appendix A of Freidberg (1987). Making use of (10.23) we write (10.22) in the form dt f 2 Po i2d V 1 f F( ) d3r + 2 f F( ) d3r (10.24) which, by integration with respect to time gives us second-order energy conservation in the form V2 -1 F( )d3r (10.28) Using (10.20) one finds, after integration by parts, the following explicit expression for V2,...

## Gk

Where QQ 7VS jm with S vah-oL* n, L* L jm and A QQ3 2 (kS1 4), and g(k) denotes ip' ip computed from the external solution for z 0. In the external region p satisfies which generalizes (10.146). By obtaining g(k) from (10.179) and solving (10.178) numerically one finds the dispersion relation explicitly in the Fig. 10.10 Current density jy (z) of the magnetic field model (10.176) for the Harris sheet (k 0) and for k 0.6 with v chosen as 0.4, 0.2, and 0.1, corresponding to increasing jm (from...

## JtjA 0 wYpA0 m

Thus, a thin current sheet forms if one of the two terms in (8.51) or both become large. Either the flux tube volume itself or its gradient (as a function of A), or both, must increase strongly. Although the discussion of the flux tube volume gives insight into the physical mechanism at work, it does not indicate whether or not a given external driving force causes a thin current sheet. To answer that question one has to relate the current density to the driver function p. The following example...

## Other sites

The phenomenon of activity is by no means confined to the Solar System. Of the many planets that one speculates to be present in the Universe -an increasing number is being detected in our galaxy - a significant fraction will be in conditions favourable for magnetospheric activity. Necessary conditions are a stellar wind and a sufficiently strong magnetic dipole moment. In the solar system this applies to Mercury, Earth, Jupiter, Saturn, Uranus and Neptune. The size of the magnetosphere,...