Macroscopic plasma description
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1 Macroscopic plasma description Macroscopic plasma theories are fluid theories at different levels single fluid (magnetohydrodynamics MHD) two-fluid (multifluid, separate equations for electron and ion fluids) hybrid (fluid electrons with (quasi)particle ions) Macroscopic equations can be obtained by taking velocity moments of Boltzmann / Vlasov equations v n order n order n + 1 Taking the n th moment of BE/VE introduces terms of order n +1! This leads to an open chain of equations that must be terminated by applying some form of physical intution. Note that the collision integrals can be very tricky!
2 We start from the Boltzmann equation Continuity equation and calculate its zeroth velocity moment. The first term gives trivially the time derivative of density. For physically relevant distributions ( when ) the force term integrates to zero (exercise) In absence of ionizing or recombining collisions, the collision integral is zero exercise here we get a first moment term Continuity equation Multiplying by mass or charge we get the continuity eqs for these General form of conservation law for F : (G is the flux of F)
3 Momentum equation (equation of motion) To calculate the first moment, multiply BE by momentum and integrate exercise This tells how momentum is transported in plasma Now the convective derivative of V and the pressure tensor are second moments The electric and magnetic fields must fulfill Maxwell s equations are external sources Note that the collision integral can be non-zero, because collisions transfer momentum between different particle species! One approximation of the collision integral is
4 Equations of state Determining the appropriate eq of state is non-trivial. In a real plasma the different particle populations (e, i) can have different thermodynamical behaviour Let us start from the isotropic case: In isothermal processes T = T 0 = const. : This is valid if the system has time to thermalize with an external heat bath fast compared to time scale of the process. The opposite limit is called adiabatic. In that case the system does not exchange energy with its surroundings. In this case we can show that specific heats in const. p & V is called polytropic index adiabatic: isothermal: isobaric: isometric: for 3D monoatomic gas!
5 Equations of MHD To derive MHD equations we first define single fluid variables as mass density charge density where macroscopic (fluid) velocity current density total pressure tensor are the pressure tensors of each species in centre of mass (CM) frame By summing up the continuity eqs. for different species we get the continuity eqs. for mass and charge
6 The momentum transfer equation is the sum of the corresponding equations of each species (exerc) Cf. the Euler equation of fluid dynamics + electric force + magnetic force (Ampère s force) plasma has been assumed fully ionized the collision terms sum to 0 electric force typically negligible, but the J x B force is essential non-linear terms make solutions difficult The current transfer eq. ( generalized Ohm s law ) is much more complicated. Multiply by and sum over (here only e and i):
7 To make Ohm s law more transparent, let s make some approximations: Introduce the average collision frequency as: This makes sense: Leaving out all derivatives and the magnetic field and recalling we get the simple Ohm s law Some of the terms in Ohm s law are smaller than others Typical approximations: In macroscopic considerations the quasi-neutrality implies When considering small perturbations to equilibrium, second order quantities VJ, JV, VV can be omitted Case : Ideal MHD: MHD Ohm s law Hall term neglected in basic MHD due to slow temporal variations weak spatial gradients
8 Assuming further isotropic pressure tensor in the eq. of motion, we finally get: () In addition an equation of state is needed. MHD plasma can be considered adiabatic (see previous discussion) and/or incompressible Often energy conservation needs to be expressed explicitly in the form where the total energy flux is Exercise: Take a scalar product of () and V and derive the energy equation of adiabatic ideal MHD in the conservation form:
9 Magnetohydrodynamics Convection and diffusion Frozen-in field lines Magnetohydrostatic equilibrium Magnetic field-aligned currents
10 Basic MHD (isotropic pressure) (or energy equation) Relevant Maxwell s equations; displacement current neglected ( )
11 Convection and diffusion Take curl of the MHD Ohm s law and apply Faraday s law Thereafter use Ampère s law and the divergence of B to get the induction equation for the magnetic field (Note that has been assumed constant here) Assume that plasma does not move diffusion equation: diffusion coefficient: If the resistivity is finite, the magnetic field diffuses into the plasma to remove local magnetic inhomogeneities, e.g., curves in the field, etc. Let L B the characteristic scale of magnetic inhomogeneities. The solution is where the characteristic diffusion time is
12 In case of the diffusion becomes very slow and the evolution of B is completely determined by the plasma flow (field is frozen-in to the plasma) convection equation The measure of the relative strengths of convection and diffusion is the magnetic Reynolds number R m Let the characteristic spatial and temporal scales be & and the diffusion time The order of magnitude estimates for the terms of the induction equation are and the magnetic Reynolds number is given by This is analogous to the Reynolds number in hydrodynamics In fully ionized plasmas R m is often very large. E.g. in the solar wind at 1 AU it is This means that during the 150 million km travel from the Sun the field diffuses about 1 km! Very ideal MHD: viscosity
13 Frozen-in field lines A concept introduced by Hannes Alfvén but later denounced by himself, because in the Maxwellian sense the field lines do not have physical identity. It is a very useful tool, when applied carefully. Assume ideal MHD and consider two plasma elements joined at time t by a magnetic field line and separated by In time dt the elements move distances udt and If ideal MHD is assumed, we can show (exercise) that l and B remain parallel at all times, i.e., that Note the notation change: V u In ideal MHD two plasma elements that are on a common field line remain on a common field line. In this sense it is safe to consider moving field lines. Another way to express the freezing: The magnetic flux through a closed loop defined by plasma elements is constant (proof: exercise)
14 The critical assumption was the ideal MHD Ohm s law. This requires that the ExB drift is faster than magnetic drifts (i.e., large scale convection dominates). As the magnetic drifts lead to the separation of electron and ion motions (J), the first correction to Ohm s law in collisionless plasma is the Hall term so-called Hall MHD It is a straightforward exercise to show that in Hall MHD the magnetic field is frozen-in to the electron motion and When two ideal MHD plasmas with different magnetic field orientations flow against each other, magnetic reconnection can take place. Magnetic reconnection can break the frozen-in condition in an explosive way and lead to rapid particle acceleration Example of reconnection: a solar flare
15 Magnetohydrostatic equilibrium Assume scalar pressure and time-independent equilibrium i.e., B and J are vector fields on constant pressure surfaces Solve for J: Diamagnetic current : this is the way how B reacts on the presence of plasma in order to reach magnetohydrostatic equilibrium. Write the magnetic force as: Magnetohydrostatic equilibrium after elimination of the current magnetic pressure magnetic tension
16 Assume plasma isotropic and nearly homogeneous: Plasma beta ; small beta ; large beta In case of the pressure gradient is negligible and the equilibrium is force-free field or field-aligned current Now is constant along B a nasty non-linear equation! If is constant in all directions, the equation becomes linear, the Helmholtz equation A force-free field and the current in the same direction
17 Magnetohydrodynamic waves Fluid waves Dispersion equation for MHD waves Alfvén wave modes V 1 MHD is a fluid theory and there are similar wave modes as in ordinary fluid theory (hydrodynamics). In hydrodynamics the restoring forces for perturbations are the pressure gradient and gravity. Also in MHD the pressure force leads to acoustic fluctuations, whereas Ampère s force (JxB) leads to an entirely new class of wave modes, called Alfvén (or MHD) waves. As the displacement current is neglected in MHD, there are no electromagnetic waves of classical electrodynamics. Of course EM waves can propagate through MHD plasma (e.g. light, radio waves, etc.) and even interact with the plasma particles, but that is beyond the MHD approximation. For such waves, see the next lecture.
18 Sound wave in a fluid Differentiate the equation of state for a gas where is the speed of sound The sound wave is a propagating pressure perturbation whose k is normal to the pressure front. In a magnetized plasma the sound wave can propagate with k B In magnetohydrostatics p is in balance with J x B pressure crests that consists of & If there are no torsional forces, the magnetic tension reduces to The magnetic field line is like a string, where a transverse pertubation propagates with speed
19 The Alfvén and magnetosonic waves This is the propagation velocity of the mode that Alfvén found in 1942, the Alfvén speed For perpendicular propagating perturbation combine the sound and Alfvén waves Alfvén wave where is the magnetosonic speed. Magnetosonic wave
20 Dispersion equation for ideal MHD waves eliminate p eliminate J eliminate E We are left with 7 scalar equations for 7 unknowns ( m0, V, B) Consider small perturbations and linearize
21 () () Find an equation for V 1. Start by taking the time derivative of () () Insert () and () and introduce the Alfvén velocity as a vector Look for plane wave solutions Using a few times we have the dispersion equation for the waves in ideal MHD
22 Alfvén wave modes Propagation perpendicular to the magnetic field: Now and the dispersion equation reduces to clearly And we have found the magnetosonic wave Making a plane wave assumption also for B 1 (very reasonable, why?) (i.e., B 1 B 0 ) The wave electric field follows from the frozen-in condition This mode has many names in the literature: - Compressional Alfvén wave - Fast Alfvén wave - Fast MHD wave
23 Propagation parallel to the magnetic field: Now the dispersion equation reduces to Two different solutions (modes) 1) V 1 B 0 k the sound wave 2) V 1 B 0 k V 1 This mode is called - Alfvén wave or - shear Alfvén wave
24 Propagation at an arbitrary angle e z B 0 k e y e x Dispersion equation Coeff. of V 1y shear Alfvén wave From the determinant of the remaining equations: Fast (+) and slow ( ) Alfvén/MHD waves
25 fast fast magnetosonic slow slow sound wave sound wave fast magnetosonic Wave normal surfaces: phase velocity as function of fast slow slow
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