1 Energy dissipation in astrophysical plasmas

Transcription

1 1 1 Energy dissipation in astrophysical plasmas The following presentation should give a summary of possible mechanisms, that can give rise to temperatures in astrophysical plasmas. It will be classified in 3 chapters. The first one will give a mathematical description for the contributions of dissipation and energy transport. The second chapter will treat various mechanisms that might dissipate energy. The last chapter will mention possible ways of transporting energy in a plasma. 1.1 energy equation The energy equation is a very useful tool to show, wether there are sources or sinks of energy in an atmosphere. If we look at a plane parallel atmosphere we can write the energy equation as ρ T ds dt = L, (1.1) where L is the energy loss function, and s the entropy per unit mass. For the case L = 0 the conservation of entropy is valid. If we express above equation for an ideal polytropic gas, in terms of the internal energy, it reads ρ de dt p ρ dρ dt = L. (1.2) For an ideal polytropic gas the pressure reads as p = ρ kt. E is the internal energy, and m c v and c p the specific heat for constant volume and pressure respectively. If we take the internal energy as E = c v T, and insert c p = c v + k b cp and γ = m c v for the polytropic gamma, we receive the energy equation as ρ γ γ 1 d ( ) p = L. (1.3) dt ργ The energy loss function L contains positive terms for the conductive, radiative and convective flux which are sources of energy. The negative terms can be all forms of energy dissipation, on which we will focus in the following chapters. If we now look at atmospheres containing plasma, we have to decide which sources and sinks we take into considerations. For looking at outer atmospheres, we can use the Plane Parrallel Approximation, meaning that we disregard the curvature of the atmosphere for small surface elements. We can now look at the timescales for changes in ρ, p and T, and decide wether or not we take into account the various contributions of dissipation.

2 2 ENERGY DISSIPATION 2 2 Energy dissipation 2.1 Ohmic dissipation For plasma moving at non relativistic speeds, in the presence of a magnetic field, we have j = σ (E + v B). (2.1) σ is the electric conductivity and can be approximated by σ = const T 3 2. If we consider a cube of the size l, the stored magnetic energy inside the cube is W = L 3 B2 2µ. (2.2) The inflow of electromagnetic energy into a volume, produces ohmic heat and kinetic energy (work done by the lorentz force). Thus we have (E H) = j2 σ + v (j B) (2.3) By looking at a thin shell we can disregard the pressure gradient. Then the ratio between ohmic energy and kinetic energy becomes 1. Now lets consider a typical atmosphere with a thickness of 10 km and T = 10 4 K. The typical timescale for the release of ohmic energy is τ D = L 2 µσ. With above values we receive a time of sec. Thats a very slow rate of energy dissipation, and thus we can see, that big flares can t be heated by ohmic dissipation only. On the other hand we see that the energy for small regions is released much faster, since the size of the region has a quadratic effect on the time. Thus the heating of spicules can be explained pretty well. 2.2 Acoustic wave heating Acoustic waves are longitudinal waves. In a uniform medium, sound waves steepen, because they propagate, according to a dispersion relation. Every part of the wave profile moves with a different velocity. Thus the crests possess a higher velocity than the troughs. The ambient sound speed can be calculated by c s = ( γkb T m i )1 2. (2.4) If we denote the velocity amplitude as v 1, the crests and troughs have a relative velocity of 2v 1.

3 2 ENERGY DISSIPATION 3 The trough is ahead λ λ and will be overtaken after t = 2 4v 1. The distance the wave can travel at c s before shocking is d = λcs 4v 1. Thus short period waves evolve into shock waves over much smaller distances than long period waves. For a vertically stratified atmosphere, the distance for shock formation is much smaller because the wave amplitude increases rapidly with altitude. If we take the kinetic energy constant and notice, that its proportional to ρv 1 2, we see that changes in ρ can only be 2 balanced by changes in v 1. In an isothermal atmosphere the density develops as ρ z = e Λ, z thus the velocity amplitude has to change like v z = e2λ.(λ z is the scale height) The distance for a shock to form in a vertically stratified medium is now d = 2Λ ln ( τc s (γ + 1)Λv 1 ). (2.5) The waves can now dissipate their energy through viscous, thermal or radiative losses. The flux of energy transmitted by the shocks can be seen as the work done by the pressure. F z = ν (p 2 p 1 ) v 2 t0 τ, (2.6) where subscripts 2 and 1 denote the rear and front shock, and t 0 is depending on the shock profile. Thus we can get the rate of Energy dissipation df dz = νγ(γ + 1)p 1 η 3, (2.7) 12 where η = ρ 2 ρ 1 ρ 1 is the fractional compression. In the photosphere of the sun we have a source for acoustic waves, due to granulation. These acoustic waves were proposed to heat the corona. But it turned out that the waves cant reach the corona, since they are reflected at the transition zone. Thus the acoustic waves are responsible for the vast temperature increase in the transition zone, but not for the high temperatures in the corona. 2.3 Dissipation of magnetic fields For the description of a plasma we need a combination of the equations of motion, maxwells equations and the continuity equation. We can make the following approximations : The velocity of the particles v mat c The field configuration doesnt change rapidly with time.

4 2 ENERGY DISSIPATION 4 The phase-velocity v ph c The plasma of consideration have good conductivity E B 1 and Ohm s law j = σe is valid in the resting frame of reference of the plasma. With these approximation we get the Ideal MHD, which is a linear approximation. B = 4π c j, (2.8) E = 1 c B t, (2.9) B = 0, (2.10) j = σ (E + 1 v B). (2.11) c If we now assume some velocity field, and look at the changes of the EM-field, we have MH kinematics. We can now look at a moving plasma with finite σ and solve the system of equations. We can reduce the system to following equation B t c2 (v B) = B, (2.12) 4πσ with values for v r,t and B t0 we can calculate B t. For v = 0 and B = (B x(y,t), 0, 0) we get equation B x t = c2 4πσ 2 B x y 2. (2.13) This equation is of the same type as the equation for thermal conductivity. There is an analogy between lines of constant temperature and fieldlines. Eventhough the flux through the y z plane is constant, the magnetic energy of the system decreases. Because of the finite σ, the magnetic energy is converted into heat. For the timescale of the dissipation for magnetic fields we get τ = 4πσ c 2 L 2. (2.14) Since the timescale is depending on the square of the size, we see that small field configurations dissipate fast, big ones slow.

5 2 ENERGY DISSIPATION Heating by MHD-Waves Additional to the equations 2.8 to 2.11, we have to take into account the dynamic interaction between fields and matter. At first we need the Equation of motion for matter. ρ v t + ρ (v )v = P + 1 j B + ρ g + ρν v. (2.15) c The second equation we need to add is the Equation of continuity which reads ρ t + (ρv) = 0. (2.16) According to the nature of changes in the plasma we also have to define the Equation of state, to describe adiabatic or polytropic changes. Together with equations 1 to 4 we have linear homogeneous system of partial differential equations. But since space and time coordinates do not appear explicitly in the equations, we can make an exponential ansatz (B 0 is homogenous in x-direction; σ ; v 0 = 0) 2.5 MHD-compressional-waves Ansatz: v 1 =(0,v 1y(y,t), 0); v 1y = c exp ( iωt ± ily) x = 0; z = 0; y = ±il; t = iω Since v 1y is only dependent on y, we have got a longitudinal wave. When such waves propagate, we have a periodic compression of plasma and thus we need a relation for adiabatic-changes which yields the adiabatic soundspeed P 1 = γ P0 ρ 0 ρ 1 = (v s ) 2 ρ 1, (2.17) [ v s = γ P 0 ρ ]1 2. (2.18) Solving the system of equations yields a relationship for the phasevelocity u 2 = ω2 l 2 = v2 s + v 2 a. (2.19) In case of a small magnetic field the phase velocity converges to the adiabatic sound speed, in case of a strong magnetic field, it converges to the Alfvén velocity.

6 2 ENERGY DISSIPATION Alfvén-waves Ansatz : v 1 = (0,v 1y(x,t), 0) ; v 1y = c exp ( iωt ± ilx) and the magnetic field only in x direction z = 0 ; y = 0 ; t = iω Since v 1y is depending on x and t, our solution is a transversal wave. With above ansatz we get the solution with the Alfvén velocity l 2 = 4πω2 ρ 0, (2.20) Bo 2 v a = ω l = B 0. (2.21) (4πρ 0 ) 1 2 When we look at the formula for the Alfvén velovity we recognize that its only depending on B 0 and ρ 0. Thus we can identify it as a typical variable of state for a plasma configuration.if we have distortions, they propagate along the field lines. For strong magnetic fields, v a becomes big, for high densities it becomes small. In the case where the density converges to 0 and v a converges to c, Alfvén waves develope into electromagnetic waves. For both kinds of waves mentioned in this chapter we can see that the phase velocity is independent of the frequency, meaning that the waves can propagate without dispersion!!! If we make an ansatz in an arbitrary angle to the magnetic field, we get three solutions namely Alfvén, MHD compression and soundwaves. Thus in most cases we have a combination of those wave types. 2.7 Magnetoacoustic waves Are a combination of MHD and acoustic waves.we distinguish between fast and slow mode waves. Both kinds of waves form shocks and dissipate like sound waves, with ohmic dissipation as an additional energy source. Fast waves can transport energy in all directions. In regions where v a v s they propagate with v a in all directions. Slow waves transport energy only in directions that are close to the magnetic field. Alfvén-waves dissipate much less energy than magnetoacoustic waves. But there is however a nonlinear interaction between Alfvén waves. For a weak magnetic field andv a < v s, two waves which propagate in opposite directions along a fieldline, can couple with each

7 2 ENERGY DISSIPATION 7 other. That results in the formation of a soundwave, which is able to dissipate its energy faster. In regions with a strong magnetic field an Alfvén wave can split up into another one, propagating in the opposite direction plus a sound wave in the original direction. The new formed Alfvén-wave has a lower frequency and can again split up... That can lead to a cascade and contributes to a faster energy dissipation. 2.8 Reconnection in Current Sheets A current sheet may be defined as a non propagating boundary between two plasmas. It is a tangential discontinuity without any flow. The total pressure on both sides of the sheet has to be continous p 2 + B 2 2 2µ = p 1 + B 2 1 2µ. (2.22) The magnetic strength is the same on both sides, but with opposite directions. At the center of the sheet the plasma pressure is enhanced by p 0 = B z1 2 2µ = B 2 z2 2µ. (2.23) The magnetic field vanishes completely in the center of the field. Thus a current sheet can be regarded as a discontinuity that separates 2 regions where the equations of ideal MHD hold. In the absence of flow the current sheet will diffuse away at the speed η l If we have a flow of plasma and magnetic flux towards the sheet with v i, we can distinguish between 3 cases v i v i v i > η l = η l < η l expansion (2.24) steady (2.25) contraction (2.26) Now the enhanced pressure in the center of the sheet expels material at the ends of the sheet. Thus magnetic flux is ejected with the plasma, and one effect on the sheet is, to reconnect its fieldlines. If the inflow speed is sub Alfvénic, the outflow field strength is smaller than the inflow. B 0 = v i v a B i. (2.27)

8 3 ENERGY TRANSPORT 8 Figure 1: current sheets 1 Thus magnetic energy is converted into heat and flow energy. This mechanism is very important for the heating of flares. 3 Energy transport 3.1 Conductive energy transport In high temperature regions, particles have a high velocity and kinetic energy. They may travel into cooler regions, like particles of cooler regions may travel into hotter ones.this process can be seen as a diffusion process. Therefore we have a net energy transport into cooler regions. The energy transport itself happens through collisions of particles. Conduction is depending on the following two points Density If the density is low, the particles can travel further, which makes conduction more effective. On the other hand there are less particles that collide. These two effects can cancel each other, and make conduction independent of density (which is the case in the corona or transition zone)

9 3 ENERGY TRANSPORT 9 Figure 2: current sheets Collisional crossection Is depending on the particle species and velocity. For high velocities the crossection is decreasing, which leads to increasing mean free path length. In the presence of magnetic fields, it makes sense to split up the heat flux into into a component parallel and perpendicular to the magnetic field.conduction along the magnetic field is primarily governed by electrons, across the fields primarily by ions. 3.2 Energy transport by radiation Figure 3: radiation

10 3 ENERGY TRANSPORT 10 Consider a surface area of size dσ, and radiation passing through it in a cone dω. The amount of energy going through that area is E λ = I λ cos Θdωdσdλ (3.1) Figure 4: radiation 2 If we imagine a volume element of plasma where radiation passes through, we have absorption and emission. Emission gives rise to the energy by The Absorption reduces the energy by de λ = ε λ dωdσdλds. (3.2) de λ = κ λ E λ ds = κ λ I λ dωdσdλds. (3.3) Out of the two equations for emission and absorption we get the Radiative transfer equation which reads di λ κ λ ds = I λ + ε λ κ λ. (3.4) The term ε λ κλ is called the Source function and plays a very important roll in stellar astrophysics. It can be seen as the ration between reemission and absorption in an atmosphere. The expression κ λ ds is also called the optical depth. It is necessary for the sun, to define an optical depth that we define as surface of the sun.

11 3 ENERGY TRANSPORT Convective energy transport In an adiabatic process we have an Adiabatic gamma For the pressure in the plasma we have the relation γ = c p c v. (3.5) P = const ρ γ = ρ µm h kt. (3.6) Out of above relations for the pressure we can derive an Adiabatic temperature gradient describing an adiabatic changes in the plasma. ad = 1 dt T dr 1 dp P dr = γ 1 γ. (3.7) We can now consider a bubble, which is moved by a force the distance r, undergoing an adiabatic change, keeping it in pressure equilibrium with its surrounding. Thus the temperature of the bubble will change according to the adiabatic temperature gradient. T 2Bubble = T 1Bubble + [ ] dt r. (3.8) dr adiabatic The temperature of the surrounding plasma will change according to T 2Surrounding = T 1Surroundingbubble + [ ] dt r. (3.9) dr external For a bubble to start moving on its own, we have to have a marginally lower density, and a marginally higher temperature. If we insert ρ 1B = ρ 1 δρ and T 1B = T 1 + δt into the gas law, using the constrain that ρ 1B ρ 1, we receive the Criterion for Convection. [ ] [ ] dt dt > dr adiabatic dr external (3.10) In atmospheres this might be the case in areas where the ionisation degree changes rapidly. From γ = cp c v = 1 + Nk c v follows that γ will decrease if c v = du increases. If we consider a dt non ideal gas, the inner energy might also include terms for excitation and ionisation. In areas where the ionisation degree changes rapidly we have du > 3 Nk which leads to dt 2 γ < 5 and thus 3 ad < 0, 4. That might lead to convection.