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Astrophysical Fluid Dynamics: II. Magnetohydrodynamics

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Astrophysical Fluid Dynamics: II. Magnetohydrodynamics Winter School on Computational Astrophysics, Shanghai, 2018/01/30 Astrophysical Fluid Dynamics: II. Magnetohydrodynamics Xuening Bai (白雪宁) Institute for Advanced Study (IASTU) & Tsinghua Center for Astrophysics (THCA) source: J. Stone Outline n Astrophysical fluids as plasmas n The MHD formulation n Conservation laws and physical interpretation n Generalized Ohm’s law, and limitations of MHD n MHD waves n MHD shocks and discontinuities n MHD instabilities (examples) 2 Outline n Astrophysical fluids as plasmas n The MHD formulation n Conservation laws and physical interpretation n Generalized Ohm’s law, and limitations of MHD n MHD waves n MHD shocks and discontinuities n MHD instabilities (examples) 3 What is a plasma? Plasma is a state of matter comprising of fully/partially ionized gas. Lightening The restless Sun Crab nebula A plasma is generally quasi-neutral and exhibits collective behavior. Net charge density averages particles interact with each other to zero on relevant scales at long-range through electro- (i.e., Debye length). magnetic fields (plasma waves). 4 Why plasma astrophysics? n More than 99.9% of observable matter in the universe is plasma. n Magnetic fields play vital roles in many astrophysical processes. n Plasma astrophysics allows the study of plasma phenomena at extreme regions of parameter space that are in general inaccessible in the laboratory. 5 Heliophysics and space weather l Solar physics (including flares, coronal mass ejection) l Interaction between the solar wind and Earth’s magnetosphere l Heliospheric physics NASA/JPL-Caltech Interstellar Medium Termination shock 6 Astrophysical applications Accretion disks Jets (Collisionless) shocks Galaxy clusters 7 Comparison of plasma and gas phases Property Gas Plasma Usually very high (effectively Electrical conductivity Very low infinite in most cases) Independently acting Two or three (electrons, ions Usually, one species and sometimes neutrals) Maxwellian (due to Often non-Maxwellian: many Velocity distribution frequent collisions) plasmas are collisionless Collective: organized motion by interacting with long-range Interactions Binary collisions electromagnetic fields in the form of plasma waves. 8 Outline n Astrophysical fluids as plasmas n The MHD formulation n Conservation laws and physical interpretation n Generalized Ohm’s law, and limitations of MHD n MHD waves n MHD shocks and discontinuities n MHD instabilities (examples) 9 How to describe a plasma? n The exact microscopic description l Follow the trajectories of all particles, and solve for the evolution of EM fields. l The description is exact, but is computationally hopeless. l Solutions can be chaotic (depending on initial conditions). n Kinetic theory (still microscopic) l Use statistical approach to describe particles (i.e., distribution function). l Evolution of the distribution function according to particle orbit theory. l EM fields are largely long-range, whereas weak binary interactions can be treated as “collisions”. n Fluid theory (macroscopic) l Only focus on macroscopic fluid quantities (e.g., density, bulk velocity, T). l No need to follow individual particle orbits. 10 Magneto-hydrodynamics (MHD) MHD couples Maxwell’s equations with hydrodynamics to describe the macroscopic behavior of highly conducting fluid such as plasmas. Ideal MHD involves several important approximations, as we list below. 1. Flow velocity is very non-relativistic. Can be relaxed to formulate relativistic MHD. 2. Electric conductivity is so high that can be considered as infinite. Can be relaxed to formulate non-ideal MHD. 3. Low-frequency, long-wavelength. This is the key to the fluid description of plasmas (besides collisionalities). We first formulate MHD equations from the above approximations, and then justify them (for 2 and 3). 11 Reduction of Maxwell’s equations We focus on the non-relativistic regime where flow velocity v<<c. Starting from Maxwell’s equations: [1] (implied from [4]) E L V [2] B ⇠ cT ⇠ c [3] (implied from [2]) [4] Displacement current can be dropped. ~B/L ~E/cT~ (V2/c2) (B/L) c J = B The system can adjust its current adjusted 4⇡ r⇥ instantaneously to match field configuration. 12 The induction equation B field evolves according to the induction equation: But what determines E? Under the assumption that the fluid is infinitely conducting: E0 =0 (in fluid rest frame) This leads to ideal MHD. The relation between E and E’ is given by a Lorentz transformation: where again we have assumed v<<c. With E’=0, the induction equation becomes 13 Momentum equation A conducting fluid is further subject to the Lorentz force: ~B2/L ~E2/L~(V2/c2)B2/L However, the electric force about a factor (V/c)2 smaller and can be dropped. Therefore, the MHD momentum equation simply read: 14 Summary: ideal MHD equations Continuity equation (unchanged): @⇢ + (⇢v)=0 @t r · Momentum equation (now includes the Lorentz force): Induction equation (new addition): @B = c E = (v B) @t − r⇥ r⇥ ⇥ Thermal energy equation (unchanged in the current form): Ds @s @P = +(v )s =0 or + v P + γP v =0 Dt @t · r @t · r r · 15 Outline n Astrophysical fluids as plasmas n The MHD formulation n Conservation laws and physical interpretation n Generalized Ohm’s law, and limitations of MHD n MHD waves n MHD shocks and discontinuities n MHD instabilities (examples) 16 Momentum conservation With the addition of the Lorentz force, we note magnetic magnetic tension pressure Momentum conservation becomes where the stress tensor is total pressure One may further include viscous stress introduced earlier, and gravitational stress (for self-gravitating system) depending on application. 17 Understanding the stress tensor Πzz Πzy Πzx Πyz Πxz Πyy Πxy Πyx Πxx l Imagine you have an infinitesimally small box. l Forces are exerted on each face from the outside volume. l The forces on each side have components in 3 directions. l The stress tensor then includes all 9 quantities needed to describe these forces. 18 More on magnetic tension and pressure The Lorentz force is perpendicular to B, but magnetic pressure sounds like an isotropic pressure. A better way to decompose the Lorentz force is as follows. J B ( B) B B B B2 ⇥ = r⇥ ⇥ = · r c 4⇡ 4⇡ r 8⇡ ✓ ◆ B2 B2 Curvature: = 4⇡ r? 8⇡ ✓ ◆ Magnetic tension Magnetic pressure 19 Magnetic flux conservation @B The induction equation: = (v B) @t r⇥ ⇥ V already implies B flux conservation. More importantly, ΦB(t+dt) it implies: ΦB(t) V The B flux through a co-moving fluid loop is constant (known as Alfvén’s theorem). t t+dt Proof: dl S Vdt C 20 Flux freezing: physical meaning In ideal MHD, the magnetic field are plasma are V “frozen-in” to each other. ΦB(t+dt) ΦB(t) V l The plasma can not move across B field lines. l If two plasma elements are initially connected by a field line, they will remain connected. l Magnetic topology is preserved in ideal MHD. t t+dt Physically: charged particles are tied to field lines as they gyrate. The frozen-in condition can break in non-ideal circumstances. In particular, magnetic reconnection is a process that breaks magnetic field topology. It generally involves kinetic effects beyond MHD (in collisionless plasmas) and/or dissipation by resistivity. We will briefly address this later. 21 Flux freezing with weak/strong field Strong field: matter move along Weak field: field lines are field lines (beads on a wire). forced to move with the gas. Strength of the B field is commonly Pgas 8⇡Pgas β = 2 characterized by the plasma � parameter: ⌘ Pmag B 22 Energy conservation Now we re-derive energy conservation incorporating B field. New components are: where From vector calculus: Therefore, the equation of energy conservation becomes: energy density Poynting flux of the B field 23 Energy conservation We can rewrite the energy conservation equation Using , we arrive at: It has exactly the same form as HD energy equation except that we have updated the energy density: and the stress tensor: 24 Outline n Astrophysical fluids as plasmas n The MHD formulation n Conservation laws and physical interpretation n Generalized Ohm’s law, and limitations of MHD n MHD waves n MHD shocks and discontinuities n MHD instabilities (examples) 25 Generalized Ohm’s law We relax the assumption of infinite conductivity, which helps address the conditions under which ideal MHD fails. The plasma is generally made of electrons and ions. Ions carry almost all the mass, representing the bulk plasmas. Here separate out the electrons (treated as a fluid).: pressure inertia Lorentz “collision” with gradient force the bulk plasma Being the lightest, e- almost instantly respond The balance among the other to EM fields (to avoid huge acceleration) terms determine the E field. => ignore inertia 26 Generalized Ohm’s law Electric field is found to be: Note: We arrive at the generalized Ohm’s law: e- pressure Hall term resistivity gradient 27 Generalized Ohm’s law Are additional terms in the generalized Ohm’s law important? One can show that these two terms are relevant only at microscopic scales: (ion inertial length) While we have ignored electron inertia, one can also show that that term is relevant at even smaller scales: (electron inertial length) This gives one example that MHD applies only on macroscopic scales. 28 Applicability and limitations of MHD n MHD applies to timescales much longer than 1 1 1 1 ⌧ !− , !− , ⌦− , ⌦− pe pi ce ci e-/ion plasma - frequencies e /ion cyclotron frequencies n MHD applies to length scales much larger than L λ ,r ,r ,c/! ,c/! D Le Li pe pi Debye length e-/ion Larmor radii e-/ion inertial lengths n MHD requires the particle distribution function to be (at least approximately) isotropic and Maxwellian with Ti=Te. 29 Resistive MHD We are left with the standard Ohm’s law: where electric conductivity . The induction equation now reads @B = c E = (v B) @t − r⇥ r⇥ ⇥ which finally becomes where Ohmic resistivity is given by 30 Resistive MHD l Resistivity breaks the frozen-in condition, allowing filed lines to slide through the plasma.
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