13. ASTROPHYSICAL GAS DYNAMICS AND MHD Hydrodynamics

Transcription

1 1 13. ASTROPHYSICAL GAS DYNAMICS AND MHD Hydrodynamics Astrophysical fluids are complex, with a number of different components: neutral atoms and molecules, ions, dust grains (often charged), and cosmic rays. The magnetic field generally ties all these fluids together except where gradients are very steep, as in shocks. The basic hydrodynamic equations express the conservation of mass, momentum and energy. We shall first derive these relations in the absence of gravity or magnetic fields. We shall generally assume that the particle distribution function is isotropic, so that there is no viscosity or heat flux an ideal fluid. The equations apply equally well to a collisional gas, like air, or to a collisionless plasma Mass conservation: The equation of continuity Let ρ be the mass density and M = ρ dv be the mass in volume V. Let v be the mean velocity; for simplicity, we assume that all the components of the gas that make a signficant contribution to ρ are moving at the same average velocity. Since mass is conserved i.e., it is neither created nor destroyed the rate of change of mass inside a fixed volume V is minus the rate at which mass is flowing out of the volume: M = ρv ds, (1) where ds is an element of surface area bounding the volume. Evaluating M/ and applying the divergence theorem, we find ρ dv + (ρv)dv = 0. (2) Since this is true for an arbitrary volume, it follows that ρ + ρv = 0, (3) Equation (3) is the differential form of the equation of mass conservation, and is often referred to as the continuity equation. It is the prototype of a conservation equation: (density) + (flux) = 0. (4) Recall that the convective derivative follows the change in a comoving volume element, d dt = + v. (5) Using this, we rewrite the continuity equation (3) as d dt ln 1 ρ = v, (6) which relates the rate of change of the specific volume 1/ρ to the divergence of the velocity field.

2 Momentum conservation for ideal fluids: the equation of motion (g = B = 0) We begin with the simplest case in which there are no gravitational or magnetic fields. The momentum density is ρv ρ w, where w is the velocity of an individual particle and is an average over a volume large enough to contain many particles, but small enough that it is a smooth function of position (except at shocks). The momentum flux is a tensor, ρ ww ; the ij component of this tensor is the flux of the i component of momentum in the j direction. The peculiar velocity of a particle, δw, is defined by w v + δw with δw = 0, (7) so that ρ ww = ρvv + ρ δwδw. (8) The tensor ρ δwδw is the stress tensor. We now assume that the stress tensor is isotropic, δw i δw j δ ij ; as remarked above, the fluid is then termed ideal. In an ideal fluid, there is no viscosity; the only stresses are normal to a surface i.e., they are the pressure, p = ρ δw 2 i (9) for arbitrary i. In tensor notation, we have ρ δwδw = pi, (10) where I is the identity tensor. Equivalently, p = 1 3ρTr δwδw. Hence the momentum flux for an isotropic fluid is ρ ww = ρvv + pi. (11) As an aside, we note that the average energy per particle is related to the temperature T by where µ ρ/n is the mean mass per particle. Hence, 1 2 µ δw2 i = 1 kt, (12) 2 p = ρ kt µ = nkt, (13) which is the equation of state for an ideal gas, as expected. It is convenient to define C 2 kt µ, (14) where, as we shall see below, C is the isothermal sound speed. It follows that the pressure is p = ρc 2. (15)

3 3 The equation of momentum conservation has the general form of equation (4), which, for an ideal fluid, implies (momentum density) + (momentum flux) = 0, (16) ρv + (ρvv + pi) = 0. (17) To obtain the conventional form of the hydrodynamic equation of motion, we expand the derivatives, v v ρ + ρ v + v ρv + ρv v + p = 0 (18) ( ) ( ) ρ + v ρv + ρ + v v + p = 0. (19) The terms in the first parentheses vanish due to mass conservation. The terms in the second parentheses can be written in terms of the convective derivative, so that ρ dv dt = p. (20) Note that this equation is just F = ma, per unit volume, for the fluid. It is often referred to as the Euler equation Energy conservation for an ideal fluid (g = B = 0) The energy conservation equation has a different form from the first two conservation equations since the fluid can gain energy from the radiation field and cosmic rays at a rate nγ per unit volume and lose energy to radiation at a rate n 2 HΛ per unit volume. We write the net energy loss rate as n 2 H L n2 H Λ n HΓ. Now, the energy density is the sum of the kinetic and internal energy densities 1 2 ρv2 + ρe = 1 2 ρv2 + p γ 1 + ρe I, (21) where γ is the ratio of specific heats and e I is the ionization energy per unit mass (see Lec. 5). The energy flux includes the rate at which the pressure does work, Hence, the energy conservation equation is energy flux = (energy density)v + pv. (22) (energy density) + (energy flux) = n 2 HL (23) ( ) ( 1 ρ 2 v2 + e + 1 ρv 2 v2 + e + p ) = n 2 ρ HL. (24)

4 4 Note that ρe + p is the enthalpy; for a monatomic gas (γ = 5/3), the enthalpy is 5 2 p. Equation (24) is the total energy equation. It can be derived more rigorously by first taking the dot product of v with the equation of motion and using the continuity equation to obtain the work equation, ( 1 2 ρv2) ( ) + 1 v 2 ρv2 = v p. (25) Again using the equation of continuity, the First Law of Thermodynamics (see Lec. 5) can be re-expressed as the internal energy equation ρ de dt p dρ ρ dt = ρdq dt = n2 HL (26) ρe + vρe = p v n 2 HL. (27) The total energy equation follows from adding the work and internal energy equations. A yet more general derivation of the energy equation follows from the Boltzmann equation, which allows one to treat the case of non-ideal fluids ( ) Sound waves Consider small fluctuations in the density of the gas, so that ρ = ρ 0 + δρ, etc.; we assume that the gas is initially stationary, so that v 0 = 0. The fluctuations in the pressure are determined by c 2 s δp δρ, (28) which, as we shall see, is the sound speed. For small fluctuations, c 2 s is constant. Since the fluctuations are small, they can be represented as a superposition of plane waves of the form δρ exp i(kx ωt). Linearizing the continuity equation and the equation of motion, we find iωδρ + ikρ 0 δv = 0, (29) iωρ 0 δv = ikδp = ikc 2 sδρ. (30) Solving these equations gives ω 2 k 2 c 2 s = 0, (31) so that the phase velocity of the waves is ω/k = c s, verifying that c s is the sound speed. For an adiabatic gas (p ρ γ ), the sound speed is given by c 2 s = p ( ) δ ln p = γp ρ δ ln ρ ρ = γc2. (32) An isothermal gas corresponds to the limit γ 1, so that c s = C in this case.

5 5 An important characteristic parameter of a flow is its Mach number M v c s. (33) Atmospheric flows are subsonic (M 1), but astrophysical flows are often supersonic (M > 1), resulting in shock waves Viscosity and heat conduction Real fluids are viscous and conduct heat. On a microscopic level, this is due to anisotropy of the particle distribution function. If the fluid is not isotropic, we define the viscous stress tensor π as ρ δwδw pi π, (34) so that the momentum flux (eq. 8) is ρ ww = ρvv + pi π. (35) The general form of the equation for momentum conservation is then ρv + (ρvv + pi π) = 0. (36) This result can be derived more rigorously from the Boltzmann equation (Shu 1992). If the mean free path λ is small (λ L, where L is a characteristic length scale in the system), the viscous stress tensor has the form π ij ρν v i x j, (37) where ν is the kinematic viscosity and is of order ν λv T, (38) where v T is the thermal velocity; except for electrons, v T is of order the sound speed. In a fully ionized plasma, the ions carry the momentum and dominate the viscosity. 1 1 Viscosity represents an internal friction, and generally manifests itself in shear flows. When the fluid has internal degrees of freedom (including the molecular or ionic state of the gas), a second viscosity coefficient, the bulk viscosity coefficient, is introduced. Bulk viscosity is proportional to v and is isotropic; as a result, it adds to the pressure. As discussed by Batchelor (1967), one can omit the effects of bulk viscosity from the equation of motion if one defines the pressure mechanically as the normal component of the stress. For a gas with internal degrees of freedom that are not in equilibrium, this mechanical pressure can differ from the thermodynamic pressure, p e, which is given in terms of the energy density and mass density under the assumption that the gas is in thermal equilibrium. Further discussion of the viscosity is given in Shu (1992).

6 6 How important are viscous effects? On a size scale L, the relative magnitude of inertial and viscous forces, inertial viscous ρv2 /L ρνv/l 2, (39) is given by the Reynolds number R Lv ν Lv λv T. (40) In astrophysical fluids, the size scale L is generally so large that R 1 and viscous effects can be neglected on those large scales. (A possible exception is in very hot gas, where v T λ becomes large.) Small length scales occur in shocks, however, and viscous effects are important there. When R is sufficiently large, flows become turbulent. Turbulent flows contain eddies over a range of scales (termed the inertial range ), extending from the scale on which the turbulence is driven down to the scale at which it is dissipated by viscous effects, where R is very roughly of order unity. Turbulence will be discussed further when we treat star formation. In the energy equation, anisotropic distribution functions lead to both viscous heating and thermal conduction. For small mean free paths, the conductive heat flux is κ T, where κ is the thermal conductivity. In a plasma, the electrons generally dominate the heat flux because of their large thermal velocity, although the ions can dominate if they are significantly hotter than the electrons. When heat conduction and/or viscous heating are important, they add to the energy flux in equation (24), so that ( 1 ρ 2 v2 + e ) [ ( + 1 ρv 2 v2 + e + p ) ] v π κ T = n 2 ρ HL. (41) As is the case with the momentum equation, this result can be derived more rigorously from the Boltzmann equation Gravity Let φ be the gravitational potential, so that the acceleration due to gravity is g = φ. The equation of motion becomes ρ dv dt = p ρ φ. (42) The energy equation is modified by a term ρv φ on the RHS, which represents the rate at which the gas gains energy from the field. Now, the equation of continuity implies that ( ) (ρφ) + ρ ρvφ = φ + ρv + ρ φ + ρv φ = ρ φ + ρv φ. (43) Adding this to the energy equation gives ( ) ( 1 ρ 2 v2 + e + φ + 1 ρv 2 v2 + e + p ) ρ + φ = n 2 HL + ρ φ. (44)

7 7 The presence of the term ρ φ/ on the RHS implies that a time dependent potential destroys local energy conservation. Global energy conservation is preserved, however. Note that the gravitational energy is W = 1 ρφdv = 1 ρ(r)ρ(r 2 2 G ) r r dv dv, (45) where the integral extends over all space. It follows that dw = 1 (ρφ) dv = dt 2 ρ φ dv (46) (Shu 1992). Hence, integrating the energy equation over all space (or at least over all the space containing matter) yields ( ) d 1 ρ dt 2 v2 + e dv + 2 dw = dw dt dt n 2 HL dv. (47) Since the total energy is this gives E = ( ) 1 ρ 2 v2 + e dv + W, (48) de dt = n 2 HL dv. (49) Bernoulli s Theorem For steady, adiabatic flows of an ideal fluid, the equation of motion can be integrated to give a useful constant of the motion. Recall the vector identity For A = B = v, this reduces to (A B) = (A )B + (B )A + A ( B) + B ( A). (50) v 2 = 2 [ (v )v ( v) v ]. (51) If the fluid is adiabatic, then p is a function of a single thermodynamic variable; in particular, we can write p = p(ρ). The equation of motion with gravity (eq. 42) becomes where v v2 + ( v) v = (h + φ), (52) h dp ρ. (53)

8 8 For adiabatic variations, h is equivalent to the specific enthalpy, h = e + p ρ = γp (γ 1)ρ. (54) Equation (52) is actually more general than this and applies to any barotropic fluid, one in which p = p(ρ); however, as we shall see below, this generalization is unlikely to be useful in practice. For steady flows, the dot product of equation (52) with v yields ( ) v 1 2 v2 + h + φ = 0, (55) 1 2 v2 + h + φ = constant along a streamline Bernoulli s Theorem. (56) It is instructive to derive Bernoulli s theorem from the energy equation. We shall generalize the treatment to non-ideal fluids, so that we can treat shocks. In a steady state, equation (41) becomes [ ( 1 ρv 2 v2 + e + p ) ] ρ + φ v π κ T = 0, (57) with the inclusion of gravity and in the absence of heating or cooling (L = 0). Since ρv = 0 in a steady flow, this reduces to equation (55) for an ideal fluid (π = 0, κ = 0), with h = e + p/ρ, the enthalpy. This emphasizes that Bernoulli s theorem generally applies only to adiabatic flows. In principle, it might be possible to have h e + p/ρ in equation (56), but the derivation from the energy equation shows that this would require a conspiracy on the part of the energy exchange term L. However, it is possible to derive a form of Bernoulli s equation from the energy equation that applies to non-ideal fluids, provided we restrict our attention to one-dimensional flows (Batchelor 1967). If the flow is in the z direction, say, we have ρv z = const. for steady flows, so that equation (57) implies 1 2 v2 + e + p π zz + φ κ dt = constant along a streamline. (58) ρ ρv z dz Thus, the standard form of Bernoulli s theorem applies along all parts of the streamline on which non-ideal effects are negligible, provided we set h = e + p/ρ. In particular, equation (58) provides one of the shock jump conditions, as we shall see in Lecture 14. As a simple application of Bernoulli s theorem we show that steady flows are approximately incompressible if the Mach number M is small. For simplicity, assume that gravity is negligible. We consider the largest possible change in velocity, δv v, so that Bernoulli s theorem implies δh v 2. We then have δh = δp ρ = c2 sδ ln ρ v 2 = M 2 c 2 s. (59) Hence, δ ln ρ M 2 : If the flow is very subsonic, then the density is approximately constant.

9 Magnetohydrodynamics: MHD Magnetic fields both simplify and complicate astrophysical fluid flows: they simplify them by tying the different components of the fluids together, and they complicate them by introducing an array of new phenomena. A full description of the behavior of a plasma requires determining the distribution function of each component under the influence of Maxwell s equations. MHD simplifies this problem enormously by assuming that the time variations are slow. As a result, the displacement current (1/c) E/ can be neglected and Ampere s Law becomes B = 4π c J. (60) In conventional electrodynamics, this equation is interpreted as determining the field B in terms of the source J, but in MHD this is often reversed: the current is determined by the configuration of the field. Faraday s Law, however, must be retained in its full form, since even slow (by relativistic standards) changes in the magnetic field can generate significant electric fields: E = 1 c B. (61) In MHD, it is also assumed that the flow time scale τ = L/v is long compared to the time scale for the gyromotion of the ions (and therefore of the electrons), so that the effects of the gyromotion average out: ω ci τ 1 where ω ci = eb/m i c. (62) Ohm s Law and Flux-Freezing An important constitutive relation in MHD is Ohm s Law, which relates the current to the electric field. Let CS be the rest frame of the electrons and CS the rest frame of the ions; the laboratory frame is unprimed. Ohm s Law states that the current is proportional to the electric field as measured in the frame of the electrons. Using the nonrelativistic transformation of the current (J = J = J) and of the electric field, we have J = σe = σ(e + v e c B) = σ(e + v e B), (63) c where σ is the electric conductivity. In almost all astrophysical plasmas, the conductivity is high enough that As a result, Faraday s Law becomes E = E + v e B 0. (64) c E = 1 c (v e B) = 1 c B B (65) = (v e B). (66)

10 10 When the plasma is sufficiently ionized, all the components of the plasma are locked together, so that v e v i v; this is the case of ideal MHD. In ideal MHD E + v B 0 (ideal MHD). (67) c As an exercise in vector calculus, the reader can use this equation in Faraday s Law and combine it with the equation of continuity to show that ( ) ( ) d B B = dt ρ ρ v ideal MHD. (68) This shows that if changes in velocity are confined to the plane normal to B, then B/ρ is constant: the field is frozen to the plasma. We now give a more general derivation of flux-freezing, allowing for non-ideal effects but assuming that the electric field in the frame of the electrons is negligible. Let S be an arbitrary surface that is comoving with the electrons. It changes at a rate ds dt = v e dl. (69) The rate of change of the magnetic flux Φ B ds through this surface is the rate of change of the flux through the original surface, ( B/) ds, plus the rate at which flux is swept up by the surface, dφ B dt = ds + B (v e dl). (70) Using Faraday s Law and Stokes Theorem, we find dφ = c E ds + dt B (v e dl) (71) = c E dl + B (v e dl) (72) = (v e B) dl + B (v e dl) (73) = 0, (74) where the final step follows from evaluating the triple scalar product. It follows that the magnetic flux through a surface comoving with the electrons is constant, provided the conductivity is high enough that the comoving electric field vanishes. Note that it is only necessary that E 0 on the perimeter of the surface, not the interior; even if the plasma inside the perimeter is not highly conducting or if complex phenomena such as magnetic reconnection occur there, the total magnetic flux through the comoving surface is constant. In the most general case of non-ideal MHD, the electric field in the frame of the electrons cannot be neglected. Taking the curl of Ohm s Law, we find ( ) 1 ( ve ) σ J = E + c B. (75)

11 11 Using Ampere s Law and Faraday s Law yields ( c ) B = 1 B 4πσ c + ( v e B). (76) c For constant conductivity, the LHS is proportional to ( B) = ( B) 2 B = 2 B, so that this equation becomes B + (B v e ) = η 2 B. (77) where the resistivity is η c 2 /4πσ. The term on the RHS describes the diffusion of magnetic flux. The characteristic value of the ratio of the advection term to this diffusion term is the magnetic Reynolds number, v e B/L ηb/l 2 Lv η R M. (78) (Note that R M is defined in terms of the total velocity v, not the electron velocity v e.) In astrophysical plasmas, R M is large both because the conductivity is high and the size scales are large. For R M 1, the plasma decouples from the field, whereas for R M 1, the plasma is well coupled to the field The Momentum Equation By virtue of Ampere s Law, the magnetic force per unit volume is f = 1 c J B = 1 4π ( B) B. (79) This force is exerted only on the charged component of the plasma; neutral atoms and molecules are dynamically affected by the magnetic field only indirectly, through collisions with charged particles. Applying the vector identity (50) to this case, we have so that B 2 = 2[(B )B ( B) B], (80) f = 1 8π B π (B )B. (81) The first term corresponds to a pressure B 2 /8π, whereas the second term is a tension that tends to straighten the field so that B is constant along field lines. Note that the magnetic force vanishes along the field, as it must from equation (79): B f = 1 8π (B )B π B (B )B (82) = 2 8π B (B )B + 1 4π B (B )B (83) = 0. (84)

12 12 Including the magnetic force, the equation of motion becomes ρ dv dt = p ρ φ 1 8π B π (B )B. (85) We can express this in conservation form by writing f as the divergence of a tensor. Adding a term proportional to B = 0 to the expression for f gives f = 1 [( B)B + (B )B 12 ] 4π B 2 (86) = 1 4π The equation of momentum conservation then becomes [ ) ρv + ρvv + (p + B2 I 1 ] 8π 4π BB [BB 12 IB2 ]. (87) = ρ φ. (88) Note that the time derivative of the field momentum, (E B)/c, does not appear in this equation because of our low-frequency assumption. Turbulence can generate a random component of the magnetic field. Using equation (87), one can show that the combined effects of magnetic pressure and tension for such a field lead to effective pressure of 1 3 (B2 /8π) i.e., one-third the energy density, as expected for a relativistic fluid. To see this, average the force over a volume large enough that the field is random, f = 1 4πV dv [BB 12 ] IB2 = 1 4πV ds [BB 12 IB2 ], (89) where we have used the divergence theorem in the second step. We assume that the field is random on the surface bounding V, so that ds x B x B y = 0, etc. As a result, we have ds (BB) = ds (Bxˆxˆx 2 + By 2 ŷŷ + B2 z ẑẑ) = 1 ds (B 2 I ), (90) 3 = 1 B 2 dv, (91) 3 where we used an analog of the divergence theorem in the last step. The average force is then f = 1 ( 1 4πV 3 1 ) B 2 dv = 1 B 2 dv, (92) 2 24πV implying that the pressure of due to a random field is B 2 /24π. We have taken some care with this derivation since naively averaging equation (81) under the assumption that (B )B z = 0 leads to result that is twice as large; the latter step is not valid because it does not incorporate the constraint that B = 0.

13 Energy Equation Recall that the energy density in the magnetic field is B 2 /8π, and that the energy flux is given by the Poynting vector, S P c E B. (93) 4π Including the magnetic terms in the energy equation (44), we obtain [ ( ) 1 ρ 2 v2 + e + φ + 1 ] [ ( 8π B2 + 1 ρv 2 v2 + e + p ) ρ + φ + c ] 4π (E B) = n 2 HL + ρ φ. (94) For ideal MHD, we have E = (v/c) B, so that the Poynting flux becomes S P = c [( v ) 4π c B B 4π The energy equation for ideal MHD is then ( ) [ 1 2 ρv2 + ρe + ρφ + B2 + v 8π ], (95) = 1 B (v B), (96) 4π = 1 [ vb 2 B(v B) ]. (97) ( 1 2 ρv2 + ρe + p + ρφ + B2 4π ) 1 ] (v B)B = n 2 4π HL + ρ φ. (98) Note that the term that enters the divergence is B 2 /4π, not B 2 /8π: it represents the sum of the magnetic energy density and the magnetic pressure Alfven Waves As we have seen above, oscillations in a gas are sound waves, which are pressure-driven longitudinal waves. Magnetic tension allows a qualitatively different kind of wave to propagate in a magnetized medium, an Alfven wave, which is transverse. For simplicity, we assume that the unperturbed magnetic field, B 0, is uniform and that the gas pressure is negligible. Alfven waves have an amplitude, δb, that is perpendicular to B 0 and to k. As a result, the magnetic pressure, (B 0 +δb) 2 /8π B0 2 /8π, is constant to first order and the perturbed equation of motion for a plane wave [δv exp(ik r ωt)] is ρ 0 v = 1 4π (B 0 )δb, (99) iωρ 0 δv = 1 4π (ik B 0)δB. (100) The perturbed Faraday s Law becomes δb = (δv B 0 ), (101) iωδb = ik (δv B 0 ), (102) = i(k B 0 )δv (103)

14 14 Inserting this into the equation of motion, we find that ω 2 k 2 = B2 0 µ2 4πρ 0 = v 2 Aµ 2, (104) where µ ˆk ˆB0 and v A B 0 (4πρ 0 ) 1/2 (105) is the Alfven velocity. One also finds that where the sign depends on the direction of propagation. δv = ± δb, (106) (4πρ 0 ) 1/2 It is interesting to note that Alfven waves exert an isotropic pressure (Dewar 1970; McKee & Zweibel 1995). The stress tensor for the field is P B = B2 8π I BB 4π. (107) For a fluctuating field, only the quadratic terms proportional to δb 2 survive. Since gas pressure does not contribute to Alfven waves, the total stress tensor associated with the fluctuations the wave pressure is P w = δb2 8π I δbδb + ρδvδv, (108) 4π = δb2 8π I (109) by virtue of equation (106). Hence, a gradient in the intensity of Alfven waves can produce a force parallel to the mean magnetic field. This does not violate the fact that magnetic fields cannot exert a force parallel to the field since the force along B 0 is perpendicular to δb GENERAL REFERENCES Batchelor, An Introduction to Fluid Mechanics, Chaps 1-3 Feynman, Vol II, Chaps Landau and Lifshitz, Fluid Mechanics, Chaps 1-2 Shu, Vol 2, Chaps 1-4, 21