Lecture Notes In Astrophysical Fluid Dynamics

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Lecture Notes in Astrophysical Fluid DynamicsMattia SormaniNovember 19, 20171

Contents1 Hydrodynamics1.1 Introductory remarks . . . . . . . . . . . . . . . . .1.2 The state of a fluid . . . . . . . . . . . . . . . . . .1.3 The continuity equation . . . . . . . . . . . . . . .1.4 The Euler equation, or F ma . . . . . . . . . . .1.5 The choice of the equation of state . . . . . . . . .1.6 Manipulating the fluid equations . . . . . . . . . .1.6.1 Writing the equations in different coordinate1.6.2 Indecent indices . . . . . . . . . . . . . . . .1.6.3 Tables of unit vectors and their derivatives .1.7 Conservation of energy . . . . . . . . . . . . . . . .1.8 Conservation of momentum . . . . . . . . . . . . .1.9 Lagrangian vs Eulerian view . . . . . . . . . . . . .1.10 Vorticity . . . . . . . . . . . . . . . . . . . . . . . .1.10.1 The vorticity equation . . . . . . . . . . . .1.10.2 Kelvin circulation theorem . . . . . . . . . .1.11 Steady flow: the Bernoulli’s equation . . . . . . . .1.12 Rotating frames . . . . . . . . . . . . . . . . . . . .1.13 Viscosity and thermal conduction . . . . . . . . . .1.14 The Reynolds number . . . . . . . . . . . . . . . .1.15 Adding radiative heating and cooling . . . . . . . .1.16 Summary . . . . . . . . . . . . . . . . . . . . . . .1.17 Problems . . . . . . . . . . . . . . . . . . . . . . . .2 Magnetohydrodynamics2.1 Basic equations . . . . . . . . . . . . .2.2 Magnetic tension . . . . . . . . . . . .2.3 Magnetic flux freezing . . . . . . . . .2.4 Magnetic field amplification . . . . . .2.5 Conservation of momentum . . . . . .2.6 Conservation of energy . . . . . . . . .2.7 How is energy transferred from fluid todoes no work? . . . . . . . . . . . . . .2.8 Summary . . . . . . . . . . . . . . . .2.9 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .systems. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .fields if the magnetic. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .force. . . . . . 5474850. 51. 54. 553 Hydrostatic equilibrium563.1 Polytropic spheres . . . . . . . . . . . . . . . . . . . . . . . . . . . . 563.2 The isothermal sphere . . . . . . . . . . . . . . . . . . . . . . . . . 612

3.33.4Stability of polytropic and isothermal spheres . . . . . . . . . . . . 64Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 684 Spherical steady flows4.1 Parker wind . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .4.2 Bondi spherical accretion . . . . . . . . . . . . . . . . . . . . . . . .4.3 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .5 Waves5.1 Introduction . . . . . . . . . . . . . . . . . . .5.2 Sound waves . . . . . . . . . . . . . . . . . . .5.2.1 Propagation of arbitrary perturbations5.3 Water waves . . . . . . . . . . . . . . . . . . .5.4 Group velocity . . . . . . . . . . . . . . . . .5.5 Analogy between shallow water theory and gas5.6 MHD waves . . . . . . . . . . . . . . . . . . .3. . . . . . . . . . . . . . . . . . . . . . . . . .dynamics. . . . . .707076828383858894100103106

These are the lecture notes for the course in astrophysical fluid dynamics at Heidelberg university. The course is jointly taught by Simon Glover and myself.These notes are work in progress and I will update them as the course proceeds.It is very likely that they contain many mistakes, for which I assume full responsibility. If you find any mistake, please let me know by sending an email atmattia.sormani@uni-heidelberg.deReferencesIn these notes we make no attempt at originality and we draw from the followingsources.[1] Acheson, D. J., Elementary Fluid Dynamics, (Oxford University Press, 1990)Notes: a very clear text on fluid dynamics. Highly recommended.[2] Landau, L.D., Lisfshitz, E.M. Fluid Mechanics, (Elsevier, 1987)Notes: part of the celebrated Course of Theoretical Physics series. A fantasticbook.[3] Feynman, R.P., The Feynman Lectures on PhysicsNotes: look at Chapter 40 and 41 in Vol II. The lectures are available for free onthe web at the following link: http://www.feynmanlectures.caltech.edu.[4] Shore, S.N., Astrophysical Hydrodynamics, (Wiley, 2007).Notes: a very comprehensive text with lots of wonderful historical notes.[5] Balbus, S., Lecture Notes,available les/2012-09-20/afd1415 pdf 18150.pdfand les/2012-09-20/hydrosun pdf 66537.pdfNotes: very clear and concise.[6] Clarke, C.J., Carswell, R.F., Astrophysical Fluid Dynamics, (Cambridge University Press, 2007).Notes: a modern textbook on astrophysical fluid dynamics.[7] Binney, J., Tremaine, S., Galactic Dynamics, (Princeton Series in Astrophysics,2008)Notes: comprehensive book that mostly deals with topics that we do not touch4

in this course, such as dynamics of star systems (which are collisionless) anddisk dynamics.[8] Zeldovich, Ya. B. and Raizer, Yu. P., Physics of Shock Waves and HighTemperature Hydrodynamic Phenomena, (Academic Press, New York and London, 1967).Notes: A classic on shock waves. There is an economic paperback Dover edition,first published in 2002.[9] Kulsrud, R.M. Plasma Physics for Astrophysics, (Princeton, 2004).Notes: a good book on plasma physics and magnetohydrodynamics.[10] Pedlosky, J., Geophysical Fluid Dynamics, (Springer, 1987)Notes: an excellent book if you are interested in more “terrestrial” topics.[11] MIT Fluid Mechanics films. These are highly instructive movies available forfree at http://web.mit.edu/hml/ncfmf.html. Description from the website:“In 1961, Ascher Shapiro [.] released a series of 39 videos and accompanyingtexts which revolutionized the teaching of fluid mechanics. MIT’s iFluids program has made a number of the films from this series available on the web.”Highly recommended![12] Purcell, E.M. and Morin, D.J., Electricity and Magnetism, (Cambridge University Press, 2013)Notes: a nicely updated edition of Purcell’s classic. Lots of solved problems.[13] Jackson, J. D., Classical electrodynamics, (Wiley, 1999)Notes: everything you always wanted to know (and more) about classical electrodynamics.5

11.1HydrodynamicsIntroductory remarksFluid dynamics is one of the most central branches of astrophysics. Itis essential to understand star formation, galactic dynamics (what isthe origin of spiral structure?), accretion discs, supernovae explosions,cosmological flows, stellar structure (what is inside the Sun?), planetatmospheres, the interstellar medium, and the list could go on.Fluids such as water can usually be considered incompressible, because extremely high pressures of the order of thousands of atmospheresare required to achieve appreciable compressions. Air is highly compressible, but it can behave as an incompressible fluid if the flow speed is muchsmaller than the sound speed. Astrophysical fluids, on the other hand,must usually be treated as compressible fluids. This means that we mustaccount for the possibility of large density changes.Astrophysical fluids usually consists of gases that are ultimately madeof particles. Although they are not exactly continuous fluids, most of thetime they can be treated as if they are. This approximation is valid ifthe mean free path of a particle is small compared to the typical lengthover which macroscopic quantities such as the density vary. If this isthe case, one can consider fluid elements that are i) large enough thatare much bigger than the mean free path and contain a vast number ofatoms ii) small enough that have uniquely defined values for quantitiessuch as density, velocity, pressure, etc. Effects such as viscosity andthermal conduction are a consequence of finite mean free paths, andextra terms can be included in the equations to take them into accountin the continuous approximation.Most astrophysical fluids are magnetised. Although this can sometimes be neglected, there are many instances in which it is necessary totake explicitly into account the magnetised nature of astrophysical fluids.Therefore we shall study magnetohydrodynamics alongside hydrodynamics.1.2The state of a fluidIn the simplest case, the state of a fluid at a certain time is fully specifiedby its density ρ(x) and velocity field v(x). In some cases it is necessaryto know additional quantities, such as the pressure P (x), the temperature T (x), the specific entropy s(x), or in the case of magnetised fluidsthe magnetic field B(x). Multi-component fluids can have more than6

one species defined at each point (think for example of a plasma made ofpositive and negative charged particles which can move at different velocities relative to one another), each with its own density and velocity,but we will not consider this case in this course.Equations of motion allow us to evolve in time the quantities thatdefine the state of the fluid once we know them at some given time t0 .In other words, they allow us to uniquely determine ρ(x, t), v(x, t), etcat all times once their values ρ(x, t t0 ), v(x, t t0 ), etc are known forall points in space at a particular time t0 .1.3The continuity equationConsider an arbitrary closed volume V that is fixed in space and boundedby a surface S (see Fig. 1). The mass of fluid contained in this volumeisZρ(x, t)dV,(1)M (t) VVn̂dSdVFigure 1: An arbitrary volume V .and its rate of change with time is (can you show why is it legit to bringthe time derivative inside the integral?):ZdM (t) t ρ(x, t)dV.(2)dtVWe can equate this with the mass that is instantaneously flowing outthrough the surface S. The mass flowing out the surface area elementdS dS n̂, where n̂ is a vector normal to the surface pointing outwards,is ρv · dS, thus summing contributions over the whole surface we have:IdM (t)(3) ρv · dS.dtSThe divergence theorem states that for any vector-valued functionF(x) (can you prove this?):ZDivergence theoremVdV · F ISdS · F(x)(4)Applying the divergence theorem with F ρv to the RHS of Eq. (3)and then equating the result to the RHS of Eq. (2) we obtain:ZZ t ρ(x, t)dV dV · (ρv).(5)VV7

Since this equation must hold for any volume V , the arguments insidethe integrals must be equal at all points.1 Hence we find: t ρ · (ρv) 0(6)Continuity equationThis is called continuity equation and expresses the conservation ofmass: what is lost inside a volume is what has outflown through thesurface bounding that volume.1.4The Euler equation, or F maWhat is the fluid equivalent of Newton’s second law F ma? Considera small fluid element of volume dV and mass dM ρdV . By Newton’ssecond law, Dv sum of the forces acting on the fluid element, (7)dMDtwhere the quantity Dv/Dt is the acceleration of the fluid element. Notethat this is not the same as t v(x, t). The difference between the two is asfollows: i) Dv/Dt is calculated by comparing the velocities of the samefluid element at t and t dt, which occupies different spatial positionsat different times ii) t v(x, t) is calculated by comparing velocities at thesame position in space at different times.We can find the relation between the two types of derivatives, whichholds for any property f (x, t) of the fluid and not just for velocities.Consider a fluid element that is initially at x(t) (see Fig. 2). Its velocityis v(x, t) and therefore after a time dt its new position will be x(t dt) 'x(t) vdt. To take the derivative following the fluid element, we mustcompare f at the new position at time t dt with f at the old positionat time t:Dff (x(t dt), t dt) f (x(t), t) limdt 0Dtdtf (x(t) vdt, t dt) f (x(t), t)'dt t f dt ( f ) · (vdt)'dt t f v · f1(8)(9)(10)(11)Suppose there is a point in which they are not equal. Then one could just integratein the neighbourhood of that point, contradicting the result (5). Hence they must beequal.8x(t dt)tvdx(t)Figure 2: The convectivederivative.

ThusConvective derivativeFn̂Figure 3: A hypotheticalplane slicing through thefluid. n̂ is the normal tothe plane. In the generalcase, the force F that thefluid on one side exerts onthe fluid on the other sidecan have any direction. Inthis course, we only considercases in which n̂ and F havethe same direction, and Fdoes not depend on the orientation of the plane.D t v · Dt(12)D/Dt is called the Lagrangian or convective derivative, to distinguish it from the Eulerian derivative t (see also Section 1.9).Now that we have discussed the LHS of Eq. (7), let us deal withthe RHS. What are the forces that act on a fluid element? The mostfundamental force acting on a fluid is pressure.Consider a hypothetical plane slicing through a static (v 0) fluidwith an arbitrary orientation. What is the (vector) force that materialon one side of that plane exerts on material on the other side? In themost general case the force depends on the orientation of the plane andthe force itself can have any direction (different from the orientation ofthe plane). The thing relating the force to the orientation is in generala second-rank tensor, because it relates a vector to a vector, i.e. it is a3 3 array of numbers.Fortunately, in this course we only consider forces that in such a staticsituation can always be considered isotropic, i.e. they do not depend onthe orientation of n̂ (can you think of an example where this is not true?),and are directed perpendicularly to the surface at each point, i.e. theyare directed along n̂ (this is usually not a good approximation in solids.Think of a twisted rubber: it is clearly not true inside it!). The force canthen be quantified by a single number called pressure, which is a functionof position and time, P P (x, t). The pressure force acting on a surfacearea dS is P dS.A pressure exerts a net force on a fluid element only if it is not spatially uniform, otherwise the force on opposite sides cancels out. Thepressure force acting on a fluid element is P dV.(13)You can show this by considering the forces on the side of a small cubeof volume dV dxdydz.If the fluid is not static, we assume that the pressure force is thesame. If two layers of fluid are moving relative to each other, viscousforces can also be present. In contrast to pressure, these are not directedperpendicularly to our hypothetical plane, and will be the subject ofSection 1.13.Now let’s put eveything together. First, substitute Eq. (12) into theLHS of (7). If the only force acting on the fluid is pressure, the RHS is9

simply given by (13). After dividing by dM and using that dM ρdVwe find: P t v (v · )v (14)ρEuler equationThis is the Euler equation. This was derived assuming that pressureis the only force. Other forces can be added as needed. One of obviousimportance in astrophysics is gravity. In presence of a gravitational fieldΦ the force per unit mass acting on a fluid element is Φ, thereforethe Euler equation becomes in this case t v (v · )v P Φ.ρ(15)If the field is externally imposed then Φ is a given function of x andt; if on the contrary the field is generated by the fluid itself, it must becomputed self-consistently.Other forces of astrophysical interest can arise because of the effectsof magnetic fields and viscosity. These will be considered below.1.5The choice of the equation of stateConsider the continuity equation (6) and the Euler equation (14). Thesetwo equations alone are not enough to evolve the system in time. Inother words, if one is given ρ0 (x) ρ(x, t t0 ) and v0 (x) v(x, t t0 )at t t0 , they are not enough to determine uniquely what they are atat t t0 . There are many possible solutions ρ(x, t), v(x, t t0 ) thatsatisfy them, and one does not know which one to choose. People saythat the continuity and Euler equations do not form a complete systemof differential equations, so we need one more.From the mathematical point of view, this can be understood becausewe have three unknowns functions (ρ, v, P ) and only two equations (6and 14).2 From the physical point of view, this can be understood if weconsider that to find the time evolution of a fluid element we need toknow the forces acting on it, but so far we have said nothing on how todetermine P !It is common to relate pressure and density through an equationof state. For most astrophysical applications, it is usually a very good2We have five unknowns and four equations if v is considered as three scalarfunctions vx , vy , vz . The Euler equation is a vector equation and counts as threescalar equations.10Euler equation with gravity.

approximation to consider the equation of state of an ideal gasP Ideal gasρkT,µ(16)where T is the temperature, k 1.38 10 23 JK 1 is the Boltzmannconstant, µ is the mass per particle.The addition of Eq. (16) does not complete our system of equations,because we have introduced a new equation but also a new unknown, thetemperature T (x, t). We have just exchanged one unknown quantity (P )for another (T ). The following are common ways of closing the systemof equations of astrophysical importance:Isothermal gas Assume that T constant in Eq. 16. This is called an isothermalgas. In this case P is proportional to ρ. Note that the quantitykT /µ has dimensions of a velocity squared and Eq. (16) can berewritten asP c2s ρ(17)wherec2s kT /µ constant.(18)cs is called the sound speed, for reasons that will become clear laterin the course (see Section 5.2). In general, if we follow a fluid element, it changes shape and volumeduring its motion. It can therefore perform work by expansion orreceive work by compression from its surroundings. An adiabaticfluid is one in which this work is converted into internal energy ofthe fluid in a reversible manner and there are no transfers of heat ormatter between a fluid element and its surroundings. The temperature of each fluid element is allowed to change, but only as a resultof compression and expansion. We will see later how to considerprocesses able to add or subtract heat from a fluid element, such asexchanges of heat due to thermal conduction between neighbouringfluid elements, viscous dissipation, and extra heating and coolingdue to radiative processes.We can find the equations governing an adiabatic gas as follows.The internal energy per unit mass of an ideal gas isInternal energy per unitmass of an ideal gasU Pρ(γ 1)11(19)

where γ 1 2/N is the adiabatic index and N is the numberof degrees of freedom per particle (N 3 for a monoatomic gas,N 5 for a diatomic gas). Note that an isothermal gas correspondsto the limit N in which the number of degrees of freedomgoes to infinity. In this case, the internal degrees of freedom actlike a heat bath that keeps the temperature constant. This is alsowhy U as γ 1.Consider again a small fluid element of volume dV , mass dM ρdV and internal energy is dM U. As it moves through the fluid,this parcel of gas changes volume, doing work by expansion andexchanging heat with the surroundings, just like the ideal gas systems you studied in your first thermodynamics course. The firstlaw of thermodynamics says thatDU DQ DW(20)where DU is the change in internal energy, DQ is the heat addedto the system and DW is the work done by the system (all perunit mass). We use the big D to emphasise that these are changestracked following the fluid. Differentiating (19) yields DU (DP )/ [ρ(γ 1)] (Dρ)P/[ρ2 (γ 1)]. The expansion work done by the fluid elementdue to its volume change is DW P Dυ (Dρ)/ρ2 , whereυ 1/ρ is the specific volume, i.e. the volume per unit mass. I

Fluid dynamics is one of the most central branches of astrophysics. It is essential to understand star formation, galactic dynamics (what is the origin of spiral structure?), accretion discs, supernovae explosions, cosmological ows, stellar structure (what is inside the Sun?), planet atmospheres, the interstellar medium, and the list could go on.

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