Complements on Mechanics and Electromagnetism

B.1 Equations of Motion and Continuity

We derive the general equations of motion and continuity. Although their form is very simple for spherical stars in hydrostatic equilibrium, the present general forms are useful when these simplifications do not apply.

B.1.1 Equations of Continuity and of Motion

Let us consider a volume element dxdydz with coordinates x, y, z. Let vx be the component of the velocity at a point x through the face dzdy and vx + -¡¿dx the velocity at point x + dx through the face dzdy. The difference between the matter which goes out and which enters the volume element through the face dzdy is d (pvx) dxdydz . (B.1)

The net loss of matter in the volume element for all three components of velocity is div (p v) dxdydz; it can also be written as a change of density p:

dt for any volume element. Thus, we get the equation of continuity dPP+ div(p v) = 0 or dPP + (v ■ V)p = -p divv. (B.3)

dt dt

The equation of motion expresses that the acceleration dv/dt of a mass element in a given volume results from the forces exerted on it:

where a is the acceleration exerted by the external forces F such as gravity or centrifugal force by unit of mass in the volume, P is the pressure and the dots represent the other possible forces. If all the external forces are negligible, we get the equation of Euler, which is non-linear in the velocity:

B.1.2 Remarks on Derivatives

The total variation of any quantity, for example, the total variation dp of the density p, in a moving volume element is composed of two parts:

1. The variation during the time dt at a given point (x, y, z) is (dp/dt)dt.

2. The difference of density at time t between two points separated by dr traveling at velocity v by the volume element during time dt is dxdy dP+ dzd = (dr • V) p. (B.6)

dx dy dz

The total variation dp is thus dp — -Jj- dt + (dr • V) p, which gives dp — d+(vV)p. (B.7)

This is the total variation of the density for a moving fluid element. The term (v • V) p is the advection, i.e., the projection of the density variation along the velocity; it represents the transport of matter. The derivative d is also written DD and called the total or hydrodynamical derivative.

B.1.3 Vectorial Operators in Spherical Coordinates

Spherical coordinates (r, ç) are most appropriate for describing stars, $ being the colatitude. The unit vectors in these directions are er, e$ and eç. The gradient of a scalar field & is d& 1 d& 1 d&

The divergence of a vector A of components (Ar, A—, AL) is

V • A = 1 dir (r'Ar) + d- (sin -A-) + _!_ . (B.9) r2 dr rsin — d — rsin— dç

The Laplace operator A& = V2& of a scalar function 0 is

The rotational of a vector B is 1

Vx B

B.1.4 Viscous Terms

The Euler equation does not account for the viscous forces, which we now consider. A small slab of area S floating on a viscous medium of density p and thickness d is opposing a force F to a displacement with velocity v in the plane defined by the slab. We define the tension t in the plane of the slab

where n is the dynamic coefficient of viscosity in g s-1 cm-1 and n = pv with v the kinematic coefficient of viscosity in cm2 s-1. If the x-axis is in the direction of motion and z is the vertical direction perpendicular to the slab, the tension along the x-axis by unit of slab surface is d Vx m nx

The tension is symmetric axz = azx for an isotropic medium. A fluid is said to be Newtonian if the tensions are linear and homogeneous functions of the velocity gradients; thus the linearity and symmetry imply the proportionality

the same for the other tensions. One adopts the general expression a = n (| + %) • (B'15)

The meaning of axx is that of a pressure on a surface perpendicular to the x-axis; it produces a change of velocity vx from the constraints exerted on the surface dydz. One has

B Complements on Mechanics and Electromagnetism dvx

One notices that X oii = 0, for i = 1-3; thus the viscosity would make a pressure, which is not the case. Thus, one has to take

The net viscous force on the volume element dV = dxdydz is the difference of the forces on the opposed faces of the volume element:

(JL Oxdx) dydz + (jy Oxydy^ dxdz + ^oXz dAj dydx = Y oxj dxdydz

d2 v7

3j j

2 d 3 ndx dxj dvx dvy dvz dx dy dz

Sometimes, instead of the term n/3, a term (n + Z)/3 is written. Z is said to be "the second viscosity"; it intervenes in the case of polyatomic gases submitted to fast oscillations. It accounts for the fact that the energy is transmitted more slowly to the vibration and rotation modes of molecules. This introduces additional viscous effects. The equation of motion (B.4) can now be written as

- + (v . V) v pa - VP + nV2 v + n V (div v) . (B.19)

B.1.5 Navier-Stokes Equation

If the medium is incompressible, the density p is constant and from the continuity (B.3) divv = 0. Thus, the equation of motion becomes dv . - + (v .V) v pa - VP + n V2 v,

p which is the equation of Navier-Stokes. If we divide it by p, we get

dt dt p which is fundamental in fluid mechanics. The coefficient v is the coefficient of kinematic viscosity, equal to v = n/g. According to (B.21) the characteristic timescale of a viscous effect is of the order of

Some viscosity coefficients are given below (Sect. B.4.1). In astrophysical media, the main sources of viscosity are turbulent viscosity in turbulent media and radiative viscosity (B.52) in hot low-density media. If the external and viscous forces are negligible, we obtain the Euler equation (B.5).

If the medium is compressible, as is the case for stellar media in general, the above equation still applies for motions with a small velocity v with respect to the sound velocity cS. For a medium with a compressibility App ~ ¡5 Ajr, ignoring the T effects one has for the sound speed (5.88)

The pressure behaves like AP ~ file ~ whime ~ g v2. Thus, the relative density variations behave like Agg ~ 4 and according to the equation of continuity, we g cS

neglect the term in div v in the equation of motion (B.19) if M2 = v2/cS C 1.

B.1.6 Equation of Motion with Rotation

The accelerations measured in a rotating frame (quantities with a prime) are related to those in an inertial frame (without a prime) by dv' dv dQ , „^ , , _

dt dt dt where the second and third terms on the right-hand side represent the centrifugal force (the third term is absent in a stationary situation). In a stationary rotating star, the centrifugal force at colatitude $ can also be written as (1/2) Q2V(r sin$)2. The last term is the Coriolis force.

The term dv/dt or dv/dt in the Navier-Stokes equation (B.20) or in the Euler equation (B.5) is the acceleration in the inertial system. To get the equations in the rotating frame, the change (dv/dt) ^ (dv'/dt) has to be made. The Navier-Stokes equation (B.20) becomes in the rotating frame d v' dv' , . ^ ,

= a - 1 VP + vV2 v' - Q x (Q x r') - — x r' - 2Q x v'. (B.25) q dt

The ratio of the inertial to Coriolis force is Rossby number (Sect. B.5.4). Viscosity acts locally.

B.1.7 Geostrophic Motions, Taylor-Proudman Theorem

If Q is constant or has a cylindrical symmetry, the centrifugal acceleration can be derived from a potential, say V. One has (cf. Sect. 2.1.2) -VV = Q2 ® and V = -(1/2)Q2 ®2, where W is the distance to the rotation axis. Equation (B.25) becomes for constant q (with Q also constant in time)

V + (v'- V) v' = -V P -- Q2W2 + 0 + v V2 v'- 2 Q x v'. (B.26) dt \q 2 J

The fluid is incompressible; 0 is the gravitational potential (1.36). For a stationary state without viscosity, the equation becomes

Fluid motions satisfying this equation are said to be geostrophic. This is typical of wind circulation on the Earth; the Coriolis force is perpendicular to the direction of the motion v'. The generalized pressure term in square brackets is perpendicular to the direction of the motion, i.e., the wind blows parallel along the isobars and turns clockwise around high pressures in the northern hemisphere. Along a streamline, followed by a fluid element of velocity v', the generalized pressure term is constant. On the Earth, this equation applies in altitude where viscous effects are negligible. If we take the rotational of the above equation, we have

because the second member is a gradient. This equation becomes v ■ V)Q - v'(V- Q) -(Q ■ V)v' + Q(V ■ v') = 0. (B.29)

If Q is constant as on Earth, V ■ v' = 0 due to continuity, q being constant. Thus one is left with

This is the Taylor-Proudman theorem, which says that the velocity cannot vary in the direction of the rotation axis. Thus, two particles on a line parallel to the rotation axis stay at the same distance from each other during rotation.

B.2 Maxwell Equations

Maxwell's equations are the general form of the equations of electromagnetism. Together with the equations of motion and continuity, they form the basic equations of magnetohydrodynamics. In the MKSA system, the Maxwell equations are

dt dD

For a low-density plasma B = ¡i0 i H, D = e0 eE. H is the magnetic field, B the magnetic induction, often referred to as the magnetic field, i the magnetic permeability, E the electric field, D the electric displacement, e0 the electrical permittivity of free space, qc the charge density and j the electric current density. In the Gauss system, used in astrophysics, Maxwell's equations are

c dt

c c d t with B = ¡iH, D = eE and the same definitions as above.

The first law expresses that a variable magnetic field produces an electrical field (or a voltage); this is the Faraday law. The second law expresses that the electric charges are the source of the electric displacement (Gauss law). The third says that the motion of charged particles, i.e., a current, creates a magnetic field (Ampere law). The fourth indicates that there is no free magnetic particles. The current I through a surface S is the total charge q crossing this surface by unit of time, I = dq/dt. The current density j = (dI/dS)(v/v) where dS is the surface element perpendicular to the motion of the charges and j is a vector in the direction of the motion of the positive charges of velocity v. The Ohm law relates E and j:

where a is the electrical conductivity, the inverse of the resistivity.

B.3 Statistical Mechanics: Pressure and Energy Density

Statistical mechanics provides a very useful relation between the energy density u and the pressure P exerted by a medium of particles of mass m and velocity v. The force F = dp/dt on a surface results from the change of the particle momenta during collisions supposed to be elastic. The change of the momentum of a particle impacting a surface with an angle ft with respect to the normal is Ap = 2p cos ft. Let us call N (ft, p)dftdp the number of particles by units of time and surface with momentum and impact angle in the intervals (p, p + dp) and (ft, ft + dft). The pressure is (Fig. B.1)

We call n(ft,p) dftdp the concentration of particles with ft and p in the quoted limits and have the relation

since particles arriving tangentially to the plane do not contribute. Thus the pressure is rn/2 n ~

J0 J0

For isotropic particle motions, let us call n(p) dp the concentration of particles with a momentum between p and p + dp; we have n(ft, p) dftdp 2n sin ftdft

This expression is introduced in (B.38). The integrations over ft and p are independent, which gives

3 Jo

The density of kinetic energy of particles of mass m and momentum p is u = f n(p)Ean(p) dp . (B.41)

P and u may now be related for non-relativistic and relativistic particles.

Fig. B.1 Elastic collision of particles with angles between $ and $ + d$ with respect to the normal n to a surface S

B.3.1 Non-relativistic Particles

The kinetic energy and momentum are Ecin = p2/(2m) and p = m v. The pressure becomes with (B.40)

1 /"" p2 2 /"" P =~ n(p)1— dp =- n(p) Ecin dp . (B.42)

Thus, one has

Such a relation applies, for example, to a perfect gas, where P = [k/(p mu)] q T and u = (3/2)[k/(pmu)] q T.

B.3.2 Relativistic Particles

In relativistic kinematics, the kinetic energy is

Ecin = E - Eo = mc2 - moc2 = moc2 ^- 1J > (B.44)

with ¡5 = v/c and m0 the rest mass. For small ¡5, this gives Ecin = 1 m0v2. The momentum is p = mv = mcfi. By eliminating ¡5 between p and Ecin from (B.44), one has for the total energy E

For photons or highly relativistic particles Ecin ^ pc and P (B.40) becomes

which gives

In general, P/u lies between 1/3 and 2/3, the lower limit being reached for relativistic particles and the upper limit for non-relativistic particles. As an example, the factor 1/3 applies to radiation pressure with Prad = (1/3) aT4 and radiative energy density urad = aT4 (Sect. C.1).

B.4 Expressions of Viscosity, Conductivity and Diffusion

The equation of diffusion is derived in Sect. 10.2 for a general geometry. Here, we examine the expressions of viscosity, conductivity and diffusion coefficients. This is usually done in the context of the kinetic theory of gases, with a simple plane parallel geometry (the geometry influences the form of the equations, but not the various coefficients which depend on local conditions). Transfer of momentum gives rise to viscosity, transfer of heat to conductivity and transfer of particles to diffusion. Ideally, one should study the collisions of particles and then integrate over all parameters as we did above for pressure (Sect. B.3); however in the kinetic theory of gases, one considers particles with an average velocity v and a mean free path t.

B.4.1 Viscosity from Turbulence, Radiation and Plasma

If a force F is applied tangentially on a slab at the surface of a horizontal layer of viscous liquid of vertical thickness z, a steady horizontal motion of the slab results with a velocity v in the direction of the applied force (Sect. B.1.4). The steady motion shows that due to viscosity the medium exerts an opposite equal horizontal force -F. The dynamic viscosity n is defined by dv

where F is the modulus of the viscous force by surface unity. If n is the concentration of the particles of mean mass m, the number of particles crossing a surface unity by unity of time, due to random isotropic motions, is (n/6) v (see Sect. 10.1). The excess of momentum transported upward over a distance £ through a horizontal surface in the liquid is (n/6) v m(dv/dz) £. This amount is to be counted twice, since a positive excess is transported by viscosity upward and a negative one downward; thus the total excess of momentum by units of time and surface is (n/3) v£m(dv/dz). Since F = dp/dt, one has from the identification with (B.48)

which provides the coefficient of dynamic viscosity. The numerical factor depends on the particle interactions. The kinematic viscosity v is n 1

where v is expressed in cm2 s-1 and n in g cm-1 s-1. These expressions apply to different physical cases, for example, it gives the viscosity coefficient of turbulent motions with an average velocity v and mean free path £. Turbulent viscosity is generally much larger than the radiative and microscopic viscosities. The timescale for viscous adjustment is given by (B.22).

B.4.1.1 Radiative Viscosity

The photons also transport momentum in stars; thus a radiative viscosity can also be defined. If one applies the above expression (B.49) to photons, one has v ^ c, i ^ 1/(kq), with q = u/c2 = aT4/c2; thus

A more refined development leads to [232] in g cm-1 s_1:

which makes a small difference with respect to (B.51). In the Sun at Mr/M = 0.5, the radiative viscosity v = 0.2 cm2 s-1. At Mr/M = 0.98 in the Sun, v = 1.9 cm2 s-1. The associated timescale is ~ 1014 yr.

B.4.1.2 Plasma or Molecular Viscosity and Resistivity

The dynamic viscosity n = VQ of an ionized plasma with particles of atomic mass Ai and charge Zie is [543] (see also 10.67)

15 T5/2A5/2 -, -, n = 2.2 x 10-15 4 i gcm s . (B.53)


Although it applies to a plasma, this viscosity is often called the molecular viscosity or the microscopic viscosity. A is the ratio of the Debye length (7.99) to the impact parameter in the plasma of electron concentration ne:

For a mixture of H and He with a mass fraction X of H, the viscosity is [515]

n « 2.2 x 10"15---— gcm"^"1 . (B.55)

lnA ~ 4 in the solar interior. The molecular viscosity at Mr/M = 0.5 in the Sun is 2.7 and 8.6 cm2 s"1 at Mr/M = 0.98. The magnetic diffusivity for a hydrogen plasma is n « 5.2 x 1011 lA cm2 s"1 . (B.56)

These coefficients apply to the microscopic effects. For specific hydrodynamic and magnetic instabilities, the coefficients are different (e.g., Sect. 13.5.1).

B.4.2 Conductivity

Let us consider a medium with a gradient of temperature T along the vertical z-axis, T decreasing upward while z increases. The energy q transferred by units of horizontal surface and time is

where K' is a coefficient of conductivity expressed in erg cm"1 s"1 K"1. By the same reasoning as mentioned previously, the quantity of energy transported by the (n/6) v particles which move upward and by the same number which move downward is by units of surface and time,

3 az where CV is the specific heat at constant volume per particle, accounting for the fact that nCV = q CV, where CV is, as usual, the specific heat by mass unity. Identifying the above two expressions, one gets

for the coefficient of conductivity K' (different from K in cm2 s-1 in 3.45).

B.4.3 Diffusion Coefficient

The diffusion coefficient expresses the capacity of a medium to transport particles. Expressions of the diffusion coefficient have been derived in Sect. 10.1.3 and the equation of particle diffusion due to an abundance gradient is derived in Sect. 10.2.1. The general form of the diffusion coefficient for particles of average velocity v and mean free path e is

From (B.50), one sees that the diffusion coefficient D and the coefficient v of kinematic viscosity have similar expressions, although they have a different physical meaning.

B.5 Dimensionless Numbers

Several dimensionless numbers characterize the various regimes in hydrodynamics. The values of these numbers are in general very different in the astrophysical and terrestrial conditions.

B.5.1 Reynolds Number

In the Navier-Stokes equation (B.21), one may consider the ratio of the inertial term ( v / t ) to the viscosity term to be inertial v/t i2 vi

viscous vv/i2 Vt V

where t, v and t are typical lengthscale, velocity and timescale. The above ratio is the Reynolds number vt

For Re numbers lower than a critical value Recrit, viscosity effects dominate and turbulent motions are damped. For the flow in a tube, Recrit « 2300. The value Recrit depends on the geometry and type of fluids. In stars, the scale t is so large that turbulence generally easily sets in. The value of Recrit determines the timescale for which viscous dissipation takes more time than the growth of the turbulent instability. For Re larger than Recrit, the spectrum of turbulence also extends to scales smaller than the critical one. For local conditions in stars a value of Recrit « 10 is sometimes taken [562]. A magnetic Reynolds number can also be defined; see (13.6).

B.5.2 Prandtl Number

For a medium with heat conduction and motions, the ratio of the timescales for thermal adjustment ttherm ~ t2/K (3.47) and for viscous adjustment tvisc ~ t2/v (B.22) is the Prandtl number

In stars at T = 107 K, it is ~ 10~6, i.e., thermal adjustment is much faster than viscous adjustment. Pr is a property of the medium and not of the flow, contrary to the Reynolds number. For water or air, Pr « 1. In stars, for radiative viscosity and transport one has from (B.52) and (3.46) the dimensionless ratio

K 15 CKQ2 4acT3 5 c2

B.5.3 Peclet and Nusselt Numbers

The Peclet number is the ratio of the thermal to the dynamical timescales, generally taken as ttherm t2 v v t Pe = t-therm = - - = (B.65)

tmotion K t K

If one accounts for the spherical geometry of the blob for calculating the radiative losses, one has a factor of 6 at the denominator (Sects. 5.4.2, B.5.3):

Amotion 6K 6K

This is the way we defined r (see 5.66). Thus, r = Pe/6. Different geometries give different factors; the example of a radiative slab is given in Sect. 13.4.4. For large values of Pe, the moving eddies do not have the time to lose energy on their ways and the motions are adiabatic.

Nusselt number: Nu is the ratio of the heat transferred in a moving fluid to that transferred by conduction. In stars, this is typically the ratio of the total (convec-tive+radiative) to the radiative flux.

B.5.4 The Rossby Number

The Rossby number is the ratio of the absolute values of the inertial acceleration in the equation of Navier-Stokes to the Coriolis acceleration aCor = -2 Q x v (cf. B.24) for a medium rotating with an angular velocity Q:

where ç is the angle between the rotation axis and the direction of the fluid motion. At large scales, the Coriolis force tends to be relatively important.

B.6 More on the Physics of Rotation

B.6.1 The Angular Velocity in Spherical Functions

In a rotating star, the angular velocity Q(r,ft) is developed over the star in spherical harmonics in terms of the Legendre polynomials. However, the angular velocity is also developed in order to simplify the expression of the transport of the angular momentum (Sect. 10.5.4). In this case, as shown by Mathis (see also [386]), the proper functions of Q for the angular momentum transport are not exactly spherical harmonics and they require a special attention. The development of Q(r, ft) in spherical harmonics is

where Qs(r) is the radial component of each harmonics. The functions Ps are the Legendre polynomials,

Pi(x) = 1 (5x3 -3x), P4(x) = 1 (35x4 -30x2 + 3) (B.69) 28

The Legendre polynomials obey, among other properties, to the following relations:

Pm(x)Pn(x)dx = 0, for m = n; (Pm(x))2dx = --- . (B.70)

On the other side, when we write Q(r, ft) and integrate it in expression (10.112) to get the angular momentum, we consider expression (10.104)

where Q is the mean angular velocity in the expression of the angular momentum and Q expresses the corresponding differential rotation,

0 0

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