Useful Identities and Theorems from Vector Calculus

Appendix A Useful Identities and Theorems from Vector Calculus

A.1 Vector Identities

A · (B × C) = C · (A × B) = B · (C × A) A × (B × C) = B(A · C) − C(A · B) (A × B) × C = B(A · C) − A(B · C) ∇×∇f = 0 ∇·(∇×A) = 0 ∇·( f A) = (∇ f ) · A + f (∇·A) ∇×( f A) = (∇ f ) × A + f (∇×A) ∇·(A × B) = B · (∇×A) − A · (∇×B) ∇(A · B) = (B ·∇)A + (A ·∇)B + B × (∇×A) + A × (∇×B) ∇·(AB) = (A ·∇)B + B(∇·A) ∇×(A × B) = (B ·∇)A − (A ·∇)B − B(∇·A) + A(∇·B) ∇×(∇×A) =∇(∇·A) −∇2 A

A.2 The Gradient Theorem

For two points a, b in a space where a scalar function f with spatial derivatives everywhere well-deﬁned up to ﬁrst order, b (∇ f ) · d = f (b) − f (a), a independently of the integration path between a and b.

P. Charbonneau, Solar and Stellar Dynamos, Saas-Fee Advanced Course 39, 215 DOI: 10.1007/978-3-642-32093-4, © Springer-Verlag Berlin Heidelberg 2013 216 Appendix A: Useful Identities and Theorems from Vector Calculus

A.3 The Divergence Theorem

For any vector field A with spatial derivatives of all its scalar components everywhere well-defined up to first order, (∇·A)dV = A · nˆ dS , V S where the surface S encloses the volume V .

A.4 Stokes’ Theorem

For any vector field A with spatial derivatives of all its scalar components everywhere well-defined up to first order, (∇×A) · nˆ dS = A · d , S γ where the contour γ delimits the surface S, and the orientation of the unit nor- mal vector nˆ and direction of contour integration are mutually linked by the right-hand rule.

A.5 Green’s Identities

For any two scalar functions φ and ψ deﬁned over a volume V bounded by a surface S and with spatial derivatives well-deﬁned up to second order, (φ∇2ψ +∇φ ·∇ψ)dV = φ(∇ψ) · nˆ dS , V S and (φ∇2ψ − ψ∇2φ)dV = (φ∇ψ − ψ∇φ) · nˆ dS . V S

These are known respectively as Green’s ﬁrst and second identities, the latter often simply referred to as Green’s theorem. Appendix B Coordinate Systems and the Fluid Equations

This Appendix is adapted in part from Appendix B of the book by Jean-Louis Tassoul (1978) given in the bibliography to this appendix with a number of additions, including the MHD induction equation, expressions for the operators u ·∇ and ∇2 acting on a vector ﬁeld, and for the divergence of a second rank tensor. Note also, in Sects.B.1.5 and B.2.5, the quantities in square brackets correspond to the components of the deformation tensor D jk = (1/2)(∂ j uk + ∂ku j ).

B.1 Cylindrical Coordinates (s, φ, z)

Fig. B.1 Geometric deﬁn- ition of cylindrical coordinates. The coordinate ranges are s ∈[0, ∞], φ ∈[0, 2π], z ∈[−∞, ∞]. The cylindrical radius s is measured perpen- dicularly from the cartesian z-axis. The zero point of the azimuthal angle φ is on the cartesian x-axis. The local unit vector triad is oriented such that eˆz = eˆs × eˆφ.

P. Charbonneau, Solar and Stellar Dynamos, Saas-Fee Advanced Course 39, 217 DOI: 10.1007/978-3-642-32093-4, © Springer-Verlag Berlin Heidelberg 2013 218 Appendix B: Coordinate Systems and the Fluid Equations

B.1.1 Conversion to Cartesian Coordinates x = s cos φ, y = s sin φ, s = x2 + y2, φ = atan(y/x), z = z.

eˆx = cos φ eˆs − sin φ eˆφ, eˆy = sin φ eˆs + cos φ eˆφ,

eˆs = cos φ eˆx + sin φ eˆy, eˆφ =−sin φ eˆx + cos φ eˆy, eˆz = eˆz .

B.1.2 Inﬁnitesimals

d = dseˆs + sdφ eˆφ + dz eˆz

dV = s ds dφ dz

B.1.3 Vector Operators

D ∂ ∂ uφ ∂ ∂ = + u + + u Dt ∂t s ∂s s ∂φ z ∂z ∂ f 1 ∂ f ∂ f ∇ f = eˆ + eˆφ + eˆ ∂ s ∂φ ∂ z s s z uφ Aφ uφ As (u ·∇)A = u ·∇A − eˆ + u ·∇Aφ + eˆφ + (u ·∇A ) eˆ s s s s z z 1 ∂ 1 ∂ Aφ ∂ A ∇·A = (sA ) + + z ∂ s ∂φ ∂ s s s z 1 ∂ A ∂ Aφ ∇×A = z − eˆ ∂φ ∂ s s z ∂ As ∂ Az 1 ∂(sAφ) ∂ As + − eˆφ + − eˆ ∂z ∂s s ∂s ∂φ z 1 ∂ ∂ 1 ∂2 ∂2 ∇2 = s + + ∂ ∂ 2 ∂φ2 ∂ 2 s s s s z A 2 ∂ Aφ ∇2 A = ∇2 A − s − eˆ s 2 2 ∂φ s s s ∂ 2 Aφ 2 As 2 + ∇ Aφ − + eˆφ + ∇ A eˆ s2 s2 ∂φ z z Appendix B: Coordinate Systems and the Fluid Equations 219

B.1.4 The Divergence of a Second-Order Tensor

1 ∂(sT ) 1 ∂Tφ ∂T Tφφ [∇ · T] = ss + s + zs − s s ∂s s ∂φ ∂z s

1 ∂(sTsφ) 1 ∂Tφφ ∂Tzφ Tφs [∇ · T]φ = + + + s ∂s s ∂φ ∂z s 1 ∂(sT ) 1 ∂Tφ ∂T [∇ · T] = sz + z + zz z s ∂s s ∂φ ∂z

B.1.5 Components of the Viscous Stress Tensor ∂us 2 τss = 2μ + ζ − μ ∇·u ∂s 3 1 ∂uφ us 2 τφφ = 2μ + + ζ − μ ∇·u s ∂φ s 3 ∂uz 2 τzz = 2μ + ζ − μ ∇·u ∂z 3 1 1 ∂us ∂ uφ τ φ = τφ = 2μ + s s s 2 s ∂φ ∂s s 1 ∂uφ 1 ∂uz τφ = τ φ = 2μ + z z 2 ∂z s ∂φ 1 ∂uz ∂us τzs = τsz = 2μ + 2 ∂s ∂z

B.1.6 Equations of Motion u2 ∂Φ ∂ ∂ ∂ Dus − φ =− − p + Bz Bs − Bz Dt s ∂s ∂s μ0 ∂z ∂s Bφ ∂(sBφ) ∂ Bs 1 ∂ 1 ∂τsφ ∂τsz τφφ − − + (sτss) + + − μ0s ∂s ∂φ s ∂s s ∂φ ∂z s Duφ uφu ∂Φ ∂ ∂(sBφ) ∂ + s =− − 1 p + Bs − Bs Dt s s ∂φ s ∂φ μ0s ∂s ∂φ Bz 1 ∂ Bz ∂ Bφ 1 ∂ 1 ∂τφφ ∂τφz τsφ − − + (sτφs) + + + μ0 s ∂φ ∂z s ∂s s ∂φ ∂z s 220 Appendix B: Coordinate Systems and the Fluid Equations ∂Φ ∂ Bφ ∂ ∂ Bφ Duz =− − p + 1 Bz − Dt ∂z ∂z μ0 s ∂φ ∂z Bs ∂ Bs ∂ Bz 1 ∂ 1 ∂τzφ ∂τzz − − + (sτzs) + + μ0 ∂z ∂s s ∂s s ∂φ ∂z

B.1.7 The Energy Equation

Ds 1 ∂ ∂T 1 ∂ ∂T ∂ ∂T T = Φν + Φη + χs + χ + χ Dt s ∂s ∂s s2 ∂φ ∂φ ∂z ∂z

2 2 2 2 2 2 2 2 Φν = 2μ(D + Dφφ + D + 2D φ + 2Dφ + 2D ) + (ζ − μ)(∇·u) ss zz s z zs 3 2 2 2 η 1 ∂ Bz ∂ Bφ ∂ Bs ∂ Bz 1 ∂(sBφ) ∂ Bs Φη = − + − + − 2 μ0 s ∂φ ∂z ∂z ∂s s ∂s ∂φ

B.1.8 The MHD Induction Equation

∂ Bs 1 ∂ ∂ = u Bφ − uφ B − (u B − u B ) ∂t s ∂φ s s ∂z z s s z ∂η ∂(sBφ) ∂ ∂η ∂ ∂ − 1 − Bs + Bs − Bz s2 ∂φ ∂s ∂φ ∂z ∂z ∂s B 2 ∂ Bφ + η ∇2 B − s − s s2 s2 ∂φ

∂ Bφ ∂ ∂ = uφ B − u Bφ − u Bφ − uφ B ∂t ∂z z z ∂s s s ∂η ∂ ∂ Bφ ∂η ∂(sBφ) ∂ − 1 Bz − + 1 − Bs ∂z s ∂φ ∂z s ∂s ∂s ∂φ ∂ 2 Bφ 2 Bs + η ∇ Bφ − + s2 s2 ∂φ ∂ Bz 1 ∂ 1 ∂ = (su B − su B ) − uφ B − u Bφ ∂t s ∂s z s s z s ∂φ z z ∂η ∂ B ∂ B 1 ∂η 1 ∂ B ∂ Bφ − s − z + z − + η ∇2 B ∂s ∂z ∂s s ∂φ s ∂φ ∂z z Appendix B: Coordinate Systems and the Fluid Equations 221

B.2 Spherical Coordinates (r, θ, φ)

Fig. B.2 Geometric deﬁni- tion of polar spherical coordinates. The coordinate ranges are r ∈[0, ∞], θ ∈[0, π], φ ∈[0, 2π]. The zero point of the azimuthal angle φ is on the cartesian x-axis and the zero point of the polar angle θ (sometimes called colatitude) is on the cartesian z-axis. Note that in so-called geographical coordinates, longitude ≡ φ, but latitude ≡ π/2 − θ.The local unit vector triad is oriented such that eˆr = eˆθ × eˆφ.

B.2.1 Conversion to Cartesian Coordinates x = r sin θ cos φ, y = r sin θ sin φ, z = r cos θ. r = x2 + y2 + z2, θ = atan x2 + y2/z , φ = atan(y/x).

eˆx = sin θ cos φ eˆr + cos θ cos φ eˆθ − sin φ eˆφ, eˆy = sin θ sin φ eˆr + cos θ sin φ eˆθ + cos φ eˆφ, eˆz = cos θ eˆr − sin θ eˆθ. eˆr = sin θ cos φ eˆx + sin θ sin φ eˆy + cos θ eˆz, eˆθ = cos θ cos φ eˆx + cos θ sin φ eˆy − sin θ eˆz, eˆφ =−sin φ eˆx + cos φ eˆy.

B.2.2 Inﬁnitesimals

d = dreˆr + rdθeˆθ + r sin θdφeˆφ dV = r 2 sin θ dr dθ dφ 222 Appendix B: Coordinate Systems and the Fluid Equations

B.2.3 Vector Operators

D ∂ ∂ uθ ∂ uφ ∂ = + ur + + Dt ∂t ∂r r ∂θ r sin θ ∂φ ∂ f 1 ∂ f 1 ∂ f ∇ f = eˆ + eˆθ + eˆφ ∂ r ∂θ θ ∂φ r r r sin uθ Aθ uφ Aφ (u ·∇)A = u ·∇Ar − − eˆr r r uφ Aφ uθ Ar + u ·∇Aθ − cot θ + eˆθ r r uφ Ar uφ Aθ + u ·∇Aφ + + cot θ eˆφ r r ∂ ∂( θ) ∂ 1 2 1 Aθ sin 1 Aφ ∇·A = (r Ar ) + + r 2 ∂r r sin θ ∂θ r sin θ ∂φ 1 ∂(Aφ sin θ) ∂ Aθ ∇×A = − eˆ θ ∂θ ∂φ r r sin 1 ∂ Ar ∂(Aφr sin θ) 1 ∂(rAθ) ∂ Ar + − eˆθ + − eˆφ r sin θ ∂φ ∂r r ∂r ∂θ 1 ∂ ∂ 1 ∂ ∂ 1 ∂2 ∇2 = r 2 + sin θ + r 2 ∂r ∂r r 2 sin θ ∂θ ∂θ r 2 sin2 θ ∂φ2 2A 2 ∂ Aθ sin θ 2 ∂ Aφ ∇2 A = ∇2 A − r − − eˆ r 2 2 θ ∂θ 2 θ ∂φ r r r sin r sin ∂ θ ∂ 2 2 Ar Aθ 2 cos Aφ + ∇ Aθ + − − eˆθ 2 ∂θ 2 2 θ 2 2 θ ∂φ r r sin r sin ∂ θ ∂ 2 2 Ar 2 cos Aθ Aφ + ∇ Aφ + + − eˆφ r 2 sin θ ∂φ r 2 sin2 θ ∂φ r 2 sin2 θ

B.2.4 The Divergence of a Second-Order Tensor

2 1 ∂(r Trr) 1 ∂(Tθr sin θ) 1 ∂Tφr Tθθ + Tφφ [∇ · T]r = + + − r 2 ∂r r sin θ ∂θ r sin θ ∂φ r 2 1 ∂(r Trθ) 1 ∂(Tθθ sin θ) 1 ∂Tφθ Tθr Tφφ cot θ [∇ · T]θ = + + + − r 2 ∂r r sin θ ∂θ r sin θ ∂φ r r 2 1 ∂(r Trφ) 1 ∂(Tθφ sin θ) 1 ∂Tφφ Tφr Tφθ cot θ [∇ · T]φ = + + + + r 2 ∂r r sin θ ∂θ r sin θ ∂φ r r Appendix B: Coordinate Systems and the Fluid Equations 223

B.2.5 Components of the Viscous Stress Tensor ∂u 2 τ = μ r + ζ − μ ∇·u rr 2 ∂ r 3 1 ∂uθ ur 2 τθθ = μ + + ζ − μ ∇·u 2 ∂θ r r 3 1 ∂uφ ur uθ cot θ 2 τφφ = μ + + + ζ − μ ∇·u 2 θ ∂φ r sin r r 3 1 1 ∂ur ∂ uθ τ θ = τθ = μ + r r r 2 ∂θ ∂ 2 r r r ∂ θ θ ∂ uφ τ = τ = μ 1 1 u + sin θφ φθ 2 θ ∂φ ∂θ θ 2 r sin r sin 1 ∂ uφ 1 ∂ur τφ = τ φ = 2μ r + r r 2 ∂r r r sin θ ∂φ

B.2.6 Equations of Motion

u2 + u2 ∂Φ ∂ Dur − θ φ =− − p Dt r ∂r ∂r Bφ ∂ Br ∂ Bθ ∂(rBθ) ∂ Br + − Bφr θ − − μ θ ∂φ ∂ sin μ ∂ ∂θ 0r sin r 0r r θ ∂ ∂ ∂τ τ + τ 1 sin 2 rφ θθ φφ + (r τrr) + (τ θ sin θ) + − r sin θ r ∂r ∂θ r ∂φ r

u2 cot θ ∂Φ ∂ Duθ + ur uθ − φ =− − 1 p Dt r r r ∂θ r ∂θ ∂( θ) ∂ Bφ ∂(Bφ sin θ) ∂ θ + Br rB − Br − − B μ ∂ ∂θ μ θ ∂θ ∂φ 0r r 0r sin θ ∂ ∂ ∂τ τ τ θ 1 sin 2 θφ rθ φφ cot + (r τθ ) + (τθθ sin θ) + + − r sin θ r ∂r r ∂θ ∂φ r r Duφ u uφ uθuφ cot θ ∂Φ ∂ + r + =− − 1 p θ ∂φ θ ∂φ Dt r r r sin rsin θ ∂(Bφ sin θ) ∂ θ ∂ ∂(Bφr sin θ) + B − B − Br Br − μ θ ∂θ ∂φ μ θ ∂φ ∂ 0r sin 0r sin r θ ∂ ∂ ∂τ τ τ θ 1 sin 2 φφ rφ θφ cot + (r τφ ) + (τφθ sin θ) + + + r sin θ r ∂r r ∂θ ∂φ r r 224 Appendix B: Coordinate Systems and the Fluid Equations

B.2.7 The Energy Equation

∂ ∂ Ds 1 2 T T = Φν + Φη + χr 2 ∂ ∂ Dt r r r 1 ∂ ∂T 1 ∂ ∂T + χ sin θ + χ r 2 sin θ ∂θ ∂θ r 2 sin2 θ ∂φ ∂φ 2 2 2 2 2 2 2 2 Φν = 2μ(D + Dθθ + Dφφ + 2D θ + 2Dθφ + 2Dφ ) + ζ − μ (∇·u) rr r r 3 2 η ∂(Bφ sin θ) ∂ Bθ Φη = − 2 2 ∂θ ∂φ μ0r sin θ 2 2 ∂ B ∂(Bφr sin θ) ∂(rBθ) ∂ B + r − + sin2 θ − r ∂φ ∂r ∂r ∂θ

B.2.8 The MHD Induction Equation

∂ Br 1 ∂ ∂ = ( θ (u Bθ − uθ B )) − uφ B − u Bφ ∂ θ ∂θ sin r r ∂φ r r t r sin ∂η ∂( θ) ∂ ∂η ∂ ∂(Bφr sin θ) − 1 rB − Br + 1 Br − 2 ∂θ ∂ ∂θ 2 2 θ ∂φ ∂φ ∂ r r r sin r ∂( θ) ∂ 2 2Br 2 Bθ sin 2 Bφ + η ∇ Br − − − r 2 r 2 sin θ ∂θ r 2 sin θ ∂φ

∂ Bθ 1 ∂ 1 ∂ = uθ Bφ − uφ Bθ − (ru Bθ − ruθ B ) ∂ θ ∂φ ∂ r r t r sin r r ∂η ∂(Bφ sin θ) ∂ θ ∂η ∂( θ) ∂ − 1 − B + 1 rB − Br 2 2 θ ∂φ ∂θ ∂φ ∂ ∂ ∂θ r sin r r r ∂ θ ∂ 2 2 Br Bθ 2 cos Bφ + η ∇ Bθ + − − r 2 ∂θ r 2 sin2 θ r 2 sin2 θ ∂φ

∂ Bφ 1 ∂ ∂ = ruφ B − ru Bφ − uθ Bφ − uφ Bθ ∂ ∂ r r ∂θ t r r ∂η ∂ ∂(Bφr sin θ) ∂η ∂(Bφ sin θ) ∂ θ − 1 Br − + 1 − B θ ∂ ∂φ ∂ 2 θ ∂θ ∂θ ∂φ r sin r r r sin ∂ θ ∂ 2 2 Br 2 cos Bθ Bφ + η ∇ Bφ + + − r 2 sin θ ∂φ r 2 sin2 θ ∂φ r 2 sin2 θ Appendix B: Coordinate Systems and the Fluid Equations 225

Bibliography

For more on all this stuff, see Morse, P. M., & Feshbach, H.: 1953, Methods of theoretical physics, McGraw-Hill, chap. 1 Batchelor, G. K.: 1967, An introduction to ﬂuid dynamics. Cambridge University Press, appendix 2 Arfken, G. B.: 1970, Mathematical methods for physicists, 2nd edn., Academic Press, chap. 2 Tassoul, J.-L.: 1978, Theory of rotating stars. Princeton University Press, appendix B or, for a concise introduction, Appendix A.2 of Goedbloed, H., & Poedts, S.: 2004, Principles of magnetohydrodynamics, Cambridge University Press Appendix C Physical and Astronomical Constants

C.1 Physical Constants

Physical quantity Symbol Value Units (SI) − Charge of electron e 1.602 × 10 19 C −31 Mass of electron me 9.109 × 10 kg −27 Mass of proton m p 1.673 × 10 kg −12 2 −1 −2 Permittivity of vacuum ε0 8.854 × 10 C N m −7 −2 Permeability of vacuum μ0 4π × 10 NA − Speed of light c 2.998 × 108 ms 1 − Planck constant h 6.626 × 10 34 Js − − Boltzmann constant k 1.381 × 10 23 JK 1 − − − − StefanÐBoltzmann constant σ 5.670 × 10 8 JK 4m 2s 1 − − − Gravitational constant G 6.673 × 10 11 m3kg 1s 2

C.2 Astronomical Constants

Astronomical quantity Symbol Value Units (SI) Earth mass M⊕ 5.972 × 1024 kg Earth radius R⊕ 6.378 × 106 m Astronomical unit AU 1.496 × 1011 m Solar mass M 1.989 × 1030 kg Solar radius R 6.960 × 108 m − Solar luminosity L 3.84 × 1026 Js 1 Parsec pc 3.086 × 1016 m Light-year ly 9.461 × 1015 m

P. Charbonneau, Solar and Stellar Dynamos, Saas-Fee Advanced Course 39, 227 DOI: 10.1007/978-3-642-32093-4, © Springer-Verlag Berlin Heidelberg 2013 Appendix D Maxwell’s Equations and Physical Units

Electromagnetism is, unfortunately, a subﬁeld of physics where the choice of units does not only inﬂuence the numerical values assigned to measurements, but also the mathematical form of the fundamental laws, i.e., Maxwell’s equations.

D.1 Maxwell’s Equations

The whole mess in converting SI units to the astrophysically ubiquitous CGS units all harks back to the deﬁnition for the unit of charge, as embodied in Coulomb’s law. Under the SI system we write the electrostatic force between two charges q1 and q2 located at positions x1 and x2 as 1 q q F = 1 2 rˆ [SI] , (D.1) 2 4πε0 r with electrical charge measured in coulomb, and with r ≡ x1 − x2 for notational brevity; whereas under the CGS system the constant 1/4πε0 is absorbed into the deﬁnition of the unit of charge: q q F = 1 2 rˆ [CGS] , (D.2) r2 with electrical charge now measured in “electrostatic units”, abbreviated “esu” and sometimes also called “statcoulomb”. It electrostatics it is relatively easy to switch from CGS to SI with −1 the simple substitution ε0 → 1/(4π). With electrical currents now measured in esu s in the 2 −1 CGS system, and remembering that c = (ε0μ0) ,theμ0/4π prefactor in the BiotÐSavart law now becomes 1/c : 1 I d × rˆ B = [CGS] [BiotÐSavart] . (D.3) c r2

P. Charbonneau, Solar and Stellar Dynamos, Saas-Fee Advanced Course 39, 229 DOI: 10.1007/978-3-642-32093-4, © Springer-Verlag Berlin Heidelberg 2013 230 Appendix D: Maxwell’s Equations and Physical Units

If you then now go through the process of re-constructing Maxwell’s equations under these two new forms for the fundamental relations (electric and magnetic forces), you eventually get to

∇·E = 4π e [Gauss’ law] , (D.4) ∇·B = 0 [Anonymous] , (D.5) 1 ∂ B ∇×E =− [Faraday’s law] , (D.6) c ∂t 4π 1 ∂ E ∇×B = J + [Ampere’s /Maxwell’s law]. (D.7) c c ∂t

In some sense, the CGS system is perhaps more “natural”, as it omits the introduction of new, apparently fundamental physical constants ε0 and μ0, to simply stick with the speed of light c, the only price to pay being an extraneous factor 4π in Gauss’ law. The Lorentz force and Poynting vector become, in CGS units: 1 F = q E + u × B [Lorentz force] , (D.8) c c S = (E × B) [Poynting ﬂux] , (D.9) 4π and the electrostatic and magnetic energies: 1 E = 2 V , e π E d (D.10) 8 1 E = B2dV . (D.11) B 8π

D.2 Conversion of Units

Table D.1 gives you the conversion factor ( f ) required to go from SI to cgs units, i.e., the numerical value of a quantity in cgs Unit = f × numerical value in SI units. Any “3” appearing in a given value for f is a notational shortcut for 2.99792458. For a somewhat humourous close to this rather dry Appendix, here are ﬁve different ways, actually to be found in various textbooks or research monographs, to express teslas in terms of other fundamental SI units:

Vs N kg Wb kg 1T= 1 = 1 = 1 = 1 = 1 . (D.12) m2 Am As2 m2 Cs Appendix D: Maxwell’s Equations and Physical Units 231

Table D.1 Conversion between SI ands CGS units Quantity SI name SI symbol Conv. factor f CGS name CGS symbol Length meter m 102 centimeter cm Mass kilogram kg 103 gram g Force newton N 105 dyne dyne Energy joule J 107 erg erg Charge coulomb C 3 × 109 electrostatic units esu Current ampere A 3 × 109 statampere esu s−1 Potential volt V 1/300 statvolt statvolt Electric field — V m−1 (1/3)×10−4 —statvoltcm−1 Magnetic field tesla T 104 gauss G Magnetic flux weber Wb 108 maxwell Mx

Bibliography

The bulk of this appendix is taken from Appendix C in Grifﬁths, D. J.: 1999, Introduction to electrodynamics, 3rd edn., Prentice-Hall with some additions extracted primarily from: Huba, J. D.: 2011, NRL Plasma Formulary No. NRL/PU/6790-11-551, Naval Research Laboratory Index

K-quenching. See Differential rotation: Bifurcation diagram, 172 K-quenching BMR. See Solar magnetic field: bipolar X-effect, 104. See also Differential rotation magnetic region X-loop, 90, 121 Boussinesq approximation, 21 a-effect, 97–101, 103, 104 Butterfly diagram. See Sunspots: butterfly and current helicity, 101 diagram and kinetic helicity, 100, 141 catastrophic quenching, 119 cyclonic fluid motions, 98 C flux tube, 134, 175 Calcium index, 198, 202 hemispheric dependency, 100, 108, 114, 134 Carrington Richard, 156 MHD simulations, 101, 141 Chaotic trajectories, 78, 82 parametrization, 108, 114, 115, 119, 189 Charge relaxation time, 18, 28 quenching, 101, 112, 113, 119, 138, 168 Chinese imperial courts, 154 in tachocline, 133 Compton drag effect, 30 a-tensor, 97, 100, 101, 141 Conductivity electrical, 14 radiative, 12 A thermal, 12 Alfvén radius, 204 Constructive folding, 48, 80, 103, 144 Alfvén’s theorem, 25 Continuity equation, 6 Alfvén velocity, 22, 204 Coriolis force, 94, 108, 122 Alfvén wave, 22 Coriolis number, 100, 200 Aly’s theorem, 27 Cosmogenic radioisotopes, 161, 173 Ampére’s law, 13, 14 Coulomb gauge, 26 Anelastic approximation, 22, 136, 143 Cowling’s theorem, 66, 68, 106, 130 Anti-dynamo theorems, 64–68 CP (circularly polarized) flow, 75–77, Axisymmetric magnetic field, 38, 66, 141 79, 82 Axisymmetrization, 62–64, 127 Critical dynamo number, 33, 110, 172, 173 Current helicity, 26, 101, 141, 142

B Babcock Harold D., 89 D Babcock Horace W., 89, 121 Dalton minimum, 161, 165 Babcock–Leighton mechanism, 121–127, 188 Data assimilation, 181 Battery mechanism, 29 Deformation tensor, 9 Beltrami ﬂow, 68 Democritus, 2

P. Charbonneau, Solar and Stellar Dynamos, Saas-Fee Advanced Course 39, 233 DOI: 10.1007/978-3-642-32093-4, Ó Springer-Verlag Berlin Heidelberg 2013 234 Index

D (cont.) Dynamo problem, 33, 34 Destructive folding, 48, 55, 59, 75, 103, 144 Dynamo waves, 105, 112, 114, 116, 119, 192. Differential rotation See also Differential rotation: and K-quenching, 168–170 dynamo waves and dynamo waves, 105, 111, 112, 115 early-type stars, 192, 194 from numerical simulation, 140, 192 E inductive action, 48, 49, 51, 118, 132 Eigenmode magnetic backreaction, 138, 140, 168 aX, 109–111 solar, 51, 140 a2, 189, 190 solar parametrization, 50 a2X, 191, 192 tachocline, 51, 93, 101, 118, 132, 140 diffusive, 40 time-scale, 62, 64 Roberts cell, 70 Dipolar magnetic field, 41, 43, 92, 121, 122, Electric field, 13, 14, 27 139, 191 astrophysical, 27, 28 tilted, 62, 195 Electrical charge conservation, 18, 19 Dirac Paul A. M., 29 Electrical current density, 17 Dynamo Electromagnetic skin depth, 59, 111, 120, 192 aX, 107–121, 132, 163, 165–167, 179, Electromotive force, 24, 29, 31. See also Mean 180, 206 electromotive force a2, 106, 189, 190, 208 emf. See Electromotive force a2X, 107, 143, 191, 192 Entropy equation, 12 amplitude modulation, 115, 165, 169, 170, Equipartition field strength, 112, 118, 173, 176 192, 207 Babcock–Leighton, 128–132, 165, 167, Exponential stretching, 72 172, 178–180, 206 CP flow, 76–78, 81 cycle period, 110, 113, 116, 118, 120, 130, F 132, 200, 206 Fabricius Johann, 154 energetics, 24, 33, 45, 52 Faraday Michael, 30 fast, 74–83, 143, 144 Faraday’s law, 13 fast versus slow, 74, 95, 143, 144 Fast dynamo. See Dynamo: fast flux transport, 116, 128, 133, 165, Fast mode, 22 170, 179 Ferraro’s theorem, 50 growth rate, 33, 47, 71, 74, 76, 77, 109, 191 First order smoothing approximation, 99 homopolar, 32 Fluctuation interface, 119 a-effect, 165 interface aX, 119, 120 coherence time, 165 kinematic. See Kinematic regime in meridional flow, 170 MHD simulation, 82, 136–145, 192, Fluid 193, 207 as continuum, 2, 18 non-kinematic, 168 circulation, 11 Roberts cell, 70, 73 density, 2 saturation, 51, 75, 112, 118, 138 entropy, 12 stochastic forcing, 164, 167, 176 incompressible, 6, 9 stretch–twist–fold, 47, 52, 75 internal energy, 12 time delay, 128, 132, 171, 176 mass conservation, 6, 45 turbulent, 82, 136 Newtonian, 9 Dynamo equations, 103, 104, 106 pressure, 8 a2 versus aX versus a2X, 106 stress, 3 axisymmetric, 106 thermal dilation, 6 boundary conditions, 107, 114 vorticity, 11, 68 Dynamo number, 106, 165, 171. See also Flux expulsion, 58, 70, 82, 144 Critical dynamo number Flux freezing, 25, 45, 55, 56, 210 Index 235

Force-free magnetic field, 26, 27 M FOSA. See First order smoothing Magnetic boundary conditions, 60 approximation Magnetic boundary layers, 59, 70 Fossil magnetic field. See Stellar magnetic Magnetic diffusion time, 16, 28, 38, 41, 42, 61, fields: fossil 102, 108, 123, 194 local versus global, 57 Magnetic diffusivity, 15 G parametrization, 42, 189 Galileo Galilei, 154 Magnetic energy, 24, 27, 54, 113, 136, 207 Gauss’ law, 13 Magnetic flux ropes, 90, 93, 134, 162, Gleissberg cycle, 158, 169 175, 194 Gnevyshev–Ohl rule, 158, 165, 166 Magnetic helicity, 26, 74, 75 Goldsmid Johann a.k.a. Fabricius, 154 Magnetic monopoles, 29 Granulation, 144 Magnetic network, 89 Magnetic Prandtl number, 17, 169 Magnetic pressure, 22 H Magnetic Reynolds number, 15–17, 58, Hale George E., 88, 90, 91 78, 106 Hale’s polarity rules, 90, 121 Magnetic tension, 22 Harriot Thomas, 154 Magnetic vector potential, 25, 38, 66 Helicity. See Kinetic helicity, Current helicity, Magnetic waves, 22 Magnetic helicity Magnetogram, 89 Hydrodynamical shearing, 48–50, 55 Magnetohydrodynamics. See MHD Hydrodynamical stretching, 44, 47, 72 Magnetosonic wave, 22 Malkus–Proctor effect, 140, 168 Material surface, 7 I Maunder Annie, 157 Induction equation. See MHD: induction Maunder E. Walter, 157 equation Maunder minimum, 160, 173, 200 Intermittency, 138, 173–176, 200 Maxwell’s equations, 13 spatial, 76, 82 Mean electromotive force, 96–98, 103, 141, 142 temporal, 77 Mean-field dynamo models, 95 Iterative map, 171 Mean-field electrodynamics, 95 Medieval maximum, 161 Meridional circulation, 116–118, 128, J 171, 194 Joule heating, 20 MHD, 1–34 Joy’s law, 91, 94, 121 approximation, 13, 17, 19 electrical charge density, 19 governing equations, 20 K ideal limit, 16, 26, 58 Kelvin’s theorem, 11 induction equation, 14, 25, 38, 49, 96 Kepler Johannes, 154 instabilities, 132, 175 Kinematic regime, 33, 49, 107, 113, 167, 173, Lorentz force. See Lorentz force 189, 207 singular limit, 16, 58 Kinematic theorem, 25 Kinetic helicity, 12, 68, 100, 141, 142, 207 N Navier–Stokes equation, 8 L Newton Isaac, 7 Lagrangian derivative, 7 Numerical simulations. See a-effect: MHD Lorentz force, 18, 19, 22, 31, 45, 51, 112, simulations, Differential rotation: from 138, 168 numerical simulation, Dynamo: MHD Lyapunov exponent, 78 simulation 236 Index

O Solar magnetic field Ohm’s law, 14 bipolar magnetic region, 91, 121, 125 Ohmic dissipation, 15, 20, 26, 37, 58, 61 large-scale, 90 polarity reversal, 90, 92, 121, 122, 138 small-scale, 89, 90, 95, 144 P surface evolution, 92, 93, 121, 124, Parker Eugen N., 98, 119 125, 139 Parker–Yoshimura sign rule, 112, 114, 116, 192 Space climate, 176 PDF. See Probability density function Space weather, 153, 176 Planar flow, 48, 52, 65 Spin-down. See Stars: spin-down Poincaré section, 79 Spörer Gustav, 156 Poloidal/Toroidal separation, 38, 49, 66, Spörer minimum, 161 104, 106 SSN. See Sunspot: number Potential magnetic field, 27, 39 Stagnation point, 46, 68, 72 Poynting flux, 23, 60 Stars See also Stellar magnetic fields Primordial relic, 28 convection, 187, 188, 194, 197 Probability density function, 76, 78 early-type, 43 fully convective, 188, 207 rotation, 187, 200–202, 205 Q solar type, 198 Quadrupolar magnetic field, 41, 208 spin–down, 195, 202, 205, 206, 209 Quenching. See a-effect: quenching, winds, 202 Differential rotation: K-quenching Stellar magnetic fields bCep sub-class, 195 A and B stars, 195 R Ap/Bp stars, 43, 195, 196 Reynolds number, 10 Doppler imaging, 195, 196, 208 magnetic. See Magnetic Reynolds number early A star, 193 Roberts cell, 68, 73, 75, 82, 99 early-type, 189, 194, 195 Roberts cell dynamo. See Dynamo: flux conservation, 43, 197 Roberts cell fossil, 16, 43, 188, 197, 210 Rossby number, 200 late-type, 188, 198, 202, 207 M-type stars, 207 neutron stars, 43, 210 S stability, 197 Scaling analysis, 15 starspots, 198 Scheiner Christoph, 154 trapping in interior, 191, 193, 194 Schwabe Samuel H., 155 white dwarfs, 43, 209 Second-order correlation approximation, 99, STF. See Dynamo: stretch–twist–fold 100, 103, 141 Stream function, 53, 68 Seed magnetic field, 28, 30, 33, 136, 211 Stress Separation of scales, 2 fluid, 3 Shearing. See Hydrodynamical shearing tensorial representation, 7 Slow mode, 22 viscous, 8 SOCA. See Second-order correlation Stretching. See Hydrodynamical stretching approximation Sunspots Solar activity, 160 as magnetic flux rope, 90 proxy, 161, 198 butterfly diagram, 92, 115, 133, 135, 138, Solar cycle, 89, 92, 153, 156, 157 156, 170, 174, 175 chaos, 173, 181 cycle, 90, 155. See also Solar cycle forecasting, 176–182 cycle numbering convention, 155 magnetic flux budget, 93 decay, 93, 122, 124, 125 polar fields, 92, 118, 121, 122, 179 early telescopic observations, 154, 155 precursor, 177 leading, 90, 121, 124 Index 237

magnetic nature, 88 dynamical, 9 naked-eye observations, 153 kinematic, 9 number, 155 Viscous dissipation function, 13 pores, 144 Vorticity equation, 12 proxy, 163–164 Vorticity tensor, 9 trailing, 90, 121, 124 Synoptic diagram, 92 W Waldmeier rule, 158, 166, 167 T WD. See Weber–Davis Tachocline. See Differential rotation: Weber–Davis MHD wind, 202, 204 tachocline Wolf minimum, 161 Taylor–Proudman theorem, 140 Wolf Rudolf, 155 Theophrastus, 154 Wolf sunspot number, 155 Thermal convection, 6, 82, 136, 144, 192, 207 Worcester Chronicles, 154 Turbulent diffusivity, 102, 118, 119, 189 Turbulent pumping, 98, 101, 132, 138, 143 Turnover time, 16, 53, 58, 100, 102, 116 Z Zeeman broadening, 89 Zeeman splitting, 88, 195 V Zeldovich’s theorem, 65, 68 Viscosity bulk, 9