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 addi- tions, 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 coordi- nates. 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 coordi- nates. 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 ori- ented 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.