By James E. Pringle, Andrew King

ISBN-10: 0521869366

ISBN-13: 9780521869362

Just about all traditional topic within the Universe is fluid, and fluid dynamics performs an important position in astrophysics. This new graduate textbook presents a simple realizing of the fluid dynamical tactics proper to astrophysics. the maths used to explain those strategies is simplified to convey out the underlying physics. The authors conceal many themes, together with wave propagation, shocks, round flows, stellar oscillations, the instabilities brought on by results corresponding to magnetic fields, thermal using, gravity, shear flows, and the fundamental techniques of compressible fluid dynamics and magnetohydrodynamics. The authors are administrators of the united kingdom Astrophysical Fluids Facility (UKAFF) on the collage of Leicester, and editors of the Cambridge Astrophysics sequence. This e-book has been built from a direction in astrophysical fluid dynamics taught on the college of Cambridge. it's appropriate for graduate scholars in astrophysics, physics and utilized arithmetic, and calls for just a easy familiarity with fluid dynamics.

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**Example text**

An analogy with traffic flow is described in Witham (1974, Chap. 3) and in Billingham & King (2000, Chap. 7). 1 Consider waves in a uniform compressible medium with uniform magnetic field B and sound speed cs . 91) where (ux , uy , uz ) is a constant vector. 92) and find two similar equations for uy and uz . If ux = 0, show that the form of the motion is an incompressible (Alfvén) wave with phase velocity VA cos θ. 93) where = ω/(kVA ) is the dimensionless phase velocity and β = cs /VA is a dimensionless measure of the strength of the field.

As shown in Fig. 3, the characteristic curve C+ which passes through the point (x0 , t0 ) starts at the point A1 with position x1 < x0 at time t = 0. Thus the Riemann invariant J+ at A is determined by the initial t = 0 values of u and cs at point A1 . That is, J+ (x0 , t0 ) = J+ (x1 , 0). 60) Similarly, the characteristic curve C− which passes through the point (x0 , t0 ) starts at the point A2 with position x2 > x0 at time t = 0. Thus the Riemann invariant J− at A is determined by the initial t = 0 values of u and cs at A2 .

We then substitute these into the relevant B = B0 + b(r, t), where |b| equations, using the equilibrium conditions that ∇ρ0 = 0, ∇p0 = 0, u0 = 0 and curl B0 = 0. We also assume that the perturbations are adiabatic so that, as before, we may write p = cs2 ρ , where cs2 = γ p0 /ρ0 is uniform and constant. 25) ∂b = ∇ ∧ (u ∧ B0 ). 26) becomes We note that eq. 26) implies that ∂ (div b) = 0. 27) We differentiate eq. 24) with respect to time, and use eqs. 26) to eliminate ∂ρ /∂t and ∂b/∂t to obtain a linear equation for the velocity perturbation u: ρ0 ∂ 2u = cs2 ∇{ρ0 div u} − B0 ∧ {∇ ∧ [∇ ∧ (u ∧ B0 )]}.