Physics: An Introduction (Second Edition): Solutions to Exercises

the first term on the left-handside of (56) is O(Q 1), the second term is O(Q2), and the third term is O(P Q 2). Thus, the neglect of the latter two terms is valid provided

When p ≫ Q 1/2 , Equation (54) reduces to

Suppose that we neglect the third term on the left-hand side. We obtain

The solution to this equation that is well behaved as p is U(0,

2 Q 1/4 p), where U(a, x) is a parabolic cylinder function. The small argument asymptotic expansion of this function yields

Matching to the solution (58) reveals that when p ≪ Q 1/2 Finally, comparison with Eqs. (9.47) and (9.49) in the book yields

Thecharacteristiclengthscaleof the solution (62) is

/4 Thus, theneglectofthe final term on the left-hand side of Equation (60) is valid provided that Q

, or

Combining the inequalities (59) and (65), the criterion for the validity of Equation (64) is obtain

Suppose that we neglect the second term on the left-hand side of Equation (60).

The solution to this equation that is well behaved as p → ∞ is where Kν(x) is a modified Bessel function. The small argument asymptotic expansion of this function yields

Matching to the solution (58) reveals that " when t ≪ Q 1/2 . Finally, comparison with Eqs. (9.47) and (9.48) in the book yields

The characteristic lengthscaleof the solution (69) is p ∼ P 1/6 . Thus, the neglect ofthe second term on the left-hand side of Equation (60) is valid provided that Q ≪ P P 1/3 , or

Combining the inequalities (59) and (72), the criterion for the validity of Equation (71) is

9.4 As before, the inflow velocity is of Ohm’s law yields

Continuity of plasma flow inside the sheet gives assuming incompressibleflow. In the presenceof strong plasma viscosity, the magnetic pressure gradient within the reconnecting layer is is balanced against the viscous force (µ ∇2V), instead of inertia, so

Finally, specified in Eq. (9.25) in the book. Equations (74) and (75) yield

Equation (77) gives specified in Eq. (9.24) in the book. The continuity equation (76) implies that