Fundamentals of Plasma Physics

12.7 Assignments 381

sheet. Note that( Bout∼∆ψ/ 2 πrLandBin∼∆ψ/ 2 πrδ;also note that∇·

r−^2 ∇ψ

)

can be expressed in terms ofBinandδ.)

(j) By eliminatingUinandUoutbetween Eqs.(12.99), (12.100), and (12.101) show
that
δ
L

=

Uin

va,in

=

1

√

Sin

where the inflow region Lundquist numberSinis defined as

Sin=

μ 0 va,inL

η

(k) If the initial separation between the two current loops is also of the order of
L,how much time is required for the two loops to merge, thus completing the
reconnection process? Structures in the solar corona have scale lengthsL∼
1-100 megameters (10^8 m) and it is observed that Alfvén wave disturbances
propagate with velocitiesvA∼ 106 m/s. The temperature of these structures is
10-100 eV. Assuming a nominal temperature of 50 eV, use Spitzer resistivityto
calculate the nominal value ofSin.How long would it take for two solar current
loops to merge if they are governed by the mechanism discussed here (what is
the appropriate time unit to use seconds, days, weeks, years, ...?). Actual current
loops change topology on time scales ranging from minutes to hours – how does
this compare with the predictions of this model?