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## Effect of on For a field and flow , (4.2) becomes (4.10)

Now consider the magnetic flux between two lines at and , which are moving with the plasma. Thus, and such that The flux is  Notice that the flux is a function of time only. The spatial dependence has been integrated out. Flux changes in time due to two effects, namely because changes in time and because the end points move at speeds and reducing or increasing the range of integration. On differentiating with respect to time we must remember to differentiate not only the integrand but also the limits. Thus, Hence, we may write By (4.10) this is zero. In other words, the amount of magnetic flux remains constant, and we say that it is frozen to the plasma.

Example 4. .1Consider the effect of a flow on an initially uniform field, . Equation (4.10) becomes Thus, (4.11)

Look for a seperable solution to this linear equation of the form so that (4.11) may be expressed as Dividing by and collecting all the functions of time onto one side of the equation and all the functions of onto the other we get Consider the -dependence first of all. Integrating we obtain Similarly we may integrate the dependent terms, Hence, we obtain In theory, we need to obtain the values for the separation constant and add together all the possible solutions. However, this example is particularly simple. Satisfying the initial condition gives Therefore, Thus, the flow carries in the field lines and makes the magnetic field strength grow exponentially in time.   Next: MHD Equilibrium Structures Up: MHD Equations Previous: MHD Equations
Prof. Alan Hood
2000-11-06