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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