8/3/2019 H.K. Moffatt and P.G. Saffman- Comment on "Growth of a Weak Magnetic Field in a Turbulent Conducting Fluid wit

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Reprinted f r o mTHEPHYSICSF FLUIDSVolume 7, Number 1,January 1964

Comment on "Growth of a Weak Magnet icField in a Turbu lent Conducting Fluid withLarge M agnet ic P rand t l Number"H. K. MOFFATT

Trinity College, Cambridge, EnglandA N DP. G. S . 4 F F M A N *

Je t Propulsion Laboratory, Pasa dena, Cali fornia(Received 3 September 1963)HE effect of a uniform straining motion U = (alz i ,Tff2xz,f323)where a1 + a2 + a3 = 0, and

a1 > a2 > a3,on what may be called a "blob" ofweak magnetic field, that is a magnetic field whichis exponentially small outside a certain boundedregion, has recently been examined by Pa0.l Themagnetic field H,(x, t ) is expressed [Eq. (3.7)] as aweighted integral over all space of i ts initial value,which is prescribed statistically in terms of i ts two-point correlation. Pao shows that, i f a2 = 0, thevalue of the mean-square magnetic field at theorigin ultimately tends to a constant. He then infersthe effect of turbulence on a weak random magneticfield, and in particular concludes that , if the mag-netic Prandtl number v / X > 100, then H 3 initiallyincreases and then tends to a constant value. Thisinference and conclusion seem unjustified for thefollowing reasons.

Firstly, the magnetic field distribution was notsolenoidal. With a solenoidal initial field (i.e., onefor which there is no net flux of H across anycoordinate plane), it can be shown from Pao's Eq.(3.7) that ultimately Hl(x, t) varies with time likee - " . t , t-', e"", according as a? is greater, equal, orless than zero. The 2- and 3-components decreaseeven more rapidly. Moreover, confining the analysist o the case a2 = 0, where the decay with time isalgebraic rather than exponential, is too restrictivein view of the fact that ala2a3s negative in allknown forms of turbulence. But in any event, themean-square magnetic field at any point of a singleblob eventually decays to zero although there willbe, in general, initial amplification if X is sufficientlysmall.

Secondly, and more importantly, the analysis ofa single blob cannot in itself be conclusive, sincethe overlapping of neighboring blobs cannot beneglected. It is true that a random magnetic fieldcan be thought of as initially decomposed into anarray of nonoverlapping solenoidal blobs, each blobcontained within a region of uniform strainingmotion :

H(x,0) = H(')(x, ).I

However, the volume occupied by the blob H'"ultimately increases exponentially [as e( ' l+" ') ifa2 > 0, as t * e " l t if a2 = 0, and as e a l t i f a, < 01.Thus, more and more blobs must overlap as timepasses and, after a long time, at any point a largenumber (proportional to e (a l+o l ' ) r if az > 0, andt t e a l t if a2 = 0, and em ' $ f a2 < 0) of blobs whichwere initially distinct must overlap. If the magneticfields in the blobs are uncorrelated, the mean-squaremagnetic field at any point behaves like

if a2 > 0,, ( a i + a * ) t 2 a ~ t = e ( a ~ - a l ) 6

if a2 < 0.ea ' t e2""1 - ( n 2 - a r ) lThus in this case for any strain field (except theaxisymmetric fields having a1 = az or a2 = as,HZincreases exponentially. This result was originallyproved conclusively by Pearson' who considered theaction of irrotational uniform strain on a weakrandom vorticity field.Pearson's mathematics may equally be appliedto a magnetic field and the result (1)more rigorouslyobtained. The exponential increase is associated withthe fact tha t the total magnetic energy of a singleaccording as a2 >, =, or