H.K. Moffatt- Intensification of the Earth’s Magnetic Field by Turbulence in the Ionosphere

8/3/2019 H.K. Moffatt- Intensification of the Earth’s Magnetic Field by Turbulence in the Ionosphere

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J OUXNAL OF ~ ~ E O P H Y S I C A L ESEARCH VOLUME7,N o. 8 JULY 1962

Intensification of the Earth’s Magnetic Field byTurbulence in the Ionosphere1

H. K . MOFFATT

Department of Applied Ma them atics and Theoretical PhysicsUniversity of Cambridge, England

Abstract. If the magnetic diffusivity of a fluid in turbu lent m otion is greater than itskinematic viscosity, then, according to Batchelor, any magnetic field initially present mustultimately decay to zero. However, if a magnetic field is externally maintained, the turbulencewill generate fluctuations th at m ay be quite large if th e magnetic Reynolds num ber is largecompared with unity. Th e spectrum of these fluctuations increases up t o a wave num ber k,

marking the threshold of conduction effects, and falls off rapid ly beyon d k,. The net effect of

the turbulence can be expressed in termsof

an eddy conductivity equal to the molecular con-ductivity multiplied by the (-5/2)th power of the magnetic Reyn olds num ber.

A situation which may well arise in iono-

spheric turbulence, but which has received lit-

t le at tention so far, is that described by the

inequalities

1<< B , << R

R and R, are the turbulent Reynolds number

and magnetic Reynolds num ber, defined in te rmsof the root-mean-square velocity U’, he length-

scale of the energy-containing eddies L, and the

two diffusive constants characterizing the me-

dium, v the kinematic viscosity and X the mag-

netic diffusivity:

(1)

R = u‘L / v , R , = u‘L/h (2)

X is inversely prop ortion al to the electrical con-

ductivity U and the magnetic permeability p

of the medium:

x = (4*pfT)-’ (31

According to the work of Batchelor [lOSO],

the condition R , << R ( tha t i s , X >> v )

ensures that weak random magnetic fields will

ultimately decay to zero. The arguments used

by Batchelor were not universally accepted, but

I shall assume their validity here without elab-

1Based on a paper presented a t the InternationalSymposium on Fundamental Problems in Turbu-lence and Their R elation t o Geophysics sponsoredby the Internation al Union of Geodesy and Geo-physics and the International Union of Theoreticaland Applied Mechanics, held September 4-9, 1961,in Marseilles, France.

oration, and explore some of the consequences.

Accepting, then, th at rand om magnetic fields de-

cay to zero, let us suppose that a large-scale

weak magnetic field is applied to the system.

(This is the case of interest in ionospheric

studies, the applied field being then the earth’s

magnetic field.) It is as though a pulse of mag-

netic field were injected into the system in eachtime interval St. Each pulse is distorted by the

turbulence, being thoroughly randomized at t h e

same time, and is then destroyed at the wave

numbers at which conduction becomes impor-

tant . Th e applied f ield must be weak if its dy-

namic effect is to be neglected.

The wave number marking the onset of vis-

cous effects, k,, is known to depend only on

v and C, th e ra te of dissipation of energy pe r

unit mass, through the equation

k , = (€ /v3)”4 (4)

Since X >> v, the wave number k. marking

th e onset of conduction effects m ust b e smaller

than k,, and, provided that it is at the same

time large compared with L-’ ( the wave num-

ber characteristic of the energy-containing ed-

dies), it can only depend on z an d X through

the relationship

k , = (E/x3)1’4 (5 )

L-’ << k , << k, (6)

The relations

are equivalent to ( l ) , f we accept the well-

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3072 H. . MOFFATT

known semiempirical formula for C,

€ = U f 3 / L (7)

The picture that we contemplate is, then, the

following. A ste ady source of m agnetic energy

is maintained a t wave numb ers of t he o rder ofL-* or less. Th e associated field Ho(r) s distortedby the turbulence, each successive pulse of the

field being driven by the turbulence into thewave-number range k > k,, in which it isdestroyed b y th e overriding effect of conduction.

We seek to determine the spectral properties ofthe statistically steady magnetic field H(r, t )

(the sum of all the pulses) th a t is thu s supported.Now, in stating that the turbulence is statis-

tically steady, we implicitly assume that thereexists a source of vort icity a t low wave num bers(i.e., that some large-scale destabilizing force

generates vorticity). This vorticity is con-tinuously transferred by the nonlinear inter-action to the viscous sink at wave numbersk > k,. I n so far as viscous effects an d conductive

effects can be neglected, both vortex lines andmagnetic field lines are convected in the fluidmotion, the flux of either quantity through any

material circuit being conserved. It is therefore

fair to argue, in th e manner adopted by Batche-lor, that the steady statistical properties of the

two fields (vorticity and magnetic) that are notinfluenced by the diffusive mechanisms must beidentical. In particular, the spectra of vorticity

and magnetic field must have the same wave-number dependence in the range defined by

L-' << k << (E/X">"* (8)

Moreover, at wave numbers large compared

with L-1, if we accept Kolmogorov's picture, thevelocity and vorticity spectrum tensors areisotropic, and so therefore is the magnetic field

spectrum tensor. We can therefore define a

scalar r ( k ) , the magnetic field spectrum, whichincreases as k+l'*, like the vorticity spectrum,in the range (8). We anticipate that r ( k ) will

decrease rapidly in the range k > ke because ofthe severe conductive damping. For k < L-1,

the spectrum tensor will not in general be

isotropic, but in this range r ( k ) may be under-stood to represent the trace of the spectrumtensor averaged over the sphere of radius k inwave-number space. Th en r ( k ) dk is proportional

to the contribution to the total magnetic energy(1/8a)Hz rom the range ( k , k + dk):

(9)

Clearly th e maximum contribution comes from

the region around k = k, . Moreover, the rapid

falloff th at we antici pate for12> k , implies that ,in any realization of the flow in physical space,

th e magne tic lines of force will be approximatelystraight with small fluctuations throughout any

volume V , of fluid of dimension smaller than

The spectrum r ( k ) for k >> k , may be

determined by a perturbation method based

upon this premise. I need not give the details

her e; they closely resemble the work of Golitsyn[1960], who determined the form of the spec-

trum when it is wholly controlled by the con-

ductive damping. The result is

where C is a dimensionless constant of orderunity.

Thus T ( k ) rises as k"I8 up to k , and falls ask-u19beyond k,. Since it must be continuous inorder of magnitude near k,, and since we must

satisfy equation 9 so t h a t C may be determined,

T ( k ) may finally be written in the form

P ( k ) =L-' << k << k,

k , << k << k ,

(11)

4/9Xjj"-13kl/3

4, g ~ - 2 ~ 5 ~ ? / 3 k - 1 1 / 3IIt may fairly be supposed that r(k)does not

behave too erratically for k < L-1 and tha t it

falls off very rapidly for k > k,.

The net effect of the turbulence on the field

may be computed roughly as follows. In the

absence of tu rbulence, only th e large-scale fieldHo(r)would be present with mean square valueapproximately

L - 1--No2 = S, r(lc)d k + L - ~ F ( L - ~ ) (12)

If we suppose that the spectral law l l (a) is

valid, at least in order of magnitude, right upto k = L-I, then, using ( 2 ) and ( 7 ) , (12)

becomes__

H 2 = 9 / 4 R , z z (13)

indicating the extent to which turbulence at

large magnetic Reynolds number increases the

mean square field intensity (provided alwaystha t R, < R ) .



SYMPOSIUM O N T U RBU L E N CE I N GEOPHYSICS 3073

The extent to which the dissipation of en-

ergy by Ohmic heating is increased may also

be readily calculated. Thus, if

Do= X k 21 ' ( k ) dk '+ XL-31'(L-') (14)

is the dissipation in the absence of turbulence,and

L - 1

I) = X lm21'(k) d k .i2 Xk?I'(k,) (15)

is the increased dissipation under turbulent

conditions, then

D = Rm5I2Do (16)

This result may be interpreted in terms of aneddy conductivity U,, defined so that the to ta l

dissipation, by analogy with (14), is given by

D = ( ~ T ~ u J - ' L - ~ I ' ( L - ' ) (17)

The n evidently U. can be expressed in the equiv-

alent forms

indicating first the strong dependence of U@ on

R , and second the anomalous effect th at in-crease of a,keeping other parameters constant,

induces a decrease in U#, imply because the

conduction cutoff k , is raised, so that more

thorough mixing of the magn etic field is possible.

It must be stressed that the above theory

neglects the dynamic influence of the field on

the fluid motion, a neglect that is justified onlyif th e applie d field is sufficiently weak. T he sal-ient features of the theory are, then, that the

mean squ are value of the field is increased b y

a factor of order R, and that its typical scale

of variation is of order (/i'/~)''he condition

R , >> 1 may only be satisfied in rare circum-

stances in the ionosphere where the turbulence

is intense and the conductivity high. At the

high altitudes at which this might be the case,

the molecular mean free path is so large thatthe continuum approximations on which the

theory is based must be viewed with caution.But the conclusion that intense fluctuations

m ay at such high altitudes be superimposed

on the earth's magnetic field is unlikely t o be

invalidated.

REFERENCES

Batchelor, G. K., Proc. R o y . Soc. London , A , 201,405, 1950.

Golitsyn, G. S., Soviet Phys. , Doklady , 6, 536,1960;D o k l a d y A k a d . N a u k S S S R , 152, 2, 1960.

H.K. Moffatt- Intensification of the Earth’s Magnetic Field by Turbulence in the Ionosphere

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