Table Of ContentBurgulence and Alfv´en waves heating mechanism of solar corona
T.M. Mishonov∗ and Y.G. Maneva†
Department of Theoretical Physics, Faculty of Physics,
University of Sofia St. Kliment Ohridski,
5 J. Bourchier Boulevard, BG-1164 Sofia, Bulgaria
(Dated: February 5, 2008)
Heating of magnetized turbulent plasma is calculated in the framework of Burgers turbulence
[A.M. Polyakov, Phys. Rev. E. 52, 6183 (1995)]. Explicit formula for the energy flux of Alfv´en
waves along the magnetic field is presented. The Alfv´en waves are considered as intermediary
between the turbulent energy and the heat. The derived results are related to a wave channel
of heating of the solar corona. If we incorporate amplification of Alfv´en waves by shear flow the
suggested model of heating can be applied to analysis of the missing viscosity of accretion discs
and to reveal why the quasars are the most powerful sources of light in the universe. We suppose
7
thattheLangevin-Burgers approach toturbulencewehaveapplied in thecurrent work can bealso
0
helpfulforothersystemswherewehaveintensiveinteractionbetweenastochasticturbulentsystem
0
and waves and can be used in many multidisciplinary researches in hydrodynamicsand MHD.
2
n PACSnumbers:
a
J
I. INTRODUCTION by Zeldovich7, but didn’t become very popular. Yet in
6
1 the last decade the incorporation of the quantum field
For more than 60 yearswe are still facing the perplex- theories gave incitement to the theory of turbulence.
2 Kolmogorov9 power laws have been derived and grad-
ing astrophysical problem why the temperature of the
v ually the Burgers approach converted from a sophisti-
solar corona is two orders of magnitude higher than the
9 cated high energy physics problem8 to a standard tool
temperature of the photosphere; for review and refer-
0
for investigation of the turbulence over different physi-
6 ences see for example Ref. [3, sec. 5.1]. The purpose of
cal phenomena. The concept of random driver in the
9 the current work is to investigate the wave mechanism
0 of heating of the solar corona, according to which the special case of formation of spicules and the problem of
6 energyisbeing transportedbymagneto-hydrodynamical heating of space plasma in generalwas consideredin the
0 recent works Ref. [15]; here we would like to emphasize
waves, generated in the stochastic convective region. In
/ on the similarity between propagation of Alfv´en waves
h theframeworkofthisscenariothefamouscorrelationbe-
p tween the solar activity and the coronal emission in the inaninfinite magnetizedplasma andkink wavesin solar
- X-ray range can be easily explained. The novelty in the spicules16; for other papers devoted on the subject see
o
Ref. [14]. Thereexistalsodifferentwaysofgenerationof
r model is the incorporation of the Burgers approach to
t the turbulence in the problem for the calculation of the waves by turbulence where the influence of the random
s
driver is negligible12.
a spectral density of the Alfv´en waves. In the framework
v: of this model we derived an explicit formulas for: 1) the In the next section we will consider the application of
i spectral density of the MHD waves, 2) the total energy the Burgersapproachto the theory ofcreationof Alfv´en
X density and 3) the dissipated power per unit volume. In waves from the turbulence and the stochastic granula-
r perspective the results derived here can be generalized tion. In the β ∼1 regionof the solar atmosphere Alfv´en
a
for a realistic model of the turbulence and the distribu- waves can also be generated by resonant conversion of
tion of density, temperature and magnetic field. In the acoustic oscillations17.
current work the model task for calculation of the wave
power, emitted by a turbulent half-space and transmit-
ted in a non-moving magnetized plasma, is considered III. WAVE AMPLITUDES AND SPECTRAL
as an illustrative example. The homogeneous magnetic DENSITIES
field is oriented perpendicularly to the interface of both
half-spaces.
According to the Burgers idea6 the influence of the
turbulent vortices on the fluid can be modeled by intro-
duction of a random volume density of an external force
II. BURGULENCE F(t,r) on the right hand side of the Navier-Stokesequa-
tion
The Burgers approach to the turbulence gives an op-
portunity for approximate treatment of plenty of prob- ρ(∂t+V·∇)V=−∇p+η∆V+j×B+F. (1)
lems; as a tutorial lecture onBurgulence and a source of
main references we recommend Ref. [5]. This approach Analyzing low frequency and long wavelength phenom-
totheturbulencehasbeenintroducedintheastrophysics ena one can approximately suppose the random force to
2
be a white noise with a δ-function correlator5,6,7,8 turbulence in the Burgers approach. The excited Alfv´en
waves then propagate along the magnetic filed lines and
hF(t1,r1)F(t2,r2)i=Γ˜ρ2δ(t1−t2)δ(r1−r2)11, (2) reach the corona where the direction of the inequality
could be opposite.
where Γ˜ is the Burgersparameter. For the other notions
Let us consider the evolution of the amplitudes of
we use standard notations: p is the pressure, η is the
standing plane waves
viscosity, B is the magnetic field.
The current j is given by Ohm’s law V(t,r)=V υ~ (t)sin(k·r), (12)
A k
j=σE′, E′ =E+V×B, (3) B′(t,r)=B0bk(t)cos(k·r), (13)
F(t,r)=ρV f (t)sin(k·r),
A k
where σ is the electrical conductivity and E′ expresses p′ =ρV p cos(k·r), (14)
the effective electric field acting on the fluid. A k
Inquasi-stationaryapproximation,whenthe∂ Eterm hfp∗(t1)fk(t2)i=Γδ(t1−t2)δp,k11. (15)
t
is negligible, the evolution of the magnetic field in a
This is a technical tool for evaluation of the average
plasma with constant resistivity is
energy density used in many textbooks on statistical
physics. Duetothenegligibledependanceonthebound-
∂ B = rot(V×B)−ν rotrotB. (4)
t m
ary conditions usage of running waves would lead to the
We consider an incompressible fluid same result.
Substitution of this ansatz in Eq. (10) gives a separa-
divV=0 (5) tionofthevariablesandforthewaveamplitudeswehave
a system of ordinary differential equations
with wave amplitudes for the velocity
k b −k b
z x x z
V=(Vx,Vy,Vz), V ≪VA (6) dt~υ =pk+f −VAkzby−kybz−νkk2~υ, (16)
0
and the magnetic field
d b=V k ~υ−ν k2b, (17)
B=B +B′, B′(t,r)=(B′,B′,B′), B′ ≪B (7) t A z m
0 x y z 0 k·b=0, (18)
being small compared to the Alfv´en speed and the ex- k·~υ =0, (19)
ternal magnetic field. The z-axes is chosen in a vertical
direction along the constant magnetic field where for the sake of brevity the wave-vectorindices are
omitted. Time differentiation of the incompressibility
B0 =(0,0,B0)=B0ez. (8) equation Eq. (19) and substitution in Eq. (16) gives the
explicit form of the pressure
Analogouslyforthe pressurewe alsosuppose smalldevi-
ations from a constant value p k·f
0 p=− −V b (20)
k2 A z
p=p +p′. (9)
0
Further the substitution of the pressurefrom Eq.(20) in
The linearized system of magnetohydrodynamic (MHD) Eq. (16) gives
equations then reads
~υ˙ = −V k b−ν k2~υ+f ,
∂ B′ −∂ B′ A z k ⊥
∂tV=−∇ρp′+ Fρ+µB0ρ∂zzBxy′ −∂xyBzz′+νk∆V, b˙ = VAkz~υ−νmk2b, (21)
0 0
where
∂ B′=B ∂ V+ν ∆B′,
t 0 z m
divB′ =0, divV=0, (10) f⊥ =Πˆ⊥·f (22)
where is the transverse projection of the external force with
respecttothewave-vectorkbythepolarizationoperator
ν ≡ ε0c2, ν ≡ η (11) Π⊥
m k
σ ρ
ky2+kz2−kxky −kykz
are respectively the magnetic and kinematic viscosities. k⊗k k2 k2 k2
Here we would like to emphasize that the incompress- Πˆ⊥ =11− k2 =kx2k+2kz2−kkx2ky −kky2kz. (23)
ibility condition divV = 0 is applicable in the convec- kx2+ky2 −kxkz −kykz
k2 k2 k2
tive zone of the sun, where the sound speed significantly
exceeds the Alfv´en speed c ≫ V . In this region the This method for elimination of the pressure is used in
s A
convective circulation is treated as the influence of the Ref. [11] dedicated on amplification of Alfv´en waves in
3
thepresenceofshearflow. Acontinuationofthisresearch thetechnicaldetailsofthissimplederivationwillbeomit-
in the axial symmetric case is presented in Ref. [13]. ted. Here we suppose that the wave energy does not
Let us investigate the eigen-modes ∝ exp(−iωt) dis- increase significantly over one period
carding for a moment the influence of the external force
f. For the case ofsmall dissipationthe secularequations Γ≪ω (34)
A
for all the components are the same
and the brackets stand for a period and noise averaging.
−iω+ν k2 −V k
k A z =0, (24) In static regime the condition for dynamic equilibrium
(cid:12)(cid:12) VAkz −iω+νmk2(cid:12)(cid:12) requires that the power received from the noise Eq. (31)
(cid:12) (cid:12) has to be equal to the dissipated power
with eigen-v(cid:12)alues (cid:12)
γ
d hEi=−γhEi (35)
ω ≈ω −i , (25) t
A
2
where ω = V k is the frequency of the Alfv´en waves and this gives the dimensionless static averaged energy
A A z
of the oscillator
and
γ =νk2, ν ≡νm+νk (26) E = Γ =hυ2i . (36)
st 2γ x st
is the attenuation coefficient.
In regime of negligible magnetic viscosity the MHD Nowwearegoingtousethismodelexampleforderiva-
systemEq.(21)turnsintothesecondorderordinarydif- tion ofthe spectraldensity of Alfv´enwaves. For the vol-
ferential equation ume density of the averagewave energy we have
~υ¨=ν k2~υ˙ −ω2~υ+f˙ . (27)
k A ⊥ B′2 V2
hEi= + . (37)
If we introduce the x-component of the displacement (cid:10)2µ (cid:11) (cid:10)2ρ(cid:11)
0
t
x(t)= υ (t′)dt′ (28) This result may be rewritten as
x
Z
0
B2 V2 1
weobtainaneffectiveoscillatorequationunderanexter- hEi= 0 hb2i+ Ah~υ2i= h~υ2ip , (38)
4µ 4ρ 2 B
nal force 0
x¨=ν k2x˙ −ω2x+f , (29) where we have preformed a volume averaging
k A ⊥
which in case of negligible kinematic viscosity has an ef- 1
sin2(k·r) = , (39)
fective energy 2
(cid:10) (cid:11)
E = 1 x˙2+ω2x2 . (30) and used the definitions for the magnetic pressure
2 A
(cid:0) (cid:1)
B2
This is the non-perturbed by the friction and the weak p ≡ 0 (40)
random noise energy density. A harmonic oscillator un- B 2µ0
deranexternalforceisawell-knownmechanicalproblem,
see Ref. [1], Chap. V, Eq. (22,11). When the external and Alfv´en speed
force is a white noise a simple averaging gives that the
energy is a linear function of time; in other words the VA2 = B02 . (41)
random noise gives a constant power to the oscillator 2ρ 2µ
0
1 1 Γ
d hEi=d x˙2+ ω2x2 =d υ2 = , (31) In Eq. (38) we can substitute the static squared am-
t t(cid:28)2 2 A (cid:29) t x 2 plitude hυ2i from Eq. (36) and the attenuation coef-
(cid:10) (cid:11) x st
ficient from Eq. (26). Taking into account the y and
where
z-components of the Alfv´en wave velocities results in an
Γ˜ additional triplication of the energy. Then for the static
Γ≡ (32) value of the density of the wave energy we finally derive
VV2
A
is the Burgers parameter in the correlator for the plane- 3Γp
E ≡hEi = B. (42)
wave amplitudes of the transverse projection of the ran- k st 4 νk2
dom force in Eq. (15)
Thisspectraldensityisthe maindetailforthe statistical
hfp∗,⊥(t1)fk,⊥(t2)i=Γδ(t1−t2)δp,k11; (33) analysis made in the next section.
4
IV. ENERGY DENSITY, HEATING RATE AND Multiplication of the spectral density with decay γ gives
ENERGY FLUX the heating rate and summation in the wave-vectors’
space gives the dissipated by the Alfv´en waves per unit
In order to calculate the total static energy density volume power
we have to perform a summation of the spectral density
Eq. (42) over all wave-vectors Q=V E γ d3k = Γ˜VA p . (51)
Z k k(2π)3 8π2ν3 B
d3k
E = E =V E . (43)
tot k Z (2π)3 k Perhaps magnetic field dependance of the heating rate
Xk
can explain the correlation between the magnetic flux
The wave-vectors’cut-off is determined by the condition and X-ray luminosity10. In order to apply this result we
fortheexistenceofAlfv´enwaves,i.e. bytheregionwhere need to know the correlation between the magnetic field
the Alfv´en frequency ω equals the attenuation coeffi- and the radius of the active region.
A
cient γ The parameter with dimension length R which we in-
troduced due to the infrared divergencies participate in
VAkz =νk2. (44) the final formulas for the total energy density Eq. (49)
and energy flux Eq. (50). It seems very plausible if this
This equation sets the natural cut-off kc for the vertical detail could be helpful to reveal the scaling correlations
componentofthewave-vectorkz,paralleltotheconstant ofthelengthofcoronalloopsandotherobservablequan-
magnetic field B0 tities. It is premature, however, to start a detailed dis-
cussion based on a model example. The purpose of the
V
k = A. (45) presentworkistodemonstratethattheBurgersapproach
c
ν
should be used for realistic computer simulations in the
future.
Taking into consideration that the Alfv´en waves are
spreading axially symmetric as they follow the magnetic
field lines we may rewrite the wave-vector as
V. DISCUSSION AND CONCLUSIONS
k2 =k2+k2, (46)
z ρ
Upto now,the coronalheatingproblemhasnota sat-
which fixes the maximal value of the plane components
isfactory explanation not because of a lack of interest,
of the wave-vector k as a function of the vertical com-
ρ butratherduetodifficultiesconcerningthesimultaneous
ponent cut-off k
c obtaining of reliable observational data for all processes
running on the solar surface. For determination of the
k(max) = k (k −k ). (47)
ρ z c z parameters of each model as a rule we are forced to use
p
indirectmeans. Themodel,whichwenowpresentonthe
Thesecut-offswillbeusedasboundaryconditionsinthe
arenasuffersthesamedisadvantage. Foritsconfirmation
calculations of the total wave energy, integrated over all
it is necessary to know: 1) the distribution of magnetic
possible wave-vectorsin Eq.(43), and the corresponding
field near the sunspots, 2) velocity-velocity correlator in
wave power. In this way considering the axial symmetry
horizontal direction or, which is equivalent, the correla-
we obtain
tor of the small stochastic perpendicular components of
3VΓp kc kρ(max) 1 the magnetic field, etc. In short, apart from a realistic
E = B d(πk2)dk , (48) model for the turbulence we also need a realistic model
tot (2π)3ν Zkz=R1 Zkρ=0 kz2+kρ2 ρ z for all other variables. This naturally supposes a broad
interdisciplinary collaboration, which involves the whole
where we have taken into account that due to infrared
spectrum of the solar physics, from the statistical MHD
divergencies the vertical component of the wave-vector
to the X-ray luminosity of the solar corona. The confir-
shouldalsohaveaminimalvalue,determinedbythetyp-
mation of the suggested model requires realistic Monte
icalsizeRofthegivenmagneticfieldloop. Then,having
Carlo simulations, for which the analytical calculations
inmindEq.(32),thefinalexpressionforthetotalenergy
done in the current work Eq. (50), Eq. (49), Eq. (51)
becomes
are only test examples indispensable for a more realistic
3Γ˜Rp V R treatmentofthe problem. We proposeanoversimplified,
Etot = 8π2ν3B (cid:20)ln(cid:18) Aν (cid:19)−1(cid:21). (49) and therefore solvable, model, which realistic generaliza-
tion will hopefully lead to an adequate theory for the
Hereby, for the Alfv´en waves energy flux along the mag- solar corona heating.
netic field we find A natural continuation of the current investigation is
to take into account the influence of the shear flow over
S = 1E V = 3Γ˜VARpB ln VAR −1 . (50) the magnetized turbulent plasma. Preliminary numeri-
2 tot A 16π2ν3 (cid:20) (cid:18) ν (cid:19) (cid:21) calcalculationsonthebehaviorofMHDwavesinashear
5
flowshowasignificantamplificationoftheslowmagneto- tuationalsuperconductivity4. Inthephysicsofsupercon-
sonicmode. Thewavesenergyamplificationisactuallya ductors the wave field is the effective Ginzburg-Landau
transformationofthe shearflowenergyintowaveenergy wave function and the noise is produced by Langevin
which at the end dissipates into heat. This dissipation thermal fluctuations, which come from the inhomoge-
originatesan effective viscosity in the shear flow and the neous term in the TDGL equation. Naturally, the solid
current scenariofor heating of the solar corona may also stateschemeismeticulouslydevelopedandthefinalana-
happen to be the theory for the missing viscosity in ac- lyticalresultscanbe usedforfitting ofrepeatableexper-
cretion disks. Understanding the origin of an additional imental data from current-voltage characteristics. The
viscosity in a magnetized turbulent plasma is rather im- present work represents a realization of a similar idea in
portant for the proper explanation of the shining mech- the new astrophysical plasma areal.
anism of quasars, as well as for the angular momentum
transportduring formationof compactastrophysicalob-
jects.
The theory for the coronal heating takes into consid-
Acknowledgments
eration the influence of the stochastic random noise of
the turbulence and the granulationover the evolution of
Alfv´en waves. A similar mathematical scheme is used in Discussions and support by D. Damianov, R. Erdelyi,
the condensed matter physics for calculation of the fluc- I. Rousev and I. Zhelyazkovare highly appreciated.
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