Spectrum of small-scale plasma structures in the photosphere
In this report we consider possibility of formation of small-scale plasma structures in the turbulent flows of photospheric gas on the Sun and analyse dependence of their spectrum and intensity on height and the magnetic field strength. It was shown that in the height range 150–350 km the slope of t...
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irk-123456789-1104092017-01-05T03:02:39Z Spectrum of small-scale plasma structures in the photosphere Kyzyurov, Yu.V. Space plasma In this report we consider possibility of formation of small-scale plasma structures in the turbulent flows of photospheric gas on the Sun and analyse dependence of their spectrum and intensity on height and the magnetic field strength. It was shown that in the height range 150–350 km the slope of the structure spectrum decreases with increasing the altitude. Under the weak magnetic field (B = 5 G), the intensity of plasma structures is unchanged with height. The increase in the magnetic field strength results in a rise in the structure intensity and in a decrease in the spectral slope. Розглядається можливість формування дрібномасштабних плазмових структур в турбулентних потоках фотосферного газу на Сонці та аналізується залежність їх просторового спектра та інтенсивності від висоти та напруженості магнітного поля. Показано, що в інтервалі висот 150–350 км нахил спектра структур, що розглядаються, із збільшенням висоти зменшується. За умов слабкого магнітного поля (В=5 Гс) інтенсивність плазмових структур з висотою не змінюється. Збільшення напруженості магнітного поля веде до зростання інтенсивності структур та зменшення нахилу спектра. Рассматривается возможность формирования мелкомасштабных плазменных структур в турбулентных потоках фотосферного газа на Солнце и анализируется зависимость их пространственного спектра и интенсивности от высоты и напряженности магнитного поля. Показано, что в интервале высот 150‑350 км наклон спектра рассматриваемых структур с увеличением высоты уменьшается. При слабом магнитном поле (В=5 Гс) интенсивность плазменных структур с высотой не меняется. Увеличение напряженности магнитного поля приводит к росту интенсивности структур и уменьшению наклона спектра. 2007 Article Spectrum of small-scale plasma structures in the photosphere / Yu.V. Kyzyurov // Вопросы атомной науки и техники. — 2007. — № 1. — С. 81-83. — Бібліогр.: 9 назв. — англ. 1562-6016 PACS: 94.05.–a; 96.60.Mz; 47.27.–i http://dspace.nbuv.gov.ua/handle/123456789/110409 en Вопросы атомной науки и техники Національний науковий центр «Харківський фізико-технічний інститут» НАН України |
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Space plasma Space plasma Kyzyurov, Yu.V. Spectrum of small-scale plasma structures in the photosphere Вопросы атомной науки и техники |
| description |
In this report we consider possibility of formation of small-scale plasma structures in the turbulent flows of photospheric gas on the Sun and analyse dependence of their spectrum and intensity on height and the magnetic field strength. It was shown that in the height range 150–350 km the slope of the structure spectrum decreases with increasing the altitude. Under the weak magnetic field (B = 5 G), the intensity of plasma structures is unchanged with height. The increase in the magnetic field strength results in a rise in the structure intensity and in a decrease in the spectral slope. |
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Article |
| author |
Kyzyurov, Yu.V. |
| author_facet |
Kyzyurov, Yu.V. |
| author_sort |
Kyzyurov, Yu.V. |
| title |
Spectrum of small-scale plasma structures in the photosphere |
| title_short |
Spectrum of small-scale plasma structures in the photosphere |
| title_full |
Spectrum of small-scale plasma structures in the photosphere |
| title_fullStr |
Spectrum of small-scale plasma structures in the photosphere |
| title_full_unstemmed |
Spectrum of small-scale plasma structures in the photosphere |
| title_sort |
spectrum of small-scale plasma structures in the photosphere |
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Національний науковий центр «Харківський фізико-технічний інститут» НАН України |
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2007 |
| topic_facet |
Space plasma |
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http://dspace.nbuv.gov.ua/handle/123456789/110409 |
| citation_txt |
Spectrum of small-scale plasma structures in the photosphere / Yu.V. Kyzyurov // Вопросы атомной науки и техники. — 2007. — № 1. — С. 81-83. — Бібліогр.: 9 назв. — англ. |
| series |
Вопросы атомной науки и техники |
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AT kyzyurovyuv spectrumofsmallscaleplasmastructuresinthephotosphere |
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2025-07-08T00:34:51Z |
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2025-07-08T00:34:51Z |
| _version_ |
1837036881892605952 |
| fulltext |
SPACE PLASMA
Problems of Atomic Science and Technology. 2007, 1. Series: Plasma Physics (13), p. 81-83 81
SPECTRUM OF SMALL-SCALE PLASMA STRUCTURES
IN THE PHOTOSPHERE
Yu.V. Kyzyurov
Main Astronomical Observatory, National Academy of Sciences of Ukraine, Kiev, Ukraine,
e-mail: kyzyurov@mao.kiev.ua
In this report we consider possibility of formation of small-scale plasma structures in the turbulent flows of
photospheric gas on the Sun and analyse dependence of their spectrum and intensity on height and the magnetic field
strength. It was shown that in the height range 150–350 km the slope of the structure spectrum decreases with
increasing the altitude. Under the weak magnetic field (B = 5 G), the intensity of plasma structures is unchanged with
height. The increase in the magnetic field strength results in a rise in the structure intensity and in a decrease in the
spectral slope.
PACS: 94.05.–a; 96.60.Mz; 47.27.–i
INTRODUCTION
The structure and dynamics of the solar photosphere
are very important for better understanding of basic solar
phenomena such as atmospheric energy transport,
turbulent diffusion of magnetic field or chaotic excitation
of solar oscillations. The degree of photospheric gas
ionization is quite small [1, 2]. It means that electrically
charged particles in the photosphere can be considered as
passive contaminants embedded in motions of the gas.
Results of observations clearly show that the photospheric
flows include both organized and stochastic motions [3,
4]. The spectra associated with the random velocity fields
obey power laws, which are close to the spectrum of
Kolmogorov turbulence [4]. Turbulent motions of the gas
have to result in formation of random plasma structures in
the photosphere [5]. Parameters of the photosphere and
turbulent mixing depend on height [1, 2, 6]. In addition
there are regions with various magnetic field strengths in
the photosphere [7]. It is important to analyse possible
dependence of the spectrum and intensity of plasma
structures generated in turbulent photospheric flows on
the height and the magnetic field strength. This analysis is
the aim of the report. The present consideration will be
restricted to small-scale structures with length-scales
smaller than the length-scale of the mean plasma density
gradient.
BASIC EQUATIONS AND RELATIONS
To describe turbulent mixing in the solar photosphere
(which is a slow process) a three-fluid model can be used.
Since the charged particles are passive contaminants, they
have no influence on motions of neutral gas and the gas
velocity field u(x, t) may be treated as a known function
of position and time. The gas in the photosphere can be
regarded as incompressible, ∇u=0. The behaviour of
charged particles embedded in the gas flow can be
described by the following set of equations [5]:
0)v(/ =∇+∂∂ sss NtN , (1)
sssssss NNmq ∇−×Ω+=− −− 12
Tss
1 v)(/)( bvEuvτ , (2)
where the variables are chosen as density Ns and velocity
vs for each species (s≡i, e), τs is a characteristic time of
charged particle collisions with neutrals, qs is the particle
charge (qe=–qi=–e), Ωs= qsB/msc is the gyrofrequency, vTs
is the thermal velocity, ms is the particle mass, b=B/B is
the unit vector along the magnetic field B, E is the electric
field.
In the photosphere τiΩi<<1 and the assumptions of
quasi-neutrality Ne=Ni=N and isothermality Te=Ti=Tn=T
are valid.
In the case of turbulent flows the gas velocity may be
separated into mean and fluctuating parts u=u0+u1
(u0=<u>, <u1>=0, u1<u0). The same may be made for
plasma density N=N0+N1 (N0=<N>, <N1>=0, N1< N0); N1
represents plasma structures generated by the turbulent
velocity field u1.
The way of derivation of Ψ(k,ω), the spatiotemporal
spectrum of δN= N1/N0, from Eqs. (1), (2) is described in
[5]. Length-scales of random gas motions were restricted
to the inertial range of turbulence. In this range turbulence
is homogeneous and isotropic one with known statistical
properties. The spectrum tensor of the field u1 [4, 5, 8] is:
1222 )]1(4[)()()(),( −+⋅=Φ tt kEkD τωπτω αβαβ kk , (3)
k0<k<kν,
where Dαβ=δαβ–kαkβ /k2 is the projection operator,
τt(k)=(νk2+ε1/3k2/3)–1 is the decay time of eddy with a
length-scale k–1, E(k)=C1ε2/3k–5/3 is the energy spectrum
function, k0
–1 is the basic energy input scale, kν=(ε/ν3)1/4 is
the Kolmogorov dissipation wavenumber, ν is the
kinematic viscosity of the gas, ε is the rate of turbulent
energy dissipation per unit mass, the Kolmogorov
constant C1 is around 1.5 [9].
To obtain Ψ(k,ω) the only electric field E considered
was that required to prevent charge separation (due to E
electrons tend to follow ions). In addition a contribution
of the mode interaction in the process of plasma structure
generation was taking into account through the coefficient
of turbulent diffusion KT. For the structures with length-
scales smaller than LN=N0|∇N0|–1, the length-scale of ∇N0,
the following expression was derived [5]
)()]1)((4[),( 2122222 kk Qkttk τττωτωπω −++1=Ψ , (4)
LN
–1<k<kd,
here τk=(DAk2+KTk2)–1=(DAk2+ε1/3k2/3)–1, DA is the
ambipolar diffusion coefficient, Q(k)= [(n×k)2/(LNk)2+
+(b×k)2/(τiΩi)2]C1ε2/3k–11/3, n=LNN0
–1∇N0 is the unit along
mailto:kyzyurov@mao.kiev.ua
82
∇N0, kd = (ε/DA
3)1/4 is the Oboukhov-Corrsin wavenumber
known in the theory of passive scalar turbulent convection
[9], in the present case it define the structure length-scale
at which KT=DA.
From Eq.(4) we can obtain the spatial spectrum of δN
)()]/1(4[),()( 1 kkk QdP ktktN ττττπωω −
∞
∞−
+=Ψ= ∫ . (5)
Unlike [5] the inequality DA≠ν was taken into account
in the present case. Using Eq.(5) a mean-square level of
δN in the range (k1, k2) may be calculated
))/(())/(()( 3/4
1
3/4
2
2
ddN kkSkkSdPN −== ∫ kkδ , (6)
where
+−+= −−− ]3/2Pr)3[(
8
3)( 2/322 xxkLxS dN
+++−
−
+
−−
])2/Pr)1(arctg()2/Pr)1((arctg[
Pr12
3 2/12/52/1
22
xx
kL dN
Pr)))]1(2/(Pr)1(ln(Pr)1())1/(ln(2[
Pr18
3 22
++++−+
−
Ω
+ xxxxiiτ
here Pr=ν/DA is the diffusion Prandtl number.
The 1D spectrum of plasma structures in the turbulent
photospheric flow that may be measured along z-direction
may be obtained from Eq.(5) too:
== ∫ ∫⊥⊥
ζ π
ϕ
k
NzN dPdkkkP
0
2
0
)()( k
= ∫ ⊥⊥
−
⊥⊥
− Ω+
ζ
θτθ
k
ziizN dkkkkFkkfkkkfL
0
7
2
222
1
2 )()),,(),,((
4
1
where f(k⊥,kz,θ)=k⊥
2+k⊥
2cos2θ+2kz
2sin2θ, θ1 is the angle
between z and n, θ2 between z and b, kζ
2=kd
2–kz
2,
k2=k⊥
2+kz
2, F(k)=[(1+(k/kd)4/3)× (2+(k/kd)4/3+(k/kν)4/3)]–1.
Eqs. (6), (7) give an opportunity to estimate
changeability of small-scale plasma structures with
changing the height and the magnetic field strength.
CHANGEABILITY OF PHOTOSPHERIC
PLASMA STRUCTURES
To estimate changeability of the photospheric plasma
structures we shall consider the case when n and b are in
vertical direction, while the possible measurement
direction z is horizontal. Then θ1=θ2=π/2 and Eq.(7) takes
the form
∫ ⊥⊥
−
⊥
− +Ω+=
ζ
τ
k
ziiNzN dkkkkFkkkLkP
0
72222 )()2)((
4
1)( . (8)
The plasma structures are analysed near heights of 150
and 350 km. The outer scale of turbulence k0
–1=
L0=940 km is the same for both heights [6], and we
suppose that LN≈L0. The mean gas velocity on L0 is u0,
and then ε= u0
3/L0. Parameters of the photosphere taken
from [1,2,6] together with the calculated values of kν
–1
and kd
–1are presented in Table 1. Characteristics of plasma
structures calculated with use of Eqs.(6), (8) and the value
τiΩi are shown in Table 2 (γ is the power index when
PN(kz) was approximated by a simple power law kz
–γ). The
limits of integration in Eq.(6) are k1 = 2π/Lm, k2 = kν
(Lm=300 km).
Fig.1 shows the 1D spectrum PN(kz) calculated with
the use of Eq.(8) for h=150 km: line 1 is for the case of
the magnetic field B=5 G, line 2 for B=250 G, a straight
line is the power law kz
–5/3. Fig.2 represents the same for
h=350 km.
From the figures and Table 2 it seen that the small-
scale plasma structures have to be sensitive to the change
in both the height and the magnetic field strength. In the
region with a weak magnetic field, dependence of the
structure intensity on height is almost absent, though the
spectral shape changes. An increase in the magnetic field
provides a change in the rms fluctuation level and the
spectral slope.
Table 1. Parameters of the solar photosphere
Parameter h=150 km h=350 km
T, K
Nn, m–3
Ne, m–3
mi, a.u.m.
u0, km/s
kν
–1, cm
kd
–1, cm
5180
5.05×1022
6.04×1018
25
1.1
10.4
1.33
4670
1.01×1022
1.12×1018
26.3
2.05
20
2.5
1E-5 1E-4 1E-3 1E-2 1E-1 1E+0 1E+1
WAVE NUMBER, 1/m
1E-14
1E-13
1E-12
1E-11
1E-10
1E-9
1E-8
1E-7
1E-6
1E-5
1E-4
1E-3
1E-2
1E-1
1E+0
1E+1
1E+2
1D
S
P
EC
TR
U
M
,m
1
2
Fig.1. Spectrum PN(kz) at h=150 km
1E-5 1E-4 1E-3 1E-2 0.1 1 1E+1
WAVE NUMBER, 1/m
1E-14
1E-13
1E-12
1E-11
1E-10
1E-9
1E-8
1E-7
1E-6
1E-5
1E-4
1E-3
1E-2
0.1
1
1E+1
1E+2
1D
S
P
EC
TR
U
M
,m
1
2
Fig.2. Spectrum PN(kz) at h=350 km
(7)
83
Table 2. Characteristics of plasma structures
h, km B, G τiΩi <δN2>1/2, % γ
150
150
350
350
5
250
5
250
1.98×10–5
9.9×10–4
9.4×10–5
4.7×10–3
2.5
2.6
2.5
2.8
2.22
1.41
1.94
1.23
CONCLUSIONS
An analytic expression for the 1D spectrum of the
plasma structures in a turbulent flow of photospheric gas
Eq.(7) as well as the formula for estimation of the RMS
level of their intensity Eq.(6) were presented in the report.
Using the expressions it was shown that in the height
range 150–350 km the slope of the structure spectrum
decreases with increasing the altitude. Under the weak
magnetic field (B=5 G), the intensity of plasma structures
is unchanged with height. The increase in the magnetic
field strength results in a rise in the structure intensity and
in a decrease in the spectral slope.
The obtained results seem to be important for better
understanding of basic solar phenomena, such as
generation of the random component of magnetic field or
chaotic excitation of solar oscillations.
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