Non-thermal fluctuations in plasma density near the temperature minimum of the solar atmosphere
We consider the formation of plasma-density fluctuations by non-thermal motions of gas near the temperature minimum of the solar atmosphere. For the inertial wavenumber range of turbulent velocity field of gas, an analytic expression for the one-dimensional (1D) spectrum of the fluctuations is obtai...
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irk-123456789-796392015-04-04T03:01:45Z Non-thermal fluctuations in plasma density near the temperature minimum of the solar atmosphere Kyzyurov, Yu.V. MS2: Physics of Solar Atmosphere We consider the formation of plasma-density fluctuations by non-thermal motions of gas near the temperature minimum of the solar atmosphere. For the inertial wavenumber range of turbulent velocity field of gas, an analytic expression for the one-dimensional (1D) spectrum of the fluctuations is obtained. This expression is used to predict the shape of the 1D horizontal spectrum of the fluctuations under three values of magnetic field strength: 5, 50, and 250 G. Also, estimates of the rms level of the fluctuations are made. It is shown that the increase in magnetic field has to alter the shape of the horizontal spectrum and to increase the level of the plasma-density fluctuations. 2005 Article Non-thermal fluctuations in plasma density near the temperature minimum of the solar atmosphere / Yu.V. Kyzyurov // Кинематика и физика небесных тел. — 2005. — Т. 21, № 5-додаток. — С. 183-186. — Бібліогр.: 10 назв. — англ. 0233-7665 http://dspace.nbuv.gov.ua/handle/123456789/79639 en Кинематика и физика небесных тел Головна астрономічна обсерваторія НАН України |
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MS2: Physics of Solar Atmosphere MS2: Physics of Solar Atmosphere |
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MS2: Physics of Solar Atmosphere MS2: Physics of Solar Atmosphere Kyzyurov, Yu.V. Non-thermal fluctuations in plasma density near the temperature minimum of the solar atmosphere Кинематика и физика небесных тел |
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We consider the formation of plasma-density fluctuations by non-thermal motions of gas near the temperature minimum of the solar atmosphere. For the inertial wavenumber range of turbulent velocity field of gas, an analytic expression for the one-dimensional (1D) spectrum of the fluctuations is obtained. This expression is used to predict the shape of the 1D horizontal spectrum of the fluctuations under three values of magnetic field strength: 5, 50, and 250 G. Also, estimates of the rms level of the fluctuations are made. It is shown that the increase in magnetic field has to alter the shape of the horizontal spectrum and to increase the level of the plasma-density fluctuations. |
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Article |
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Kyzyurov, Yu.V. |
| author_facet |
Kyzyurov, Yu.V. |
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Kyzyurov, Yu.V. |
| title |
Non-thermal fluctuations in plasma density near the temperature minimum of the solar atmosphere |
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Non-thermal fluctuations in plasma density near the temperature minimum of the solar atmosphere |
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Non-thermal fluctuations in plasma density near the temperature minimum of the solar atmosphere |
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Non-thermal fluctuations in plasma density near the temperature minimum of the solar atmosphere |
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Non-thermal fluctuations in plasma density near the temperature minimum of the solar atmosphere |
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non-thermal fluctuations in plasma density near the temperature minimum of the solar atmosphere |
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Головна астрономічна обсерваторія НАН України |
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2005 |
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MS2: Physics of Solar Atmosphere |
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http://dspace.nbuv.gov.ua/handle/123456789/79639 |
| citation_txt |
Non-thermal fluctuations in plasma density near the temperature minimum of the solar atmosphere / Yu.V. Kyzyurov // Кинематика и физика небесных тел. — 2005. — Т. 21, № 5-додаток. — С. 183-186. — Бібліогр.: 10 назв. — англ. |
| series |
Кинематика и физика небесных тел |
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AT kyzyurovyuv nonthermalfluctuationsinplasmadensitynearthetemperatureminimumofthesolaratmosphere |
| first_indexed |
2025-07-06T03:40:01Z |
| last_indexed |
2025-07-06T03:40:01Z |
| _version_ |
1836867336736342016 |
| fulltext |
NON-THERMAL FLUCTUATIONS IN PLASMA DENSITY NEAR
THE TEMPERATURE MINIMUM OF THE SOLAR ATMOSPHERE
Yu. V. Kyzyurov
Main Astronomical Observatory, NAS of Ukraine
27 Akademika Zabolotnoho Str., 03680 Kyiv, Ukraine
e-mail: kyzyurov@mao.kiev.ua
We consider the formation of plasma-density fluctuations by non-thermal motions of gas near
the temperature minimum of the solar atmosphere. For the inertial wavenumber range of turbulent
velocity field of gas, an analytic expression for the one-dimensional (1D) spectrum of the fluctua-
tions is obtained. This expression is used to predict the shape of the 1D horizontal spectrum of
the fluctuations under three values of magnetic field strength: 5, 50, and 250 G. Also, estimates
of the rms level of the fluctuations are made. It is shown that the increase in magnetic field
has to alter the shape of the horizontal spectrum and to increase the level of the plasma-density
fluctuations.
INTRODUCTION
Intensive studies of the solar atmosphere structure and motions are motivated by their importance for obtaining
a better understanding of basic solar phenomena such as atmospheric energy transport, turbulent diffusion of
magnetic fields or chaotic excitation of solar oscillations (see, e.g., [1–3, 8]). Observational data show that gas
motions near the temperature minimum of the solar atmosphere have non-thermal (turbulent) nature [2]. It was
established that the photospheric flows include both organized and stochastic motions. Spectra associated with
the stochastic velocity fields obey power laws, which are consistent with the spectrum of Kolmogorov turbu-
lence [1, 8]. It is generally agreed that the atmosphere near the temperature minimum (T = 4170 ◦ K) is weakly
ionized (hydrogen number density Nn = 2.1·1021 m−3, when electron number density Ne = 2.5 · 1017 m−3) [4, 9].
In many cases it can be considered as a mixture of ion-electron plasma and the neutral gas. One can say that
the plasma is a passive contaminant embedded in flows of the gas. The electrically charged components have
no influence on the atmospheric gas motions near the temperature minimum excluding sunspots and active
regions with very strong magnetic fields. The non-thermal motions of the photospheric gas have to result
in non-thermal fluctuations in plasma density. The formation of these fluctuations (with scales smaller than
the granular size, l < Lg) in the lower solar atmosphere was theoretically considered in [5, 6]. Expressions
for their three-dimensional (3D) and one-dimensional (1D) omnidirectional spectra were obtained and analysed
in [5, 7]. However, in experiments a 1D spectrum along the given direction is usually measured.
The aim of this report is to obtain the 1D spectrum of the plasma fluctuations induced by turbulent gas
motions in the solar atmosphere near the temperature minimum, and to consider the shape of the spectrum
and the rms fluctuation level under changes in the magnetic field strength.
BASIC ASSUMPTIONS AND EQUATIONS
Turbulent mixing of the weakly ionized plasma near the temperature minimum of the solar atmosphere is a slow
process. The time-scales of the process are larger than an interval between ion-neutral collisions t � τi and
the length-scales are larger than the ion mean free path l � Λi. It can be described by fluid equations for ions,
electrons and neutrals. The neutral gas motion can be regarded as given (the gas is unaffected by collisions
with the electrically charged components). The behaviour of charged components is of interest in this work.
Then, a simplified system of equations for electrons and ions may be used [5]:
∂Ns/∂t + �∇(Nsvs) = 0, (1)
τ−1
s (vs − u) = qsm
−1
s E + ωBs(vs × b) − v2
Ts
�∇Ns/Ns. (2)
In this set of equations, the variables are chosen as density Ns and velocity vs for each species (s ≡ i, e), u is
the velocity of neutral gas, τs is a characteristic time of charged particle collisions with neutrals, qs is the particle
c© Yu. V. Kyzyurov, 2004
183
charge (qe = −qi = −e), ωBs = 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.
Assumptions Te ≈ Ti ≈ Tn = T and Ne ≈ Ni = N are valid for slow processes near the temperature
minimum. In addition, τiωBi � 1 in the quite regions of the solar atmosphere. If the only electric field
considered is that required to prevent charge separation (due to the electric field E electrons tend to follow
ions), and eliminating E from (2), we then obtain the local drift velocity of ions (or plasma in general)
vi(x, t) ≈ u(x, t) + βi(u × b) − Da∇2N, (3)
here βi = τiωBi, Da is the ambipolar diffusion coefficient.
Using (3), the equation of continuity (1) then becomes
∂N/∂t + �∇ · (Nu) + βib · �∇× (Nu) − Da∇2N = 0. (4)
The equation (4) describes the plasma-density behaviour when plasma is embedded in the flow of neutral gas
and relates the plasma number density, N , with the neutral gas velocity, u.
If the flow of neutral gas is turbulent, u and N can be divided into their ensemble mean parts u0 = <u>,
N0 = < N > and fluctuations around them u1, N1 (< u1 > = 0, < N1 > = 0): u = u0 + u1 (u0 > u1), and
N = N0 + N1 (N0 > N1). Here we consider the case when the length-scales of random ingredients u1 and N1
are close to each other and smaller than the length-scales of mean quantities u0 and N0. The same can be said
about the time-scales. If l is the length-scale of random fluctuations, Lg stands for a granular size (the length-
scale of mean gas velocity), and LN = N0|�∇N0|−1 is the length-scale of mean plasma-density gradient, then
evidently the following inequalities are valid: l � LN ≤ Lg.
The velocity of gas motions near the atmosphere temperature minimum is usually small compared to
the sound velocity, hence, the neutral gas can be considered as incompressible: �∇ · u = �∇ · u1 = 0. Then,
we can write the equation for the relative plasma-density fluctuations, δN = N1/N0, in the form [5, 6]:
∂δN/∂t + �∇ · (δN u1) − Da∇2δN = −L−1
N (u1 · n) − βib · (�∇× u1), (5)
where n = LN(�∇N0/N0) is the unit vector along the mean plasma-density gradient.
The equation (5) describes the process of formation of plasma fluctuations with scales l < LN . It may be
seen that the first term on the RHS of (5) is more important at larger scales l > βiLN , and the second is more
important for smaller scales l < βiLN . The process in which the neutral gas turbulence in conjunction with
the background plasma-density gradient produce plasma fluctuations by mixing regions of high and low density
dominates at the larger scales, while the interaction of the plasma embedded in the turbulent motions of neutral
gas with the magnetic field is more important for generation of the small-scale fluctuations.
SPECTRA OF PLASMA DENSITY FLUCTUATIONS
Under the assumption that the random fields u1(x, t) and δN(x, t) are stationary functions of x and t with
Fourier transforms: u1j(x, t) =
∫
dkdω u1j(k, ω) ei(kx−ωt), and δN(x, t) =
∫
dkdω δN(k, ω) ei(kx−ωt), the Fourier
transform of (5) is
(Dak2 − iω) · δN(k, ω) + ikj
∫
dk′dω′ δN(k′, ω′) · u1j(k − k′, ω − ω′)
= −L−1
N (n · u1(k, ω)) − iβik(u1 × b). (6)
The convolution term on the LHS of (6) represents the contribution of mode interactions in the process of
plasma fluctuation generation. If we take it into account through the coefficient of turbulent diffusion Dt, then
δN(k, ω) = (−L−1
N n − iβi(b× k)) ((Da + Dt) k2 − iω)−1 · u1(k, ω). (7)
For statistically homogeneous and stationary random fields the following relations are valid (see, e.g., [10]):
〈u1i(k, ω) · u∗
1j(k
′, ω′)〉 = Φij(k, ω) δ(k − k′) δ(ω − ω′), (8)
〈δN(k, ω) · δN∗(k′, ω′)〉 = Ψ(k, ω) δ(k − k′) δ(ω − ω′), (9)
where the (∗) denotes a complex conjugate, Φij(k, ω) is the spatiotemporal spectrum tensor of the field u1,
Ψ(k, ω) is the spatiotemporal spectrum of the relative fluctuations in plasma density.
184
The tensor Φij(k, ω) can be presented in the form [1, 10]:
Φij(k, ω) = (δij − kikj k−2) [4π2k2(1 + ω2τ2
k )]−1τkE(k), (10)
here E(k) = C1ε
2/3k−5/3 is the energy spectrum function in the inertial range k0 < k < kd (the wavenumber
k0 = 2π/Lg represents the outer scale of turbulence or the basic energy input scale, in our case the granular
size Lg, kd ≈ ν−3/4ε1/4 is the viscous wavenumber that corresponds to the viscous length-scale ld = 2π/kd
at which viscous dissipation is adequate to dissipate the energy at the rate ε, ν is the kinematic viscosity of
the gas, the Kolmogorov constant C1 is around 1.5 [10]), τ−1
k = νk2 + ε1/3k2/3 is the decay rate of eddy with
the wavenumber k.
Taking (7)–(10) into account, we obtain an expression for the spatiotemporal spectrum of δN [5, 7]:
Ψ(k, ω) =
L−2
N k−2(n × k)2 + β2
i (b × k)2
4π2k2((Da + Dt)2k4 + ω2) (1 + ω2τ2
k )
· τk E(k). (11)
The spatial spectrum of the fluctuations S(k) is related with Ψ(k, ω) as S(k) =
∫∞
−∞ dω Ψ(k, ω). As in
the case of most fluids and gas, near the atmosphere temperature minimum the Schmidt number can be regarded
about unity, i.e., Da ≈ ν, then (Da + Dt) k2 ≈ νk2 + ε1/3k2/3, and after integration, we have
S(k) =
L−2
N k−2(n × k)2 + β2
i (b × k)2
8π k2(νk2 + ε1/3k2/3)2
· E(k), (12)
the 3D power spectral density of δN . Using (12) we may obtain the rms level of the fluctuations:
〈(δN)2〉1/2 = 〈(N1/N0)2〉1/2 =
(∫
dkS(k)
)1/2
=
(∫ k2
k1
S0(k) dk
)1/2
, (13)
where
S0(k) =
L−2
N + β2
i k2
2k3(1 + (k/kd)4/3)2
(14)
is the 1D omnidirectional spectrum of the fluctuations, S0(k)dk represents the contribution to the power level
of relative plasma-density fluctuations from the wavenumber range (k, k + dk).
From the expression (13) for the 3D spectrum we can obtain 1D spectrum that may be measured along
z-direction, S1(kz). If we use the cylindrical polar coordinates (k⊥, ϕ, kz), k2 = k2
⊥ + k2
z , then:
S1(kz) =
∫ √
k2
d−k2
z
0
k⊥dk⊥
∫ 2π
0
S(k)dϕ =
∫ √
k2
d−k2
z
0
L−2
N f(k⊥, kz , A1) + β2
i k2 f(k⊥, kz , A2)
8 k7(1 + (k/kd)4/3)2)
k⊥dk⊥, (15)
where k0 < kz < kd, f(k⊥, kz, A) = k2
⊥ + k2
⊥ cos2 A + 2k2
z sin2 A (A1 is the angle between z and n, and A2
between z and b). If we consider the region of solar atmosphere with n ‖ b, then for the horizontal direction
z ⊥ n, b (A1 = A2 = 90◦), and the 1D spectrum takes the form:
S1(kz) =
∫ √
k2
d
−k2
z
0
(L−2
N + β2
i k2) (k2
⊥ + 2k2
z)
8 k7(1 + (k/kd)4/3)2)
k⊥dk⊥. (16)
To estimate the rms level of relative fluctuations (13), and the shape of expected spectra (14), (16), we adopt
the following values of parameters near the temperature minimum of the atmosphere [3, 4, 9]: the mean ion
mass mi = 25.3 a.m.u.; the length-scale of the mean plasma-density gradient LN ≈ Lg ≈ 940 km, the mean gas
velocity on the outer scale of turbulence (associated with the granular size Lg) u0 ≈ 1.1 km s−1; the energy dis-
sipation rate ε ≈ u3
0L
−1
g ≈ 1.5 ·103 m2 s−3; Da ≈ ν ≈ 11 m2 s−1; and the viscous length-scale ld = 2π/kd ≈ 6 m.
The level and the spectral shapes are estimated under three values of the magnetic field strength B: 5, 50, and
250 G. An increase in B results in the rise in βi = τiωBi, which respectively takes the values: 5.5·10−4, 5.5·10−3,
and 2.7 ·10−2. The rms fluctuation level in the wavenumber range km ≤ k ≤ kd (km = 2πL−1
m , Lm = 300 km) is
2.5, 2.8, and 6.6 %, respectively. Figure 1 shows the shapes of the 1D fluctuation spectra. Figure 1A illustrates
the omnidirectional spectra S0(k) (14), Figure 1B shows the horizontal spectra S1(kz) (16) (the values of B are
indicated near the lines), the dashed line represents the slope of the Kolmogorov spectrum k−5/3.
185
Figure 1. 1D spectra of plasma-density fluctuations expected near the temperature minimum of the solar atmosphere
for three values of the magnetic field strength B indicated near the lines
CONCLUSIONS
In this work the analytic expression (15) for the 1D spectrum of small-scale plasma-density fluctuations, which
may be generated by the microturbulence of gas near the temperature minimum of the solar atmosphere, was
derived. This expression was used to predict the shape of the 1D horizontal spectrum (16) of the fluctuations
for three values of magnetic field strength: 5, 50, and 250 G (see Fig. 1B), when parameters of the turbulence
and the atmosphere were unchanged. The rms level of the fluctuations was estimated as well. It is shown
that the increase in the magnetic field has to alter the shape of the 1D spectrum (see Fig. 1) and to increase
the level of relative fluctuations in plasma density. Under the usual conditions for the temperature minimum of
the quiet atmosphere, the fluctuation level for wavenumbers inside the inertial range of turbulence takes values
2.5% (B = 5 G), 2.8% (B = 50 G), and 6.6% (B = 250 G).
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