Pis'ma v ZhETF, vol. 99, iss. 6, pp. 365-370 © 2014 March 25
Whether the Sun's supergranulation possesses a scaling?
V. P. Goncharov+1), V. I. Pavlovx + Obukhov Institute of Atmospheric Physics of the RAS, 109017 Moscow, Russia
x UFR des Mathématiques Pures et Appliquées - LML CNRS UMR 8107, 59655 Villeneuve d'Ascq, France
Submitted 20 January 2014 Resubmitted 17 February 2014
The Sun's supergranulation is considered as a collective effect provided by a statistical ensemble of narrow upflow jets. The responsibility for their occurrence rests with the Rayleigh-Taylor blow-up instability. It is this mechanism that is a trigger for the collapse of thermal irregularities and leads to the formation of such jets. The scaling laws of turbulence generated by these jets are discussed.
DOI: 10.7868/S0370274X14060022
1. Introduction. Similarity (scaling) is a special symmetry where a change in scale of independent variables can be compensated by a similarity transformation of other variables. Restricting the type of functional dependencies among variables and parameters, the scaling simplifies the analysis and allows to derive useful scaling laws. For the Sun's supergranulation where these laws are masked by strong turbulence the study of full frequency-wavenumber spectra is the most rational way to detect the scaling.
The Sun's supergranulation was discovered more than fifty years ago [1] in speed fluctuations, however even today it continues to remain one of the most intriguing problems of modern physics. The supergranulation as a collective effect of large-scale auto-organization is first of all a feature of the surface velocity field at the surface of the quiet Sun. The typical example of supergranulation cells is presented in Fig. 1.
Notwithstanding numerous attempts to explain why and how this phenomenon originates, a full understanding of it has not yet been reached [3]. Existing theoretical models of supergranulation are basically of two types: those that postulate that it is driven by thermal buoyancy, and those that do not. The most of models are very qualitative and either rely on extremely simplified theoretical frameworks or on simple dynamical toy models designed after phenomenological considerations. The looseness of theoretical models, combined with the incompleteness of observational constraints and shortcomings of numerical simulations, has made it difficult to either validate or invalidate any theoretical argument so far.
1)e-mail: v.goncharov@rambler.ru; Vadim.Pavlov@univ-lille1.fr
Fig. 1. The supergranulation horizontal velocity field as obtained by using the balltracking technique (from [2]). The size of the image is 200" x 200"
The scaling duality can be argued from simple considerations. Indeed, if a physical quantity f admits scaling in some range of space and time, its presentation looks like f (x,t) ~ xmtn. Then its Fourier transform also has scaling f (k, w) ~ k-mw-n in the correspondent range of wavelengths k and frequencies w.
Currently, full frequency-wavenumber spectra for the Sun's supergranulation are absent. In up-to-date investigations, this phenomenon is treated only in terms of pure wave spectra for studying the power-laws of spectrum tails. Unfortunately, this information is not enough
to identify the similarity parameter for spectral scaling. Because of this, it would be useful to consider a simple estimation model which would allow to analyze the spectral scaling in more detail.
In this work we consider the scenario which assumes that supergranulation cells spring up because hot narrow jets reach the Sun's apparent surface called the photosphere. In order to explain the formation of such jets, we suggest the simplest idealized model - vertically homogeneous layer of inviscid incompressible fluid which is under the action of gravity between two horizontal, impenetrable, stress-free boundaries (see Figs. 2a and b).
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(a)
(b)
w
//////////// TTTTTT7TTTT7
Fig. 2. The collapsing upward jet (a) and its spreading (b) on the upper boundary
Suppose that due to heating at the bottom boundary there arise islets of more light fluid. As known [4-6], the further dynamics of these fragments turns out to be unstable and has the tendency to collapsing, i.e., to the formation of infinitely narrow jets in finite time. In particular, in the self-similar stage of the radially symmetric collapse these fragments look like fluid ellipsoids, whose vertical semi-axes tend to infinity, while contact areas with the bottom boundary tend to zero.
Once an infinitely narrow jet reaches the upper boundary, the inverse process - the liquid spreading along the upper boundary is initiated (see Fig. 2b). It is obvious that both collapsing and spreading are described by the same idealized model. The distinction is only in the buoyancy sign and that the infinitely narrow jet is a final state in the one case and an initial one in the other. For this reason, in the spreading regime the infinitely narrow jet passes through all stages of the collapse, but in a reverse order.
In this work the solar supergranulation is presented as a collective effect provided by a statistical ensemble of narrow upflow jets. Note that the horizontal flow pattern generated by such mechanism on the upper boundary is no different from the Benard convection only if swirling jets are absent in this ensemble. Otherwise, some of cells must have the vortex structure of the flow. Because this fact has been established experimentally
[2, 7], it can serve as one of the arguments in favor of the blow-up mechanism of the solar supergranulation.
Another undeniable experimental fact which deserves attention is the availability of power-law approximation for spectrum tails of the horizontal velocity and the intensity of thermal radiation at the solar surface. It is well known that such type of spectral behavior can be caused by singularities. This idea was first proposed in [8] to determine the turbulence-like spectrum generated by the singularities on the fluid surface. Later the very same idea, but only for other models, was used in [9-12].
2. Estimation model and scaling. As an estimation model we consider the set of two-dimensional equations
atv + (v-V)v = Jwr-V(fer), (1)
dth + V(hv)=0, dtT + (v -V)t = 0. (2)
The notation is as follows: x are the Cartesian horizontal coordinates; V is the horizontal gradient operator; dt = d/dt; v(x,t) is the depth-averaged horizontal velocity. For a detailed derivation of this model, see [5, 6].
Equations (1), (2) describe the depth-averaged flow in active boundary layer of thickness h(x,t) and relative buoyancy t(x,t) = gAg/go, where g is gravity and Ag(x,t) is the density deviation from the background constant value g0. When density variations are produced only by temperature ones, AT, the relative buoyancy can be computed as
t = -gpAT,
(3)
where 3 is thermal expansion coefficient.
Equations (1), (2) are Hamiltonian and can be obtained from first principles with use of the Poisson brackets
[vi,v'k} = h ^ (divk - dkVi), {h, vk} = -dkS, {t, v'k} = -h-1Sdkt,
(4)
(5)
and the Hamiltonian H
i J cix (hv2 + h2T) .
Here and in what follows, primed field variables mean the dependence on the primed spatial coordinates, S = = S(x — x') is the Dirac delta function, all the trivial Poisson brackets are omitted for the sake of space, and all the integrals are taken over the whole area occupied by the 2D-flow.
In addition to H any Hamiltonian system can also has other conserved quantities G. For example, for the model above, total mass Q and total buoyancy N
Q
dx h, N
dx hr,
are the simplest among them.
The presence of additional conserved quantities implies the existence of symmetry transformations under which the equations of motion are invariant. Because Poisson brackets are known a one-parameter symmetry transformation can be formulated as the equations
de v = {v,G}, deh = {h,G}, de t = {t,G}, (6)
where e is a transformation parameter. In the trivial case e = t and G = H, Eqs. (6) merely reproduce Eqs. (1), (2).
Among a large number of symmetries inherent in the considered model, our interest is only with the scale symmetry. Such symmetries are closely related with so-called virial theorems. In our case, this theorem can be directly deduced from the equations of motion and is formulated in terms of two integrals
V = J dx h (x • v), I = J dx hx2,
where V is the virial and I is the moment of inertia.
According the virial theorem, the time-dependent variables V and I obey the equations
dl dG
— = 2V = 2 —, dt dV1
dV dG
(7)
Thus, the quantity
V2
G=--IH
is the integral of motion.
Substituting (8) into (6) and using the Poisson brackets (4), (5), we obtain the following symmetry equations
dev = {v, G} = -V (1 + x •V) v + 2Hx - Idtv, deh = {h, G} = -VV(hx) - Idth, deT = {t, G} = -V(x • V)t - IdtT.
Since these equations are linear and mutually independent, their solutions are easily found and treated as the scale transformation. This transformation changes not only the dependent variables but also the independent ones and looks as follows
, x ., fdt (9)
x
I
v
(10)
Here, for convenience, we do not indicate the functional dependence, assuming in what follows e = 0 and that the primed (new) variables depend on the primed arguments x', t', while the unprimed (old) variables are functions of the unprimed arguments x, t.
Transformation (9), (10) brings Eqs. (1), (2) to the form
dy + (v' ■ V')v' - 2Gx' = ^h'V'r' - V'(h'T'), (11) d'h' + V' • (h'v') = 0, d'T + (v' • V')t' = 0, (12)
where V' = d/dx' and dt = d/dt'. At the same time, it transforms the virial V and the moment of inertia I into the integrals of motion
V' = J dx' h'(x'v') =0, I' = J dx' h'x'2 = 1. (13)
3. Self-similar solutions. We will restrict ourselves in studying only the radially symmetric self-similar solutions which in scaling (primed) coordinates correspond to the state of res
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