**10. The thickness of disk in nuclear area of Burgers vortex**

So far we have considered the behavior of a whirlwind in a disk plane. However the whirlwind of Burgers is in 3D formation. We will discuss now a question on a thickness of a disk in the area where the Burgers vortex is located. For this purpose we will address a z-projection of the Navier–Stokes Eq. (72). Integrating this equation taking into account the formula for speed vz, we will receive dependence enthalpy from the z coordinate:

$$\mathbf{h}\left(\mathbf{z}\right) = \mathbf{c}\_{\ast 0}{}^2 - \left(4\mathbf{A}^2 + \Omega\_0{}^2\right) \mathbf{z}^2 / 2\mathbf{z}$$

where cs0 is a sound speed at the vortex center (enthalpy, h0 ¼ cs0 2 , at the center of vortex is estimated by Clapeyron equation) and Ω<sup>0</sup> is an angular speed of rotation of local frame of reference. Whence we obtain half thickness of a disk at the kernel area of a whirlwind:

$$\mathbf{z}\_{\rm 0} = \mathbf{c}\_{\rm 0} \left( 2 \mathbf{A}^2 + \Omega\_0^{\rm 2} / 2 \right)^{-1/2}. \tag{88}$$

The question arises whether the disk thickness in area of vortex localization changed. On radius of R0 the half-thickness of Keplerian disk from (4) is of order zK ffi cs0*=*2Ω0. Therefore the relative thickening

$$\frac{\Delta z}{Z\_K} \equiv \frac{Z\_0}{Z\_K} - 1 = \frac{2\sqrt{2}}{\sqrt{\left(1 + 4A^2/\Omega\_0^2\right)}} - 1,\tag{89}$$

is positive if A <1*:*3Ω0. This condition is carried out in all areas of a typical protoplanetary disk. Therefore, the disk in the area of localization of a whirlwind of Burgers is thicker.

## **11. Discussion and conclusion**

First let's pay attention to the nontrivial structure of monopoly and dipole vortices in a rotating and gravitating pure gas disk. Monopole vortices (33) with mass distribution (32) are localized formations and can have positive and negative velocity circulation, and Г*:*Г>0vortex, characterized by low pressure, has negative excess mass density of substance, in contrast of Г< 0 vortex of higher pressure, with the positive excess mass density (see **Figure 3**).

More interesting are properties of solitary dipole vortex - modon (60), (61) with mass distribution (65) in short-scale and long-scale limits. There exist two types of mass distribution in dipole vortex. Anti-symmetrically located one almost round condensation and one rarefaction (**Figure 6**) characterizes the first type. The second type is characterized by the anti-symmetrical located two condensations and two rarefactions, and second condensation-rarefaction pair has sickle-form

Therefore, during an order of <sup>10</sup><sup>6</sup> year, for meter-sized rigid particles, in the

vortex trunk the mass amount comparable with mass of Venus accumulates. Finally, note that the disk in the Burgers vortex localization area is thicker.

*Vortices in Rotating and Gravitating Gas Disk and in a Protoplanetary Disk*

*DOI: http://dx.doi.org/10.5772/intechopen.92028*

**Author details**

**39**

Martin G. Abrahamyan

Yerevan Haybusak University, Armenia

provided the original work is properly cited.

\*Address all correspondence to: haybusaksci@gmail.com

© 2020 The Author(s). Licensee IntechOpen. This chapter is distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/ by/3.0), which permits unrestricted use, distribution, and reproduction in any medium,

**Figure 10.** *The isodense picture of the galaxy Markaryan 266 with two nuclei, rotating in the opposite direction [36].*

(**Figure 7**). Circulation of substance in different parts of modon occurs in opposite direction (**Figure 4**)!

Now it is difficult to judge about a way of evolution of these structures, for example, whether monopole vortices lead to the formation of planets in circumstellar disks, or the formation of stars or clouds in the galactic gas disk? Or, if it could transformed the dipole vortices to well-known double objects, such as double stars, double nuclei in galaxies (as Mrk 266 [36], **Figure 10**), as well as in giant molecular clouds, or a planet with a companion in circumstellar disk, or not?

As for dusty protoplanetary disks, long-lived anticyclonic vortical structures can capture the �10 cm to meter-sized particles and grow up them into planetesimals. Let's estimate an order of magnitudes of time (86), and mass (87) for planetesimal formation by Burgers vortex for a model of a disk of radius 30 AU and mass 0*:*5 M<sup>⊙</sup> round a star of solar mass: M ≈ M⊙. Taking R0 ¼ 20 AU we will obtain estimations Ω<sup>0</sup> ≈ 8 � <sup>10</sup>�9s�<sup>1</sup> and <sup>Σ</sup> � 1600 g*=*cm2. For a typical protoplanetary disk at considering distance the vertical scale height is of order H ≈ 108km and sound speed cs ≈ HΩ<sup>0</sup> ≈ 0*:*8 km*=*s.

Let the maximum rotation speed of a vortex be �10 m/s at distance r0 <sup>≈</sup> <sup>10</sup>10m from its center, and converging speed of a stream be vr ¼ A � r0 ≈ 5 m*=*s. Then we will have

$$\mathbf{a} \approx \mathbf{1} \mathbf{0}^{-9} \mathbf{s}^{-1}, \mathbf{A} \approx \mathbf{5} \cdot \mathbf{1} \mathbf{0}^{-10} \mathbf{s}^{-1}.$$

The condition (85) is carried out with a large supply for protoplanetary disks. The molecular viscosity of gas, estimated by the formula *ν* � λ*cs*, in which λ is the mean free path of molecules, *cs* is the speed of a sound, does not play an appreciable role in processes of a protoplanetary disk. For this reason, the "α-disk" model [29] is used, in which turbulent viscosity is represented by the expression <sup>ν</sup> � <sup>α</sup>csH <sup>≈</sup> <sup>α</sup>H2 Ω0. The dimensionless parameter α is constant value of an order α � 10�<sup>2</sup> . The scale of viscous length thus makes L<sup>ν</sup> ≈ 106km, so Burgers vortex of big sizes cannot be destroyed by viscosity. Keplerian shear length makes Lshear ≈ 6 � 109km. Hence, vortices with the sizes reff < Lshear can have circular form.

Taking ρ ∗ *=*ρ ≈ 10<sup>10</sup> in a midplane of a disk, using in (87) and (86) also the average value for viscosity from stability condition (85), we will receive the estimations:

$$\mathbf{M\_p} \approx \mathbf{10^{28}g}; \boldsymbol{\tau} \sim \mathbf{3} \cdot \mathbf{10^6} (\mathbf{m/D}) \text{yrs.}$$

*Vortices in Rotating and Gravitating Gas Disk and in a Protoplanetary Disk DOI: http://dx.doi.org/10.5772/intechopen.92028*

Therefore, during an order of <sup>10</sup><sup>6</sup> year, for meter-sized rigid particles, in the vortex trunk the mass amount comparable with mass of Venus accumulates. Finally, note that the disk in the Burgers vortex localization area is thicker.
