**2. Magnitude of some parameters of circumstellar disks**

A typical circumstellar disk is a few hundred AU (astronomical unit, 1 AU = 1.5 10<sup>13</sup> cm) in size. It is mainly composed of hydrogen and helium gas. We consider a vortex in such axially symmetrical viscous accretion disk with effective temperature T and gas density ρ, of almost Keplerian rotation.

The sound speed in gas is estimated by

$$\mathbf{c}\_{\mathbf{s}} = \sqrt{\gamma \mathbf{k} \mathbf{T} / \mathbf{m}\_{\mathbf{H}}} \approx (\gamma \mathbf{T} / \mathbf{100K})^{1/2} \mathbf{k} \mathbf{m} / \mathbf{s},\tag{3}$$

where γ ¼ 1*:*4 is the gas adiabatic index, k is Boltzmann constant, mH is hydrogen atom mass.

In a vertical direction, the gas is in hydrostatic balance with a characteristic scale height:

$$H \sim \frac{c\_s}{\Omega} \approx 0.03 \left(\frac{T}{100K}\right)^{1/2} \left(\frac{M\_\odot}{M}\right)^{1/2} \left(\frac{R}{AU}\right)^{3/2} AU. \tag{4}$$

The thickness-to-radius ratio (aspect ratio) is usually � 1/10 and increases slowly with radius, R. The superficial density of the gas in a disk can be estimated as Σ ≈ 2H ρ.

In "α-model" of accretion disk [29], the expense of gas occurs with a speed dm*=*dt ¼ 3πνΣ, where ν is the kinematic viscosity of gas, ν ¼ α cs H.

The dynamic time scale of a disk is

$$
\tau \sim \frac{1}{\Omega} \approx \frac{1}{5} \left(\frac{M\_{\odot}}{M}\right)^{1/2} \left(\frac{R}{AU}\right)^{3/2} yr \tag{5}
$$

For Keplerian disk, radial momentum equation solution yields to a difference between the speeds of rigid particles and surrounding gas [30]. In a thin gas disk (cs ≪ ΩR), rigid particles drift relative to gas with a speed

$$\frac{\Delta v}{c\_s} \sim \frac{c\_s}{\Omega R} \approx 0.03 \left(\frac{T}{100K}\right)^{1/2} \left(\frac{M\_\odot}{M}\right)^{1/2} \left(\frac{R}{AU}\right)^{1/2}.\tag{6}$$

At cs � 1 km*=*s, typical drift speed is of order 30 m/s. The characteristic scale of drift time [18, 19] almost by two orders surpasses the dynamic time: τ<sup>d</sup> � r*=*Δv � ð Þ <sup>R</sup>*=*A*:*E*:* 102 yr.

For a characteristic time <sup>τ</sup><sup>s</sup> � <sup>Σ</sup>*=*αΩρ <sup>∗</sup> [31], where <sup>ρ</sup> <sup>∗</sup> is the mass density of a particle, dust settled on a midplane of a disk. The characteristic time between collisions of rigid particles among themselves is estimated as τcol � Dρ ∗ *=*Σ ∗ Ω, where D is the diameter of a particle and Σ<sup>∗</sup> is the superficial density of rigid particles in a disk which is more than by two orders less than a disk Σ.

Viscous dissipation and orbital shear limit the sizes of a vortex. Viscous dissipation destroys vortices of sizes less than the viscous scale [32]:

$$L\_{\nu} = \frac{ac\_{\text{s}}H}{v\_{\theta}} \approx 0.003 \left(\frac{a}{0.01}\right) \left(\frac{0.1c\_{\text{s}}}{v\_{\theta}}\right) \left(\frac{M\_{\odot}}{M}\right)^{1/2} \left(\frac{R}{AU}\right)^{3/2} AU,\tag{7}$$

where cs is the sound speed. Evidently, Eq. (9) is the equation of state of disk

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

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

Perturbations of the disk in a rotating with angular velocity Ω<sup>0</sup> � Ωð Þ r0 cylindrical coordinate system (**Figure 2**) are described by 2D hydrodynamic equations<sup>1</sup>

**V**<sup>0</sup> � **e**φð Þ Ω � Ω<sup>0</sup> r,

Φ is the sum of perturbations of gravitational potential U and enthalpy

In Eq. (10) we have used the radial equilibrium condition for the disk:

Taking into account Eq. (9), the continuity Eq. (11) can be written as

2

Taking operator **curl** on Eq. (10) and then by combining the equation of conti-

<sup>1</sup> Here and below the bar indicates the differentiation of equilibrium parameters of the disk on the radial

dH*=*dt þ cs

Ω2

where the velocity was presented in the form of

and the Poisson equation is

nuity, after simple transformation, we obtain

d**v***=*dt þ 2Ω0**e**<sup>z</sup> � **v** þ **e**φvrrΩ<sup>0</sup> þ ∇Φ ¼ 0, (10)

dρ*=*dt þ ρ∇ð Þ¼ **v** 0, (11)

Φ � U þ H, (12)

ΔU ¼ 4πGρ, (14)

r ¼ dΦ0*=*dr*:* (15)

∇**v** ¼ 0*:* (16)

<sup>d</sup>*=*dt <sup>¼</sup> *<sup>∂</sup>=*∂<sup>t</sup> <sup>þ</sup> V0*∂=*r∂<sup>φ</sup> <sup>þ</sup> **<sup>v</sup>**∇; (13)

d*=*dt curlz**v** þ 2Ω þ rΩ<sup>0</sup> f g ½ �*=*ρ ¼ 0*:* (17)

:

substance.

*The local frame of reference 1.*

**Figure 2.**

coordinate r.

**25**

where v<sup>θ</sup> is the rotational speed of a vortex.

The Keplerian shear flow forbids the formation of circular structures with the sizes larger than the shear length scale:

$$\mathcal{L}\_{\text{shear}} = \sqrt{v\_{\theta} \left| \frac{d\Omega}{dR} \right|^{-1}} \approx 0.05 \left( \frac{v\_{\theta}}{0.1c\_{\circ}} \right)^{1/2} \left( \frac{M\_{\odot}}{M} \right)^{1/4} \left( \frac{R}{AU} \right)^{5/4} AU. \tag{8}$$

The vortices, whose sizes surpass Lshear, are extended in an azimuthal direction that allows them to survive longer. In [33] we have shown the possibility of formation in a disk extended in an azimuthal direction three-axis ellipsoidal vortex, with a linear field of circulation, similar to Riemann S ellipsoids [34]. However, in a disk round the central star of solar mass, at distance 30 AU, the vortex of characteristic speed of rotation, 0.01cs, can be circular and have the size of an order of �1 AU.

In a gas disk, drag force on rigid particles from gas is exposed, which, depending on size of a particle, is expressed either by Stokes or Epstein's formula (see, e.g., [25]).

Here our main results obtained by investigations of the linear and nonlinear perturbation equations of differentially rotating gravitating gaseous disk in geostrophic and post-geostrophic approximations are presented [35], as well as the results on formation of planetesimals by Burgers vortex in a protoplanetary disk [26].
