**1. Introduction**

Nonlinear equations describing dynamics of 2D vortices are important in the physics of the ocean and the atmosphere, in plasma physics, and in astrophysics. The same type of nonlinear equations describes these vortical structures. In fluid dynamics, Hasegawa-Mima equation is well-known [1].

$$\frac{\partial}{\partial t}(\mathbf{1} - \Delta)\boldsymbol{\upmu} - \boldsymbol{v}\_0 \frac{\partial \boldsymbol{\upmu}}{\partial \boldsymbol{\upmu}} - (\mathbf{e}\_{\boldsymbol{\upmu}} \times \nabla \boldsymbol{\upmu})\nabla \Delta \boldsymbol{\upmu} = \mathbf{0},\tag{1}$$

vortex trunk is almost uniform, then falls down under hyperbolic law, and at distance reff ¼ 4*:*5r0 (effective radius of vortex) makes 1/3 of the maximum value of v<sup>θ</sup> (**Figure 1**). The asymptotic behavior of Burgers vortex in small and big distances

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

km*=*s, (3)

*AU:* (4)

*yr* (5)

cs <sup>¼</sup> <sup>√</sup>γkT*=*mH <sup>≈</sup> ð Þ <sup>γ</sup>T*=*100K <sup>1</sup>*=*<sup>2</sup>

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

<sup>1</sup>*=*<sup>2</sup> *M*<sup>⊙</sup>

*M*<sup>⊙</sup> *M*

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

<sup>1</sup>*=*<sup>2</sup> *R*

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

100*K*

dm*=*dt ¼ 3πνΣ, where ν is the kinematic viscosity of gas, ν ¼ α cs H.

<sup>Ω</sup> <sup>≈</sup> <sup>1</sup> 5

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

*M*

<sup>1</sup>*=*<sup>2</sup> *R*

*AU* <sup>3</sup>*=*<sup>2</sup>

*AU* <sup>3</sup>*=*<sup>2</sup>

from the vortex center represents the Rankin vortex [27, 28].

T and gas density ρ, of almost Keplerian rotation. The sound speed in gas is estimated by

*Rotational velocity profiles of Burgers and Rankin vortices.*

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

*<sup>H</sup>* � *cs*

The dynamic time scale of a disk is

<sup>Ω</sup> <sup>≈</sup> <sup>0</sup>*:*<sup>03</sup> *<sup>T</sup>*

*<sup>τ</sup>* � <sup>1</sup>

(cs ≪ ΩR), rigid particles drift relative to gas with a speed

hydrogen atom mass.

height:

**23**

**Figure 1.**

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

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

which describes the nonlinear Rossby waves in the atmosphere [2] and drift nonlinear waves in plasma [3]. Here *ψ*ð Þ *x*, *y*, *z* is a stream function: *v* ¼ *ez* � ∇*ψ*. In plasma physics *ψ* is the electric potential, and constant *v0* is defined by equilibrium density gradient.

The exact solution of the equation, describing a stationary solitary dipole vortex (modon) drifting along the y-axis on rotating shallow water, was obtained in [4]. The same type of solutions later received a large number of similar equations [5–10].

Nonlinear vortex disturbances of uniformly rotating gravitating gaseous disk were considered in [9]. For short-scale (much smaller than the Jeans wavelength: λ ≪ λJ) and long-scale λ ≫ λ<sup>J</sup> perturbations, nonlinear equation turns into Eq. (1).

IR, submillimeter, and centimeter radiation of protoplanetary disk analyses shows that vortices serve as incubators for the growth of dust particles and formation of planetesimals [11–14]. The initial stage of growth probably proceeds through the nucleation of submicron-sized dust grains from the primordial nebula, which then forms the monomers of fractal dust aggregates up to �1 mm to �10 cm for characteristic time of an order of 103 years [15, 16]. The best astrophysical evidence for grain growth to specified sizes is the detection of 3.5 cm dust emission from the face-on disk of radius 225 AU round classical T Tauri star TW Hya [17]. When the planetesimals reached a size of about 1 km, they began to attract other smaller bodies due to their gravity.

In models of protoplanetary disks, gas practically moves on sub-Keplerian speeds. Rigid particles, under the action of a head wind drag, lose the angular momentum and energy. As a result, the �10 cm to meter-sized particles drift to the central star for hundreds of years, that is, much less than the lifetime of a disk which makes several millions of years [18, 19].

Long-lived vortical structures in gas disk are a possible way to concentrate the �10 cm to meter-sized particles and to grow up them in planetesimal. Similar effect of vortices on the Earth was observed in special laboratories and also in the ocean [20].

In some areas of the stratified protoplanetary disks, the current has a 2D turbulent character. An attractive feature of such hydrodynamic current consists in the fact that in it, through a background of small whirlpools, long-living vortices will spontaneously be formed without requirement of special initial conditions [21–23]. In laboratory experiments [24, 25], formation of Burgers vortex, which will be considered here, is often observed in 2D turbulent flows. Anticyclonic vortices in a protoplanetary disk merge with each other and amplify, while cyclonic ones are destroyed by a shear flow [26].

In cylindrical system of coordinates (r, θ, z), the Burgers vortex is defined as

$$\mathbf{v}\_{\mathbf{r}} = -\mathbf{A}\mathbf{r}, \mathbf{v}\_{0} = \alpha \mathbf{r}\_{0}^{\ 2} \left[ \mathbf{1} - \exp\left(-\mathbf{r}^{2}/\mathbf{r}\_{0}^{2}\right) \right] / \mathbf{r}, \quad \mathbf{v}\_{\mathbf{z}} = 2\mathbf{A}\mathbf{z}.\tag{2}$$

This is a vortex with a converging stream of substance to its center with gradient –Α, ω and r0 as the circulation and the size of a trunk of a vortex. Rotation of a

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

**Figure 1.** *Rotational velocity profiles of Burgers and Rankin vortices.*

**1. Introduction**

density gradient.

bodies due to their gravity.

destroyed by a shear flow [26].

ocean [20].

**22**

which makes several millions of years [18, 19].

vr ¼ �Αr, v<sup>θ</sup> ¼ ωr0

Nonlinear equations describing dynamics of 2D vortices are important in the physics of the ocean and the atmosphere, in plasma physics, and in astrophysics. The same type of nonlinear equations describes these vortical structures. In fluid

*<sup>∂</sup><sup>y</sup>* � ð Þ *<sup>e</sup><sup>z</sup>* � <sup>∇</sup>*<sup>ψ</sup>* <sup>∇</sup>Δ*<sup>ψ</sup>* <sup>¼</sup> *<sup>0</sup>*, (1)

*∂ψ*

which describes the nonlinear Rossby waves in the atmosphere [2] and drift nonlinear waves in plasma [3]. Here *ψ*ð Þ *x*, *y*, *z* is a stream function: *v* ¼ *ez* � ∇*ψ*. In plasma physics *ψ* is the electric potential, and constant *v0* is defined by equilibrium

The exact solution of the equation, describing a stationary solitary dipole vortex (modon) drifting along the y-axis on rotating shallow water, was obtained in [4]. The same type of solutions later received a large number of similar equations [5–10]. Nonlinear vortex disturbances of uniformly rotating gravitating gaseous disk were considered in [9]. For short-scale (much smaller than the Jeans wavelength: λ ≪ λJ) and long-scale λ ≫ λ<sup>J</sup> perturbations, nonlinear equation turns into Eq. (1). IR, submillimeter, and centimeter radiation of protoplanetary disk analyses shows that vortices serve as incubators for the growth of dust particles and formation of planetesimals [11–14]. The initial stage of growth probably proceeds through the nucleation of submicron-sized dust grains from the primordial nebula, which then forms the monomers of fractal dust aggregates up to �1 mm to �10 cm for characteristic time of an order of 103 years [15, 16]. The best astrophysical evidence for grain growth to specified sizes is the detection of 3.5 cm dust emission from the face-on disk of radius 225 AU round classical T Tauri star TW Hya [17]. When the planetesimals reached a size of about 1 km, they began to attract other smaller

In models of protoplanetary disks, gas practically moves on sub-Keplerian speeds. Rigid particles, under the action of a head wind drag, lose the angular momentum and energy. As a result, the �10 cm to meter-sized particles drift to the central star for hundreds of years, that is, much less than the lifetime of a disk

Long-lived vortical structures in gas disk are a possible way to concentrate the

In some areas of the stratified protoplanetary disks, the current has a 2D turbulent character. An attractive feature of such hydrodynamic current consists in the fact that in it, through a background of small whirlpools, long-living vortices will spontaneously be formed without requirement of special initial conditions [21–23]. In laboratory experiments [24, 25], formation of Burgers vortex, which will be considered here, is often observed in 2D turbulent flows. Anticyclonic vortices in a protoplanetary disk merge with each other and amplify, while cyclonic ones are

In cylindrical system of coordinates (r, θ, z), the Burgers vortex is defined as

2 *=*r 2 0

*<sup>=</sup>*r, vz <sup>¼</sup> <sup>2</sup>Αz*:* (2)

<sup>2</sup> <sup>1</sup> � exp �<sup>r</sup>

–Α, ω and r0 as the circulation and the size of a trunk of a vortex. Rotation of a

This is a vortex with a converging stream of substance to its center with gradient

�10 cm to meter-sized particles and to grow up them in planetesimal. Similar effect of vortices on the Earth was observed in special laboratories and also in the

dynamics, Hasegawa-Mima equation is well-known [1].

ð Þ 1 � Δ *ψ* � *v*<sup>0</sup>

*∂ ∂t*

*Vortex Dynamics Theories and Applications*

vortex trunk is almost uniform, then falls down under hyperbolic law, and at distance reff ¼ 4*:*5r0 (effective radius of vortex) makes 1/3 of the maximum value of v<sup>θ</sup> (**Figure 1**). The asymptotic behavior of Burgers vortex in small and big distances from the vortex center represents the Rankin vortex [27, 28].
