**3. Model and basic equations for linear perturbations**

Consider at first a gravitating pure gas disk of mass density ρð Þr , which rotates with angular velocity Ωð Þr around the z-axis. Explore 2D perturbations in plane of the disk, ignoring its vertical structure. Present any characteristic functions of the disk as f0ð Þþ r f r, ð Þ φ, t , where f0ð Þr describes the equilibrium state and f r, ð Þ φ, t is a small but finite perturbation.

We will consider isentropic perturbations (S = constant) and therefore enthalpy H(S, P) = H(P),

where P is the pressure and

$$\mathbf{d}\mathbf{H} = \mathbf{d}\mathbf{P}/\mathfrak{p} = \mathbf{c}\_{\mathfrak{s}}{}^{2}\mathbf{d}\mathfrak{p}/\mathfrak{p},\tag{9}$$

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

**Figure 2.** *The local frame of reference 1.*

Δ*v cs*

*Vortex Dynamics Theories and Applications*

ð Þ <sup>R</sup>*=*A*:*E*:* 102

yr.

*<sup>L</sup><sup>ν</sup>* <sup>¼</sup> *<sup>α</sup>csH vθ*

sizes larger than the shear length scale:

s

Lshear ¼

a small but finite perturbation.

where P is the pressure and

H(S, P) = H(P),

**24**

� *cs*

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

particles in a disk which is more than by two orders less than a disk Σ.

0*:*01 � � 0*:*1*cs*

<sup>≈</sup> <sup>0</sup>*:*<sup>05</sup> *<sup>v</sup><sup>θ</sup>*

tion destroys vortices of sizes less than the viscous scale [32]:

<sup>≈</sup> <sup>0</sup>*:*<sup>003</sup> *<sup>α</sup>*

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

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi *vθ d*Ω *dR* � � � �

� � � � �1

**3. Model and basic equations for linear perturbations**

100*K*

� �1*=*<sup>2</sup> *M*<sup>⊙</sup>

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 �

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

Viscous dissipation and orbital shear limit the sizes of a vortex. Viscous dissipa-

*vθ* � � *M*<sup>⊙</sup>

The Keplerian shear flow forbids the formation of circular structures with the

0*:*1*cs*

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].

Consider at first a gravitating pure gas disk of mass density ρð Þr , which rotates with angular velocity Ωð Þr around the z-axis. Explore 2D perturbations in plane of the disk, ignoring its vertical structure. Present any characteristic functions of the disk as f0ð Þþ r f r, ð Þ φ, t , where f0ð Þr describes the equilibrium state and f r, ð Þ φ, t is

We will consider isentropic perturbations (S = constant) and therefore enthalpy

2

dρ*=*ρ, (9)

dH ¼ dP*=*ρ ¼ cs

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

*M*

*M*

� �<sup>1</sup>*=*<sup>4</sup> *R*

� �<sup>1</sup>*=*<sup>2</sup> *R*

*AU* � �<sup>3</sup>*=*<sup>2</sup>

> *AU* � �<sup>5</sup>*=*<sup>4</sup>

*M*

� �1*=*<sup>2</sup> *R*

*AU* � �1*=*<sup>2</sup>

*:* (6)

*AU*, (7)

*AU:* (8)

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

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> :

$$\mathbf{d}\mathbf{v}/\mathbf{d}\mathbf{t} + 2\Omega\_0 \mathbf{e}\_\mathbf{z} \times \mathbf{v} + \mathbf{e}\_\mathbf{q} \mathbf{v}\_\mathbf{r} \Omega^\prime + \nabla \Phi = \mathbf{0},\tag{10}$$

$$\mathbf{d}\rho/\mathbf{dt} + \rho \nabla(\mathbf{v}) = \mathbf{0},\tag{11}$$

where the velocity was presented in the form of

$$\mathbf{V}\_0 \equiv \mathbf{e}\_{\boldsymbol{\upphi}} (\boldsymbol{\Omega} - \boldsymbol{\Omega}\_0) \mathbf{r},$$

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

$$
\Phi \equiv \mathbf{U} + \mathbf{H},
\tag{12}
$$

$$\mathbf{d}/\mathbf{dt} = \partial/\partial\mathbf{t} + \mathbf{V}\_0 \partial/\mathbf{r} \partial\mathbf{q} + \mathbf{v} \nabla;\tag{13}$$

and the Poisson equation is

$$
\Delta \mathbf{U} = 4 \pi \mathbf{G} \rho,\tag{14}
$$

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

$$
\boldsymbol{\Omega}^2 \mathbf{r} = \mathbf{d} \boldsymbol{\Phi}\_0 / \mathbf{d} \mathbf{r}.\tag{15}
$$

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

$$\mathbf{d}\mathbf{H}/\mathbf{dt} + \mathbf{c}\_{\ast}{}^{2}\nabla\mathbf{v} = \mathbf{0}.\tag{16}$$

Taking operator **curl** on Eq. (10) and then by combining the equation of continuity, after simple transformation, we obtain

$$\mathbf{d} / \mathbf{d} \mathbf{t} \{ [\mathbf{curl}\_z \mathbf{v} + 2\Omega + \mathbf{r} \Omega'] / \mathbf{p} \} = \mathbf{0}. \tag{17}$$

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

The expression in the curly brackets in this equation is a generalized vortencity. The equation shows that for 2D isentropic perturbations, generalized vortencity is conserved along the current lines. So for stationary perturbations, generalized vortencity is an arbitrary function of ψ:

$$(\mathbf{curl}\_{\mathbf{z}}\mathbf{v} + \mathbf{2}\boldsymbol{\Omega} + \mathbf{r}\boldsymbol{\Omega}')/\boldsymbol{\rho} = \mathbf{B}(\boldsymbol{\Psi}).\tag{18}$$

where kR ¼ Ω0*=*cs is the Rossby wave number, κ<sup>0</sup>

β � 3Ω<sup>0</sup>

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

<sup>0</sup> � κ<sup>0</sup> 2 ρ0

If to take the relationship of density perturbations with perturbations of gravi-

<sup>ρ</sup> x, y � � <sup>¼</sup> <sup>Δ</sup>U x, y � �*=*4πð Þ <sup>G</sup> ,

2

ΔU*=*2Ω<sup>0</sup> ω<sup>J</sup>

<sup>2</sup> <sup>¼</sup> k2 *=*kJ 2

*=*kJ

� �<sup>h</sup> <sup>þ</sup> xb � �*:* (29)

� �Δφ <sup>þ</sup> xb n o*:* (30)

0, when the vortencity is constant and is equal to Γ*=*ρ0π*a*<sup>2</sup>

<sup>2</sup> <sup>þ</sup> <sup>x</sup><sup>β</sup> � �, (27)

�<sup>1</sup> Δψ � <sup>κ</sup><sup>0</sup>

<sup>2</sup> � <sup>4</sup>πGρ<sup>0</sup> is the square of the Jeans frequency. The order of magnitude of |H/U| can be estimated using the definition

Φ.

j j <sup>H</sup>*=*<sup>U</sup> <sup>≈</sup> k2

cs 2 *=*ω<sup>J</sup>

<sup>0</sup> <sup>Δ</sup><sup>h</sup> � <sup>κ</sup><sup>2</sup>

<sup>0</sup> <sup>1</sup> � <sup>κ</sup><sup>2</sup>

By selection of function B, we can explore the stationary vortex solutions of

Let's take a look at the simplest case of uniformly rotating disk of homogeneous

*<sup>R</sup>* <sup>¼</sup> <sup>Γ</sup>*=πα*2, *<sup>R</sup>*<sup>≤</sup> *<sup>a</sup>* 0, *R*≥ *a*

*<sup>K</sup>*1ð Þ*<sup>s</sup> <sup>I</sup>*0ðÞþ*<sup>s</sup> <sup>I</sup>*1ð Þ*<sup>s</sup> <sup>K</sup>*0ð Þ*<sup>s</sup>* , *<sup>R</sup>*≤*<sup>a</sup>*

*<sup>K</sup>*1ð Þ*<sup>s</sup> <sup>I</sup>*0ðÞþ*<sup>s</sup> <sup>I</sup>*1ð Þ*<sup>s</sup> <sup>K</sup>*0ð Þ*<sup>s</sup>* , *<sup>R</sup>*≥*<sup>a</sup>*

which gives a circularly symmetric solution for relative perturbed density of mass

<sup>1</sup> � *<sup>K</sup>*1ð Þ*<sup>s</sup> <sup>I</sup>*0ð Þ *sR=<sup>a</sup>*

*I*1ð Þ*s K*0ð Þ *sR=a*

where Γ is the velocity circulation. We assume that the velocity circulation Γ differs from zero only in a circle of radius *a*ð Þ ≪ r0 around point O. Using now the polar coordinates R, ð Þθ : x ¼ Rcosθ, y ¼ Rsinθ (**Figure 2**), the equation for disturbances

�

Limit Hj j ≪ j j U corresponds to the large-scale perturbations �λ ≫ λJ. Then

where kJ ¼ ωJ*=*cs is the Jeans wavenumber. Eq. (28) shows that the case Hj j ≫ j j U

0k2 R*=*Ω<sup>2</sup> 0

0*=*ω<sup>2</sup> J

square of the epicyclical frequency, and

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

tational potential using Poisson equation

<sup>2</sup> <sup>¼</sup> dP0*=*dρ<sup>0</sup> ð Þ and j j ΔΦ <sup>¼</sup> k2

where ω<sup>J</sup>

Eqs. (29) and (30).

density, <sup>β</sup> <sup>¼</sup> 0, <sup>κ</sup><sup>2</sup>

(29) can be written as

**27**

cs

instead of Eq. (25), we obtain the equation

Bð Þ¼ ψ ρ<sup>0</sup>

where k is wavenumber of perturbations

describes the small-scale disturbances: matching k<sup>2</sup>

B hð Þ¼ <sup>ρ</sup>�<sup>1</sup>

<sup>B</sup>ð Þ¼ <sup>ϕ</sup> <sup>ρ</sup>�<sup>1</sup>

<sup>Δ</sup>*<sup>h</sup>* � <sup>4</sup>*k*<sup>2</sup>

8 >>>><

>>>>:

ψ ¼ H*=*2Ω<sup>0</sup> � h, and Eq. (25) takes the form

ψ ¼ U*=*2Ω<sup>0</sup> � ϕ, and Eq. (27) turns into

<sup>0</sup> <sup>¼</sup> <sup>4</sup>Ω<sup>2</sup>

*ρ ρ*0 ¼ � <sup>Γ</sup> *πascs* <sup>2</sup> <sup>¼</sup> <sup>2</sup>Ω<sup>0</sup> <sup>2</sup>Ω<sup>0</sup> <sup>þ</sup> r0Ω<sup>0</sup>

<sup>0</sup>*=*2Ω0ρ0*:* (26)

, (28)

<sup>2</sup> ≫ 1, orλ ≪ λJ. In this case

(31)

(32)

0 � � is the

In a uniformly rotating (Ω ¼ const*:*) gravitating disk, no drifting stationary vortex solution can be obtained without specifying the function Bð Þ ψ , because Eq. (18) can be represented as a Jacobean Jf g ψ, curl ð Þ <sup>z</sup>**v** þ 2Ω *=*ρ ¼ 0, which satisfies the arbitrary circularly symmetric vortex disturbance around point O.
