**4. Vortices in the geostrophic approximation**

This approach assumes that Coriolis forces and gravity balance the pressure gradient in the disk.

Then from the equation of motion (17), we get the perturbed geostrophic velocity:

$$\mathbf{v\_G} = (\mathbf{1}/2\Omega\_0)\mathbf{e\_z} \times \nabla \Phi. \tag{19}$$

Using the last, Eq. (18) takes the form

$$(\mathbf{1}/2\mathbf{\Omega}\_0\mathbf{\rho})\,\Delta\Phi+(\mathbf{2}\mathbf{\Omega}+\mathbf{r}\mathbf{\Omega}\_0\mathbf{'})/\mathbf{\rho}=\mathbf{B}(\boldsymbol{\Psi}).\tag{20}$$

In further analysis of this topic, we will introduce the local Cartesian coordinate system (X,Y) such that (**Figure 2**)

$$
\partial / \partial \mathbf{x} = \partial / \partial \mathbf{r}, \partial / \partial \mathbf{y} = \partial / \mathbf{r} \partial \mathbf{q}, \tag{21}
$$

and will explore the vortical perturbations around a point O in a linear approximation.

The stream function ψ x, y of perturbed velocity (19) is expressed through perturbations Φ by the following formula:

$$
\Psi = \Phi / 2\Omega\_0. \tag{22}
$$

Imagine around a point O function ρð Þ x and Ωð Þ x in the form of

$$
\rho(\mathbf{x}) = \rho\_0(\mathbf{r}\_0) + \mathbf{x}\rho\_0' + \rho(\mathbf{x}, \mathbf{y}); \Omega(\mathbf{x}) = \Omega\_0 + \mathbf{x}\Omega\_0'. \tag{23}
$$

In this case V0 ≈ r0Ω<sup>0</sup> 0x.

Perturbations of density and enthalpy (9) in linear approach are connected by the following formula:

$$
\rho(\mathbf{x}, \mathbf{y}) = \rho\_0 \mathbf{H}(\mathbf{x}, \mathbf{y}) / \mathbf{c}\_\*^2,\tag{24}
$$

Then Eq. (13) with an accuracy to a constant term will be in the form of

$$\mathbf{B}(\boldsymbol{\Psi}) = \rho\_0 \, ^{\cdot 1} \{ \Delta \boldsymbol{\Psi} \text{-} \mathbf{\kappa}\_0 \, ^2 \mathbf{k}\_{\mathbb{R}} \, ^2 \mathbf{H} / 2 \mathbf{\Omega}\_0 \mathbf{\Omega}^2) + \mathbf{x} \boldsymbol{\mathfrak{\beta}} \}, \tag{25}$$

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

where kR ¼ Ω0*=*cs is the Rossby wave number, κ<sup>0</sup> <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> 0 � � is the square of the epicyclical frequency, and

$$
\mathfrak{P} \equiv \mathfrak{W}\_0' - \kappa\_0^2 \mathfrak{p}\_0' / 2 \,\mathfrak{Q}\_0 \mathfrak{p}\_0. \tag{26}
$$

If to take the relationship of density perturbations with perturbations of gravitational potential using Poisson equation

$$
\rho(\mathbf{x}, \mathbf{y}) = \Delta \mathbf{U}(\mathbf{x}, \mathbf{y}) / 4\pi(\mathbf{G}),
$$

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

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

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

This approach assumes that Coriolis forces and gravity balance the pressure

Then from the equation of motion (17), we get the perturbed geostrophic

In further analysis of this topic, we will introduce the local Cartesian coordinate

ð Þ 1*=*2Ω0ρ ΔΦ þ 2Ω þ rΩ<sup>0</sup>

and will explore the vortical perturbations around a point O in a linear

Imagine around a point O function ρð Þ x and Ωð Þ x in the form of

ρ x, y

<sup>0</sup> þ ρ x, y

Perturbations of density and enthalpy (9) in linear approach are connected by

*=*cs

H*=*2Ω0Ω<sup>2</sup>

<sup>¼</sup> <sup>ρ</sup>0H x, y

Then Eq. (13) with an accuracy to a constant term will be in the form of

2 kR 2

‐<sup>1</sup> Δψ‐κ<sup>0</sup>

ρð Þ¼ x ρ0ð Þþ r0 xρ<sup>0</sup>

0x.

Bð Þ¼ ψ ρ<sup>0</sup>

the arbitrary circularly symmetric vortex disturbance around point O.

**4. Vortices in the geostrophic approximation**

Using the last, Eq. (18) takes the form

system (X,Y) such that (**Figure 2**)

The stream function ψ x, y

In this case V0 ≈ r0Ω<sup>0</sup>

the following formula:

**26**

perturbations Φ by the following formula:

curlz**v** þ 2Ω þ rΩ<sup>0</sup> ð Þ*=*ρ ¼ Bð Þ ψ *:* (18)

**v**<sup>G</sup> ¼ ð Þ 1*=*2Ω<sup>0</sup> **e**<sup>z</sup> � ∇Φ*:* (19)

*<sup>∂</sup>=*∂<sup>x</sup> <sup>¼</sup> *<sup>∂</sup>=*∂r, *<sup>∂</sup>=*∂<sup>y</sup> <sup>¼</sup> *<sup>∂</sup>=*r∂φ, (21)

of perturbed velocity (19) is expressed through

; <sup>Ω</sup>ð Þ¼ <sup>x</sup> <sup>Ω</sup><sup>0</sup> <sup>þ</sup> <sup>x</sup>Ω<sup>0</sup>

2

Þ þ <sup>x</sup><sup>β</sup> , (25)

ψ ¼ Φ*=*2Ω0*:* (22)

<sup>0</sup>*:* (23)

, (24)

<sup>0</sup> ð Þ*=*ρ ¼ Bð Þ ψ *:* (20)

vortencity is an arbitrary function of ψ:

*Vortex Dynamics Theories and Applications*

gradient in the disk.

approximation.

velocity:

$$\mathbf{B}(\boldsymbol{\Psi}) = \rho\_0^{-1} \{ \Delta \boldsymbol{\Psi} - \kappa\_0^{-2} \Delta \mathbf{U} / 2 \Omega\_0 \left. \alpha \right\|^2 + \mathbf{x} \boldsymbol{\mathfrak{P}} \}, \tag{27}$$

where ω<sup>J</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 cs <sup>2</sup> <sup>¼</sup> dP0*=*dρ<sup>0</sup> ð Þ and j j ΔΦ <sup>¼</sup> k2 Φ.

where k is wavenumber of perturbations

$$|\mathbf{H}/\mathbf{U}| \approx \mathbf{k}^2 \mathbf{c}\_s^2 / \alpha \mathbf{l}^2 = \mathbf{k}^2 / \mathbf{k} \mathbf{l}^2,\tag{28}$$

where kJ ¼ ωJ*=*cs is the Jeans wavenumber. Eq. (28) shows that the case Hj j ≫ j j U describes the small-scale disturbances: matching k<sup>2</sup> *=*kJ <sup>2</sup> ≫ 1, orλ ≪ λJ. In this case ψ ¼ H*=*2Ω<sup>0</sup> � h, and Eq. (25) takes the form

$$\mathbf{B(h)} = \rho\_0^{-1} \{ \Delta \mathbf{h} - \left( \kappa\_0^2 \mathbf{k}\_\mathbb{R}^2 / \Omega\_0^2 \right) \mathbf{h} + \mathbf{x} \mathbf{b} \}. \tag{29}$$

Limit Hj j ≪ j j U corresponds to the large-scale perturbations �λ ≫ λJ. Then ψ ¼ U*=*2Ω<sup>0</sup> � ϕ, and Eq. (27) turns into

$$\mathbf{B}(\boldsymbol{\Phi}) = \rho\_0^{-1} \left\{ \left( \mathbf{1} - \mathbf{x}\_0^2 / \alpha\_{\parallel}^2 \right) \Delta \boldsymbol{\upmu} + \mathbf{x} \boldsymbol{\mathbf{b}} \right\}. \tag{30}$$

By selection of function B, we can explore the stationary vortex solutions of Eqs. (29) and (30).

Let's take a look at the simplest case of uniformly rotating disk of homogeneous density, <sup>β</sup> <sup>¼</sup> 0, <sup>κ</sup><sup>2</sup> <sup>0</sup> <sup>¼</sup> <sup>4</sup>Ω<sup>2</sup> 0, when the vortencity is constant and is equal to Γ*=*ρ0π*a*<sup>2</sup> 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 (29) can be written as

$$
\Delta h - 4k\_R^2 = \begin{cases}
\Gamma/\pi \alpha^2, & R \le a \\
0, & R \ge a
\end{cases} \tag{31}
$$

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

$$\frac{\rho}{\rho\_0} = -\frac{\Gamma}{\pi a \kappa c\_s} \begin{cases} 1 - \frac{K\_1(s) I\_0(sR/a)}{K\_1(s) I\_0(s) + I\_1(s) K\_0(s)}, & R \le a \\\\ \frac{I\_1(s) K\_0(sR/a)}{K\_1(s) I\_0(s) + I\_1(s) K\_0(s)}, & R \ge a \end{cases} \tag{32}$$

**Figure 3.**

*3D image of relative density perturbations of whirlwind in the range* 0 ≤*R=a*<1*:*4 *for* 2*а*kR ¼ 20*.*

where s � 2*a*kR. This is a monopole vortex with the following perturbed velocity field:

$$v\_{\theta} = \frac{\Gamma}{\pi a s} \begin{cases} \frac{K\_1(s) I\_1(sR/a)}{K\_1(s) I\_0(s) + I\_1(s) K\_0(s)}, & R \le a \\\\ \frac{I\_1(s) K\_1(sR/a)}{K\_1(s) I\_0(s) + I\_1(s) K\_0(s)}, & R \ge a. \end{cases} \tag{33}$$

**v**<sup>I</sup> � ð Þ 1*=*2Ω<sup>0</sup> **e**<sup>z</sup> � d**v***=*dt*:* (36)

<sup>2</sup> **ez** � <sup>d</sup>½ � **<sup>e</sup>**<sup>z</sup> � <sup>∇</sup><sup>Φ</sup> dt*:* (37)

<sup>L</sup> � *<sup>∂</sup>=*∂<sup>t</sup> <sup>þ</sup> ð Þ <sup>1</sup>*=*2Ω<sup>0</sup> ð Þ <sup>∇</sup><sup>Φ</sup> � <sup>∇</sup> <sup>z</sup>*:* (40)

LΔU � ½α LΔΦ þ β∂Φ*=*r∂φ ¼ 0, (43)

Ω0; ω<sup>J</sup>

2*β α* 1 *r ∂H*

2*β α* � 2 1 *r ∂H*

∇**v**<sup>G</sup> ¼ 0, (38)

<sup>2</sup> LΔΦ*:* (39)

<sup>∇</sup>**v**<sup>I</sup> <sup>¼</sup> 0, (41)

<sup>2</sup> � <sup>4</sup>πGρ<sup>0</sup> (44)

*<sup>∂</sup><sup>φ</sup>* <sup>¼</sup> <sup>0</sup>*:* (45)

*<sup>∂</sup><sup>φ</sup>* <sup>¼</sup> <sup>0</sup>*:* (46)

<sup>2</sup> LΔΦ*:* (42)

Substituting Eq. (35) to Eq. (36) and taking approximation d*=*dt ≪ Ω (slowly varying perturbations), dropping the term **v**I∇ in the expression (13) for d*=*dt, we get.

∇**v**<sup>I</sup> ¼ � 1*=*4Ω<sup>0</sup>

*<sup>=</sup>*dt <sup>þ</sup> <sup>ρ</sup>þρ<sup>0</sup>

Lρ � ρ0*=*4Ω<sup>0</sup>

Here we have served the terms that are of second order in perturbed amplitude

Using the Poisson equation, we get from Eq. (42) the basic nonlinear equation

; α<sup>0</sup> � dα*=*dr; β ¼ α<sup>0</sup>

On the limit ∣H∣≪∣U∣ that corresponds to large-scale disturbances: λ ≫ λJ,

Eqs. (45) and (46) have the same structure differing only by their coefficients,

In a Cartesian coordinate system (X,Y) (**Figure 2**), we will look for stationary solutions of Eq. (45) (and (46)) in a small neighborhood of the guiding center O

ð Þ ∇*H* � ∇ *<sup>z</sup>* <sup>Δ</sup>*<sup>H</sup>* <sup>þ</sup>

ð Þ ∇*U* � ∇ *<sup>z</sup>* <sup>Δ</sup>*<sup>U</sup>* <sup>þ</sup>

In view of the assessment (28), for short-scale perturbations (λ ≪ λJ), Eq. (42)

**v**<sup>I</sup> ¼ 1*=*4Ω<sup>0</sup>

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

With the use of Eqs. (19) and (34), we find

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

The continuity equation now takes the form

or, using Eqs. (14), (20), and (21)

and neglected terms of highest order.

α � ω<sup>J</sup> 2 *=*2Ω<sup>0</sup> 2

> *∂ ∂t* þ 2 2Ω<sup>0</sup>

*∂ ∂t* þ 2 2Ω<sup>0</sup>

and are Hasegawa-Mima type (see Eq. (1)).

**6. A solitary dipole vortex**

**29**

d ρþρ<sup>0</sup>

where

where

takes the form

Eq. (42) turns into [9].

Note that the vortices with positive and negative velocity circulation Γ have different properties. Whirlwind with positive circulation is characterized by low pressure, with negative excess mass density of substance. Vortex with negative circulation has a higher pressure and relatively tight formation with the positive excess mass density.

To illustrate these results, we will take into account the fact that the Rossby wavenumber usually is of the order of the inverse thickness of the disk. Considering that the size of the vortex *a* as the disk thickness order, we will get for the Bessel function argument 2*a*kR ≈ 12. **Figure 2** shows 3D image relative density perturbations in monopole whirlwind occupying the region 0≤ *R=a*≤ 1*:*35, for the value of the argument s ¼ 12. This vortex is a retrograde-circulating rarefaction around the center O condensation in the case of Γ<0 and prograde-circulating rarefaction in the case of Γ>0 (**Figure 3**). The decrease of density in the area *R* >*a* of larger vortex is steeper. If the size of the vortex tends to zero, we get a simple classic case of point vortex.

For long-scale perturbations (33), the Rankin vortex velocity profile is given [27, 28]:

$$v = \frac{\Gamma}{2\pi\chi a} \begin{cases} R/a, & R \le a \\ a/R, & R \ge a. \end{cases} \tag{34}$$

where <sup>γ</sup> <sup>¼</sup> <sup>1</sup> � <sup>Ω</sup><sup>2</sup> *=*πGρ0.

## **5. Vortices in the post-geostrophic approximation**

In this section, we will get nonlinear perturbation equation, taking into account the inertia term in the equation of motion (10) for homogeneously rotating disk. The cross product of Eq. (10) with ez: ez � Eq. (10), gives

$$\mathbf{v} = \mathbf{v}\_{\mathbb{G}} + \mathbf{v}\_{\mathbb{I}},\tag{35}$$

where the first term is geostrophic speed (19) and the second is

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

$$\mathbf{v}\_{\mathrm{l}} \equiv (\mathbf{1}/2\Omega\_0)\mathbf{e}\_{\mathrm{z}} \times \mathbf{d}\mathbf{v}/\mathrm{d}\mathbf{t}.\tag{36}$$

Substituting Eq. (35) to Eq. (36) and taking approximation d*=*dt ≪ Ω (slowly varying perturbations), dropping the term **v**I∇ in the expression (13) for d*=*dt, we get.

$$\mathbf{v}\_{1} = \left(\mathbf{1}/4\Omega\_{0}{}^{2}\right)\mathbf{e}\_{z} \times \mathbf{d}[\mathbf{e}\_{z} \times \nabla \Phi] \mathbf{d}t.\tag{37}$$

With the use of Eqs. (19) and (34), we find

$$
\nabla \mathbf{v}\_{\rm G} = \mathbf{0},
\tag{38}
$$

$$
\nabla \mathbf{v}\_{\rm l} = -\left(\mathbf{1}/4\Omega\_0 \mathbf{^2}\right) \mathbf{L}\Delta \Phi. \tag{39}
$$

where

where s � 2*a*kR. This is a monopole vortex with the following perturbed velocity

*3D image of relative density perturbations of whirlwind in the range* 0 ≤*R=a*<1*:*4 *for* 2*а*kR ¼ 20*.*

*K*1ð Þ*s I*1ð Þ *sR=a*

*I*1ð Þ*s K*1ð Þ *sR=a*

Note that the vortices with positive and negative velocity circulation Γ have different properties. Whirlwind with positive circulation is characterized by low pressure, with negative excess mass density of substance. Vortex with negative circulation has a higher pressure and relatively tight formation with the positive

To illustrate these results, we will take into account the fact that the Rossby wavenumber usually is of the order of the inverse thickness of the disk. Considering that the size of the vortex *a* as the disk thickness order, we will get for the Bessel function argument 2*a*kR ≈ 12. **Figure 2** shows 3D image relative density perturbations in monopole whirlwind occupying the region 0≤ *R=a*≤ 1*:*35, for the value of the argument s ¼ 12. This vortex is a retrograde-circulating rarefaction around the center O condensation in the case of Γ<0 and prograde-circulating rarefaction in the case of Γ>0 (**Figure 3**). The decrease of density in the area *R* >*a* of larger vortex is steeper. If the size of the vortex tends to zero, we get a simple classic case

For long-scale perturbations (33), the Rankin vortex velocity profile is given

In this section, we will get nonlinear perturbation equation, taking into account the inertia term in the equation of motion (10) for homogeneously rotating disk.

**v** ¼ **v**<sup>G</sup> þ **v**I, (35)

�

*R=a*, *R*≤*a a=R*, *R*≥*a:*

*<sup>v</sup>* <sup>¼</sup> *<sup>Γ</sup>* 2*πγα*

*=*πGρ0.

**5. Vortices in the post-geostrophic approximation**

The cross product of Eq. (10) with ez: ez � Eq. (10), gives

where the first term is geostrophic speed (19) and the second is

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

(33)

(34)

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

*<sup>v</sup><sup>θ</sup>* <sup>¼</sup> <sup>Γ</sup> *πas*

*Vortex Dynamics Theories and Applications*

8 >>>><

>>>>:

field:

**Figure 3.**

excess mass density.

of point vortex.

where <sup>γ</sup> <sup>¼</sup> <sup>1</sup> � <sup>Ω</sup><sup>2</sup>

[27, 28]:

**28**

$$\mathbf{L} \equiv \partial / \partial \mathbf{t} + (\mathbf{1} / 2\mathbf{\Omega}\_0)(\nabla \Phi \times \nabla)\_\mathbf{z}. \tag{40}$$

The continuity equation now takes the form

$$\mathbf{d}(\rho\_+\rho\_0)/\mathbf{dt} + \left(\rho\_+\rho\_0\right)\nabla\mathbf{v}\_\mathbf{l} = \mathbf{0},\tag{41}$$

or, using Eqs. (14), (20), and (21)

$$\mathbf{L}\boldsymbol{\rho} - \left(\boldsymbol{\rho}\_{0}/4\boldsymbol{\Omega}\_{0}\boldsymbol{\epsilon}^{2}\right)\mathbf{L}\boldsymbol{\Delta}\boldsymbol{\Phi}.\tag{42}$$

Here we have served the terms that are of second order in perturbed amplitude and neglected terms of highest order.

Using the Poisson equation, we get from Eq. (42) the basic nonlinear equation

$$
\mathbf{L}\Delta \mathbf{U} - \mathfrak{H}\mathfrak{a}\,\mathbf{L}\Delta \Phi + \mathfrak{H}\mathfrak{O}/\mathfrak{r}\mathfrak{gl} = \mathbf{0},\tag{43}
$$

where

$$\mathfrak{a} \equiv \mathfrak{w}\_{\uparrow}^{2} / 2\mathfrak{Q}\_{0}{}^{2}; \mathfrak{a}' \equiv \mathfrak{d}\mathfrak{a} / \mathfrak{dr}; \mathfrak{f} = \mathfrak{a}'\mathfrak{Q}\_{0}; \mathfrak{w}\_{\uparrow}^{2} \equiv 4\pi\mathfrak{G}\rho\_{0} \tag{44}$$

In view of the assessment (28), for short-scale perturbations (λ ≪ λJ), Eq. (42) takes the form

$$
\left(\frac{\partial}{\partial t} + \frac{2}{2\Omega\_0} (\nabla H \times \nabla)\_x\right) \Delta H + \frac{2\beta}{\alpha} \frac{1}{r} \frac{\partial H}{\partial \rho} = 0. \tag{45}
$$

On the limit ∣H∣≪∣U∣ that corresponds to large-scale disturbances: λ ≫ λJ, Eq. (42) turns into [9].

$$
\left(\frac{\partial}{\partial t} + \frac{2}{2\Omega\_0} \left(\nabla U \times \nabla\right)\_x\right) \Delta U + \frac{2\beta}{a-2} \frac{1}{r} \frac{\partial H}{\partial \rho} = \mathbf{0}.\tag{46}
$$

Eqs. (45) and (46) have the same structure differing only by their coefficients, and are Hasegawa-Mima type (see Eq. (1)).

## **6. A solitary dipole vortex**

In a Cartesian coordinate system (X,Y) (**Figure 2**), we will look for stationary solutions of Eq. (45) (and (46)) in a small neighborhood of the guiding center O

with a radius of *a* ≪ r0 in the form of a vortex drifting in y-direction at a constant speed u. Introducing the wave variable

$$
\eta = \mathbf{y}\text{-ut}\tag{47}
$$

For long-scale perturbations

*vR* ¼ � <sup>1</sup> 2Ω

> *<sup>v</sup><sup>θ</sup>* <sup>¼</sup> <sup>1</sup> 2Ω ∂Φ *<sup>∂</sup><sup>R</sup>* <sup>¼</sup> *<sup>u</sup>*

(61) on the circle R ¼ *a*.

ate formula (62).

**Figure 4.**

**31**

*The stream lines of solitary dipole vortex [9].*

*const:* ¼

while for small-scale disturbances

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

∂Φ *<sup>R</sup>∂<sup>θ</sup>* <sup>¼</sup> *<sup>u</sup>* s

U ! H ¼ cs

8 >>>><

>>>>:

8 >>>><

>>>>:

8 >>><

>>>:

<sup>2</sup> <sup>¼</sup> <sup>2</sup>Ωð Þ *<sup>a</sup>*

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

2

<sup>1</sup> � *<sup>s</sup>* 2

> <sup>1</sup>ð Þ *sR=a RK*1ð Þ*s*

The current lines are determined by dR*=*vr ¼ Rdθ*=*v<sup>θ</sup> that gives

*a* þ *s* 2 *g*2

<sup>1</sup> � *<sup>s</sup>* 2 *g*2 � � *R*

*K*1ð Þ *sR=a K*1ð Þ*s*

*aK*0

<sup>1</sup> � *<sup>s</sup>* 2 *<sup>g</sup>*<sup>2</sup> <sup>1</sup> � *<sup>g</sup>*

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

2 *=*u � �ð Þ ln <sup>j</sup><sup>α</sup> � <sup>2</sup><sup>j</sup> <sup>0</sup>

<sup>ρ</sup>∣ρ0, and s2 <sup>¼</sup> <sup>2</sup>Ω<sup>2</sup>

*<sup>g</sup>*<sup>2</sup> <sup>1</sup> � *aJ*1ð Þ *gR=<sup>a</sup> RJ*1ð Þ*g* � � � � sin *<sup>θ</sup>*, *<sup>R</sup>*≤*<sup>a</sup>*

*J* 0 <sup>1</sup>ð Þ *gR=a J*1ð Þ*g* � � � � cos *<sup>θ</sup>*, *<sup>R</sup>*≤*<sup>a</sup>*

Moreover, the condition (57) is derived from the requirements of continuity

**Figure 4** shows the current lines of drifting solitary dipole vortex, the appropri-

*J*1ð Þ *gR=a J*1ð Þ*g* � � sin 2*θ*, *<sup>R</sup>*≤*<sup>a</sup>*

sin 2*θ*, *R*≥*a:*

From Eqs. (56) and (19), we get the velocity field of a vortex in the form

*<sup>a</sup>=*<sup>u</sup> � �ð Þ ln<sup>α</sup> <sup>0</sup>

sin *θ*, *R*≥*a*

*cos θ*, *R*≥*a*

, (58)

*:* (59)

(60)

(61)

(62)

Eq. (46) can be rewritten in the form

$$\left\{\partial/\partial\mathbb{\eta}-\mathbf{A}(\nabla\mathbf{U}\times\nabla)\_{\mathbf{z}}\right\}\delta\mathbf{U}=\Lambda\partial\mathbb{U}/\partial\mathbb{\eta},\tag{48}$$

or in the form of the Jacobean

$$\mathbf{J}(\mathbf{U}\cdot\mathbf{x}/\mathbf{A}, \Delta\mathbf{U} + \Lambda\mathbf{x}/\mathbf{A}) = \mathbf{0},\tag{49}$$

where

$$\mathbf{A}(\mathbf{A})^{-1} = \mathbf{2}\mathbf{u}\mathbf{2}, \Lambda = -\mathbf{4}\mathbf{\Omega}\_0 \mathbf{^2} \mathbf{A}(\ln|\mathbf{a} - \mathbf{2}|)'.\tag{50}$$

On basis of Eq. (49)

$$
\Delta \mathbf{U} + \Lambda \mathbf{x}/\mathbf{A} = \mathbf{F}(\mathbf{U} - \mathbf{x}/\mathbf{A}),
\tag{51}
$$

where F is an arbitrary function. As we are interested in the restricted solutions, then in the limit of large values η, solution U should vanish for arbitrary values x; therefore

$$\mathbf{F}(-\mathbf{x}/\mathbf{A}) = -\Lambda \mathbf{x}/(\mathbf{A}).\tag{52}$$

We will assume that the function F (51) in the equation is linear not only for large η but across the whole plane (x, η). In general, F can be represented as ∝ ð Þ U � x*=*A . Introducing polar coordinates R, θ : x ¼ Rcosθ, η ¼ R sin θ, we can write Eq. (49) in the form

$$(\Delta + \mathbf{k}^2)\mathbf{U} = \mathbf{A}^{-1}(\mathbf{k}^2 - \Lambda)\mathbf{R}\cos\ \theta, \quad \mathbf{R} \le a,\tag{53}$$

$$(\Delta - \mathbf{p}^2)\mathbf{U} = \mathbf{0}, \quad \mathbf{R} \ge \mathbf{a},\tag{54}$$

where k and p are real constants. Soon the sense of splitting the (R, θ) plane into two parts will be obvious. Eq. (54) turns out to be uniform, because for a restricted solution, we have U ! 0 for large R. This condition implies

$$\mathbf{p}^2 = -\Lambda.\tag{55}$$

Eqs. (53) and (54) have the following stationary solution [9]:

$$U(\mathbf{R},\theta) = \Omega \mu a \begin{cases} \left[ \left( 1 - \frac{s^2}{g^2} \right) \frac{R}{a} + \frac{s^2}{g^2} \frac{f\_1(\mathbf{g}R/a)}{f\_1(\mathbf{g})} \right] \cos \theta, & R \le a \\\\ -\frac{K\_1(sR/a)}{K\_1(s)} \cos \theta & R \ge a \end{cases} \tag{56}$$

where J1 and K1 are Bessel and Macdonald functions, respectively, and *g* ¼ *ka* and *s* ¼ *pa* are connected by "dispersion equation" which is transcendental

$$(\mathbf{J})\_1(\mathbf{g})(\mathbf{K})\_3(\mathbf{s}) + (\mathbf{J})\_3(\mathbf{g})(\mathbf{K})\_1(\mathbf{s}) = \mathbf{0}.\tag{57}$$

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

For long-scale perturbations

with a radius of *a* ≪ r0 in the form of a vortex drifting in y-direction at a constant

*<sup>∂</sup>=*∂<sup>η</sup> � <sup>A</sup>ð Þ <sup>∇</sup><sup>U</sup> � <sup>∇</sup> <sup>z</sup>

ð Þ <sup>A</sup> �<sup>1</sup> <sup>¼</sup> 2uΩ,<sup>Λ</sup> ¼ �4Ω<sup>0</sup>

*<sup>η</sup>* <sup>¼</sup> <sup>y</sup>‐ut (47)

� �δ<sup>U</sup> <sup>¼</sup> <sup>Λ</sup>∂U*=*∂η, (48)

2

where F is an arbitrary function. As we are interested in the restricted solutions, then in the limit of large values η, solution U should vanish for arbitrary values x;

We will assume that the function F (51) in the equation is linear not only for large η but across the whole plane (x, η). In general, F can be represented as ∝ ð Þ U � x*=*A . Introducing polar coordinates R, θ : x ¼ Rcosθ, η ¼ R sin θ, we can

where k and p are real constants. Soon the sense of splitting the (R, θ) plane into two parts will be obvious. Eq. (54) turns out to be uniform, because for a restricted

solution, we have U ! 0 for large R. This condition implies

Eqs. (53) and (54) have the following stationary solution [9]:

*a* þ *s* 2 *g*2

and *s* ¼ *pa* are connected by "dispersion equation" which is transcendental

� �

where J1 and K1 are Bessel and Macdonald functions, respectively, and *g* ¼ *ka*

<sup>1</sup> � *<sup>s</sup>* 2 *g*2 � � *R*

8 >>><

>>>:

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

J U<sup>ð</sup> ‐x*=*A, <sup>Δ</sup><sup>U</sup> <sup>þ</sup> <sup>Λ</sup>x*=*AÞ ¼ 0, (49)

A ln ð Þ jα � 2j <sup>0</sup>

ΔU þ Λx*=*A ¼ F Uð Þ � x*=*A , (51)

Fð Þ¼� �x*=*A Λx*=*ð Þ A *:* (52)

<sup>Δ</sup> <sup>þ</sup> k2 � �<sup>U</sup> <sup>¼</sup> <sup>A</sup>�<sup>1</sup> k2 � <sup>Λ</sup> � �Rcos <sup>θ</sup>, R <sup>≤</sup>*a*, (53)

*J*1ð Þ *gR=a J*1ð Þ*g*

cos *θ R*≥*a*

ð ÞJ <sup>1</sup>ð Þg ð Þ K <sup>3</sup>ð Þþ s ð ÞJ <sup>3</sup>ð Þg ð Þ K <sup>1</sup>ð Þ¼ s 0*:* (57)

<sup>Δ</sup> � p2 � �<sup>U</sup> <sup>¼</sup> 0, R <sup>≥</sup>*a*, (54)

<sup>p</sup><sup>2</sup> ¼ �Λ*:* (55)

cos *θ*, *R*≤*a*

(56)

*:* (50)

speed u. Introducing the wave variable

*Vortex Dynamics Theories and Applications*

or in the form of the Jacobean

where

therefore

On basis of Eq. (49)

write Eq. (49) in the form

*U*ð Þ¼ R, *θ* Ω*ua*

**30**

Eq. (46) can be rewritten in the form

$$\mathbf{s}^2 = \left(2\Omega(a)^2/\mathfrak{u}\right) \left(\ln|\alpha - 2|\right)',\tag{58}$$

while for small-scale disturbances

$$\mathbf{U} \rightarrow \mathbf{H} = \mathbf{c}\_{\mathbf{s}}^{2} \rho |\rho\_{0}, \mathbf{and} \,\mathbf{s}^{2} = \left(2\Omega^{2}a/\mathbf{u}\right) (\ln \alpha)'.\tag{59}$$

From Eqs. (56) and (19), we get the velocity field of a vortex in the form

$$v\_R = -\frac{1}{2\Omega} \frac{\partial \Phi}{R \partial \theta} = u \begin{cases} \left[ 1 - \frac{s^2}{g^2} \left( 1 - \frac{aI\_1(\mathbf{g}R/a)}{R f\_1(\mathbf{g})} \right) \right] \sin \theta, & R \le a \\\\ \frac{aK\_1'(sR/a)}{R K\_1(s)} \sin \theta, & R \ge a \end{cases} \tag{60}$$
 
$$v\_\theta = \frac{1}{2\Omega} \frac{\partial \Phi}{\partial R} = u \begin{cases} \left[ 1 - \frac{s^2}{g^2} \left( 1 - g \frac{f\_1'(\mathbf{g}R/a)}{f\_1(\mathbf{g})} \right) \right] \cos \theta, & R \le a \\\\ \frac{sK\_1'(sR/a)}{K\_1(s)} \cos \theta, & R \ge a \end{cases} \tag{61}$$

Moreover, the condition (57) is derived from the requirements of continuity (61) on the circle R ¼ *a*.

The current lines are determined by dR*=*vr ¼ Rdθ*=*v<sup>θ</sup> that gives

$$const. = \begin{cases} \left[ \left( 1 - \frac{s^2}{g^2} \right) \frac{R}{a} + \frac{s^2}{g^2} \frac{f\_1(\mathbf{g}R/a)}{f\_1(\mathbf{g})} \right] \sin 2\theta, & R \le a \\\\ \frac{K\_1(sR/a)}{K\_1(s)} \sin 2\theta, & R \ge a. \end{cases} \tag{62}$$

**Figure 4** shows the current lines of drifting solitary dipole vortex, the appropriate formula (62).

**Figure 4.** *The stream lines of solitary dipole vortex [9].*

## **7. The contours of constant density**

As shown in Section 3, Φ*=*2Ω is the current function of perturbed speed, ψ in the case of short-scale disturbances Φ ¼ H. Then we have the ψ ¼ Hð Þρ *=*2Ω; the constant values of the contours of constant density ρ match the lines of the current ψ ¼ constant: in short-scale modon, substance flows along the lines of constant density.

In the long-scale limit, Φ ¼ U and ψ ¼ U*=*2Ω. Therefore, in this case the current lines coincide with equipotentials of the gravitational field, not with the contours of constant density. The last can be found using the Poisson equation:

$$
\Delta \Psi = 2\pi \text{G\rho/\Omega}.\tag{63}
$$

and one rarefaction (**Figure 6**) characterizes the first type. The second type is characterized by the two antisymmetrically located condensations and two rarefactions, and second condensation-rarefaction pair has sickle-form (**Figure 7**). For small values g and s, the short-scale modon is of the second type, with distinctive two condensations (see **Figure 8**). In the middle part of the dispersion curve, the short-scale and long-scale modons have roughly the same structure. They have one antisymmetrical located prominent pair of condensation-rarefaction and another weak pair of sickle forms. For large values of g and s, the short-scale modon is the first type and has the character of a cyclone-anticyclone couple; the long-scale one is the second type and is characterized by a nearly round and sickle-shaped condensations. In laboratory experiments, the solitary dipole vortices on shallow water,

*Dependence of relative perturbed density from the dimensionless distance R/a in short-scale and long-scale*

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

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

**Figure 5.**

*modons.*

**Figure 6.**

**Figure 7.**

**33**

*3D image of density distribution in the first type modon.*

*3D image of density distribution in the second type modon.*

Since equipotentials U = constant, generally speaking, do not coincide with the contours of constant density, it follows that the stream lines ψ = constant do not coincide with the contours of constant density.

Define the contours of constant density of modon. Relative density perturbations in the short-scale range are expressed by the following formula:

$$
\sigma = \mathfrak{p}/\mathfrak{p}\_0 = \beta\_{\mathfrak{sw}} H/a\mathfrak{u}\mathfrak{Q},
\text{where}
\mathfrak{p}\_{\mathfrak{sw}} \equiv a\mathfrak{u}\mathfrak{Q}/\mathfrak{c}\_\*^2.\tag{64}
$$

The relative perturbed density in long-scale range turns out to be in the form

$$\sigma = \frac{1}{a\_f^2} \Delta \psi = \beta\_{lv} \begin{cases} \frac{f\_1(\mathbf{g}R/a)}{f\_1(\mathbf{g})} \cos \theta, & R \le a \\\\ \frac{K\_1(sR/a)}{K\_1(s)} \cos \theta, & R \ge a \end{cases} \tag{65}$$

where

$$\mathfrak{G}\_{\rm lw} \equiv -u\Omega s^2 / a a \rho^2 = -\left(\ln|\alpha - 2|\right)^{\circ}/\alpha. \tag{66}$$

To illustrate, consider a logarithmic model of the disk, describing in equilibrium by the following functions of potential, mass density, and angular velocity:

$$\begin{aligned} \mathbf{U}\_{0}(\mathbf{r}) &= \mathsf{V} \mathbf{v}\_{0}^{2} \ln \left( \mathbf{R}\_{\mathbf{c}}^{2} + \mathbf{r}^{2} \right), \rho\_{0}(\mathbf{r}) = \mathbf{v}\_{0}^{2} \mathbf{R}^{2} / 2 \pi \mathbf{G} \left( \mathbf{R}\_{\mathbf{c}}^{2} + \mathbf{r}^{2} \right), \\ \boldsymbol{\Omega}^{2} &= \mathbf{v}\_{0}^{2} / \left( \mathbf{R}\_{\mathbf{c}}^{2} + \mathbf{r}^{2} \right), \end{aligned} \tag{67}$$

where Rc and v0 are constants, rotation of a disk in the small area *(a* ≪ Rc*)* can be considered as uniform, and βlw for this model turns out to be equal to

$$\mathfrak{R}\_{\rm lw} \approx 8ar/\mathfrak{R}\_{\rm (c)}\,^2 \approx 8a(\mathbf{R}\_{\rm c} + \mathbf{R}\cos\theta)/\mathfrak{R}\_{\rm c}\,^2,\tag{68}$$

where we used the relation r<sup>2</sup> ≈ Rc <sup>2</sup> <sup>þ</sup> 2RRccosθ, by placing the center O in R <sup>¼</sup> Rc.

For illustrations of a perturbed density distribution in dipole vortex (65), we used the following solutions of "dispersion equation" (57): (g,s) = (4.0, 1.52); (4.2, 2.90); (4.5, 6.0); (4.7, 10.0).

The curves in **Figure 5** show perturbed density as a function of dimensionless distance *R/a* from the guiding center O in the short-scale (curves increasing towards the center) and in the long-scale (curves descending towards the center) limit.

The density distribution is antisymmetrical to the guiding center. Depending on the choice of the dispersion curve (57) range, there are two types of mass distribution in dipole vortex. One is antisymmetrically located almost round condensation,

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

**7. The contours of constant density**

*Vortex Dynamics Theories and Applications*

coincide with the contours of constant density.

*<sup>σ</sup>* <sup>¼</sup> <sup>1</sup> *ω*2 *J*

U0ð Þ¼ r ½v0

<sup>Ω</sup><sup>2</sup> <sup>¼</sup> v0 2 *=* Rc <sup>2</sup> <sup>þ</sup> <sup>r</sup>

where we used the relation r<sup>2</sup> ≈ Rc

(4.2, 2.90); (4.5, 6.0); (4.7, 10.0).

where

**32**

As shown in Section 3, Φ*=*2Ω is the current function of perturbed speed, ψ in the case of short-scale disturbances Φ ¼ H. Then we have the ψ ¼ Hð Þρ *=*2Ω; the constant values of the contours of constant density ρ match the lines of the current ψ ¼ constant: in short-scale modon, substance flows along the lines of constant density. In the long-scale limit, Φ ¼ U and ψ ¼ U*=*2Ω. Therefore, in this case the current lines coincide with equipotentials of the gravitational field, not with the contours of

Since equipotentials U = constant, generally speaking, do not coincide with the contours of constant density, it follows that the stream lines ψ = constant do not

Define the contours of constant density of modon. Relative density perturba-

σ ¼ ρ*=*ρ<sup>0</sup> ¼ *βsw H=auΩ*, whereβsw � *au*Ω*=*cs

The relative perturbed density in long-scale range turns out to be in the form

8 >>><

>>>:

*J*1ð Þ *gR=a J*1ð Þ*g*

*K*1ð Þ *sR=a K*1ð Þ*s*

To illustrate, consider a logarithmic model of the disk, describing in equilibrium

where Rc and v0 are constants, rotation of a disk in the small area *(a* ≪ Rc*)* can

For illustrations of a perturbed density distribution in dipole vortex (65), we used the following solutions of "dispersion equation" (57): (g,s) = (4.0, 1.52);

The curves in **Figure 5** show perturbed density as a function of dimensionless distance *R/a* from the guiding center O in the short-scale (curves increasing towards the center) and in the long-scale (curves descending towards the center) limit.

The density distribution is antisymmetrical to the guiding center. Depending on the choice of the dispersion curve (57) range, there are two types of mass distribution in dipole vortex. One is antisymmetrically located almost round condensation,

2 R2

*<sup>2</sup>* <sup>≈</sup> *8a*ð Þ Rc <sup>þ</sup> Rcos<sup>θ</sup> *<sup>=</sup>*3Rc

Δψ ¼ 2πGρ*=*Ω*:* (63)

cos *θ*, *R*≤*a*

cos *θ*, *R*≥*a*

*=*2πG Rc

<sup>2</sup> � �, (67)

<sup>2</sup> ¼ �ð Þ ln j j <sup>α</sup> � <sup>2</sup> }*=*α*:* (66)

<sup>2</sup> <sup>þ</sup> <sup>r</sup> <sup>2</sup> � �,

2

<sup>2</sup> <sup>þ</sup> 2RRccosθ, by placing the center O in R <sup>¼</sup> Rc.

, (68)

2

*:* (64)

(65)

constant density. The last can be found using the Poisson equation:

tions in the short-scale range are expressed by the following formula:

Δ*ψ* ¼ *βlw*

2 *=aω<sup>J</sup>*

<sup>2</sup> <sup>þ</sup> <sup>r</sup>

by the following functions of potential, mass density, and angular velocity:

<sup>2</sup> � �, <sup>ρ</sup>0ð Þ¼ <sup>r</sup> v0

be considered as uniform, and βlw for this model turns out to be equal to

βlw � �*u*Ω*s*

<sup>2</sup> ln Rc

βlw ≈ 8*ar=3R*ð Þ*<sup>c</sup>*

*Dependence of relative perturbed density from the dimensionless distance R/a in short-scale and long-scale modons.*

and one rarefaction (**Figure 6**) characterizes the first type. The second type is characterized by the two antisymmetrically located condensations and two rarefactions, and second condensation-rarefaction pair has sickle-form (**Figure 7**). For small values g and s, the short-scale modon is of the second type, with distinctive two condensations (see **Figure 8**). In the middle part of the dispersion curve, the short-scale and long-scale modons have roughly the same structure. They have one antisymmetrical located prominent pair of condensation-rarefaction and another weak pair of sickle forms. For large values of g and s, the short-scale modon is the first type and has the character of a cyclone-anticyclone couple; the long-scale one is the second type and is characterized by a nearly round and sickle-shaped condensations. In laboratory experiments, the solitary dipole vortices on shallow water,

**Figure 6.** *3D image of density distribution in the first type modon.*

**Figure 7.** *3D image of density distribution in the second type modon.*

**Figure 8.**

*"Dispersive curve"—the solution of Eq. (57). In the three ranges of the curve, 3D images described the relative perturbed density of modon in short-scale (lower row) and long-scale (upper row) limits. The blue color indicates condensations, and red color indicates rarefactions.*

R0, in other points their sum gives the tidal force **j**3Ω<sup>0</sup>

ð Þ **v**∇ **v** ¼ **j**3Ω<sup>0</sup>

where h is specific enthalpy h <sup>¼</sup> <sup>Ð</sup>

z is returning force along the z-axis.

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

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

y � **k**Ω<sup>0</sup>

2

vx ¼ �Αx � ωr0

vy ¼ �Αy þ ωr0

**9. Motion of rigid particles in Burgers vortex**

interaction of rigid particles among themselves.

gas will be described by Stokes drag force:

In the local approach, the equation stationary isentropic shear flow of dusk viscous substance is described by Navier–Stokes and continuity equations:

2

vectors. The first term in the right-hand side of Eq. (72) is tidal acceleration, the second term is vertical gravitation, the third is acceleration of Coriolis, and the last

2

2

In the Cartesian coordinate system, the Burgers vortex (2) will be presented in

vz ¼ 2Αz,

Let us study the two-dimensional dynamics of dust rigid particles in a Burgers vortex. We will neglect the influence of rigid particles on dynamics of gas and the

As we consider centimeter- to meter-sized particles, then D considerably surpass the mean free path of gas molecules; therefore, the friction of rigid particles with

**u** = (dX/dt, dY/dt) is velocity of a particle, and X and Y are particle coordinates. In a dimensionless form, the equation of motion of particles in the accepted

� � � <sup>∂</sup>h*=*∂<sup>x</sup>

� �

*<sup>f</sup>* <sup>¼</sup> <sup>β</sup>ð Þ **<sup>v</sup>** � **<sup>u</sup>** , where <sup>β</sup> � <sup>18</sup> ρν*=*<sup>ρ</sup> <sup>∗</sup> <sup>D</sup><sup>2</sup>

dux*=*dt ¼ 2uy þ γ vxj<sup>r</sup>¼ð Þ X,Y � ux

y 1 � exp �r

x 1 � exp �r

gravitation ‐Ω<sup>0</sup>

**Figure 9.**

is a viscous stress.

where r<sup>2</sup> � x2 <sup>þ</sup> <sup>y</sup>2.

approach looks like

**35**

the form

2

*The local frame of reference 2.*

2

z � 2Ω<sup>0</sup> � **v** � ∇h þ νΔ**v** (72)

2 ,

2 , (74)

, (75)

<sup>r</sup>¼ð Þ X,Y , (76)

∇ð Þ¼ ρ**v** 0, (73)

ρ�1dp � � and **i**, **j**, and **k** are Cartesian unit

2 *=*r0 <sup>2</sup> � � � � *=*r

2 *=*r <sup>2</sup> � � � � *=*r

y. The vertical component of

obviously, are the short-scale modons of the first type with the asymmetry between high- and low-pressure centers.

Let's estimate the masses of condensations in long-scale modon:

$$m\_1 = \frac{2\pi h p\_0}{J\_1(\mathbf{g})} \beta\_{\parallel \mathbf{w}} \int\_{\pi/2}^{3\pi/2} \cos\theta d\theta \Big|\_{0}^{\mathbf{x}\_1} J\_1(\mathbf{g} \mathbf{x}) \mathbf{x} d\mathbf{x} = \frac{4\pi a^2 h p\_0}{J\_1(\mathbf{g})} \beta\_{\parallel \mathbf{w}} J\_0(\mathbf{g} \mathbf{x}\_1) H\_1(\mathbf{g} \mathbf{x}\_1), \tag{69}$$

where x ¼ R*=*ð Þ *a* , h is the thickness of the gas disk, H1ð Þ gx is the Struve function of the first order, and x1 is the root of equation J1 gx1 � � <sup>¼</sup> 0. Similarly

$$m\_2 = \frac{2\pi a^2 h \rho\_0}{J\_1(\mathbf{g})} \beta\_{\parallel \mathbf{w}} \Big|\_{-\pi/2}^{\pi/2} \cos\theta d\theta \Big|\_{\mathbf{x}\_1}^1 J\_1(\mathbf{gx}) \mathbf{x} d\mathbf{x} + \frac{2\pi a^2 h \rho\_0}{K\_1(\mathbf{s})} \beta\_{\parallel \mathbf{w}} \Big|\_{-\pi/2}^{\pi/2} \cos\theta d\theta \Big|\_{1}^\infty K\_1(\mathbf{x}) \mathbf{x} d\mathbf{x}. \tag{70}$$

Numerical estimations show that the ratio of the masses of condensations in the long-scale modon, depending on values of parameters g and s, varies in the range m1*=*m2 � 2 � 30.

Now we will focus our attention on a role of vortices for the formation of planetesimals in a protoplanetary light dusty disk.

## **8. The Burgers vortex in local frame of reference**

Let's use the local approach, choosing frame of reference, rotating with a disk with angular speed Ω<sup>0</sup> at distance R0 round the central star of mass M. In this approach, assuming the effective radius of a vortex is much smaller than R0, we will choose the Cartesian system of coordinates with center O (**Figure 9**), directing the y-axis to a star and the x-axis in direction of Keplerian flow of gas. We will present the disk rotation as

$$
\Omega(\mathsf{R}) \propto \mathsf{R}^{-q}.\tag{71}
$$

In case when only the gravitation of the central star operates, rotation will be Keplerian with q = 3/2, and for homogeneously rotating disk, q ¼ 2, i*:*e*:*2≥ q≥3*=*2.

The substance stream in chosen frame of reference, has X component of speed **i**qΩ0y, centrifugal force is compensated by gravitation of the central star at distance *Vortices in Rotating and Gravitating Gas Disk and in a Protoplanetary Disk DOI: http://dx.doi.org/10.5772/intechopen.92028*

**Figure 9.** *The local frame of reference 2.*

obviously, are the short-scale modons of the first type with the asymmetry between

*"Dispersive curve"—the solution of Eq. (57). In the three ranges of the curve, 3D images described the relative perturbed density of modon in short-scale (lower row) and long-scale (upper row) limits. The blue color*

*<sup>J</sup>*1ð Þ *gx xdx* <sup>¼</sup> <sup>4</sup>*πa*<sup>2</sup>*hp*<sup>0</sup>

where x ¼ R*=*ð Þ *a* , h is the thickness of the gas disk, H1ð Þ gx is the Struve function

Numerical estimations show that the ratio of the masses of condensations in the long-scale modon, depending on values of parameters g and s, varies in the range

*J*1ð Þ *gx xdx* þ

Now we will focus our attention on a role of vortices for the formation of

Let's use the local approach, choosing frame of reference, rotating with a disk with angular speed Ω<sup>0</sup> at distance R0 round the central star of mass M. In this approach, assuming the effective radius of a vortex is much smaller than R0, we will choose the Cartesian system of coordinates with center O (**Figure 9**), directing the y-axis to a star and the x-axis in direction of Keplerian flow of gas. We will present

In case when only the gravitation of the central star operates, rotation will be Keplerian with q = 3/2, and for homogeneously rotating disk, q ¼ 2, i*:*e*:*2≥ q≥3*=*2. The substance stream in chosen frame of reference, has X component of speed **i**qΩ0y, centrifugal force is compensated by gravitation of the central star at distance

*J*1ð Þ*g*

2*πa*<sup>2</sup>*hρ*<sup>0</sup> *K*1ð Þ*s*

*βlwJ*<sup>0</sup> *gx*<sup>1</sup>

� � <sup>¼</sup> 0. Similarly

<sup>Ω</sup>ð Þ <sup>R</sup> <sup>∝</sup>R�<sup>q</sup>*:* (71)

*βlw* ð*<sup>π</sup>=*<sup>2</sup> �*π=*2

� �*H*<sup>1</sup> *gx*<sup>1</sup>

cos *θdθ*

ð<sup>∞</sup> 1

� �, (69)

*K*1ð Þ *sx xdx:*

(70)

Let's estimate the masses of condensations in long-scale modon:

ð*<sup>x</sup>*<sup>1</sup> 0

ð1 *x*1

cos *θdθ*

*indicates condensations, and red color indicates rarefactions.*

*Vortex Dynamics Theories and Applications*

of the first order, and x1 is the root of equation J1 gx1

cos *θdθ*

planetesimals in a protoplanetary light dusty disk.

**8. The Burgers vortex in local frame of reference**

high- and low-pressure centers.

*βlw* ð<sup>3</sup>*π=*<sup>2</sup> *π=*2

*βlw* ð*<sup>π</sup>=*<sup>2</sup> �*π=*2

*<sup>m</sup>*<sup>1</sup> <sup>¼</sup> <sup>2</sup>*πhp*<sup>0</sup> *J*1ð Þ*g*

**Figure 8.**

*<sup>m</sup>*<sup>2</sup> <sup>¼</sup> <sup>2</sup>*πa*<sup>2</sup>*hρ*<sup>0</sup> *J*1ð Þ*g*

m1*=*m2 � 2 � 30.

the disk rotation as

**34**

R0, in other points their sum gives the tidal force **j**3Ω<sup>0</sup> 2 y. The vertical component of gravitation ‐Ω<sup>0</sup> 2 z is returning force along the z-axis.

In the local approach, the equation stationary isentropic shear flow of dusk viscous substance is described by Navier–Stokes and continuity equations:

$$(\mathbf{v}\nabla)\mathbf{v} = \mathbf{j}\mathfrak{Q}\mathbf{2}\_0^2\mathbf{y} - \mathbf{k}\mathfrak{Q}\_0^2\mathbf{z} - \mathfrak{Q}\mathbf{2}\_0 \times \mathbf{v} - \nabla\mathbf{h} + \nu\Delta\mathbf{v} \tag{72}$$

$$\nabla(\rho \mathbf{v}) = \mathbf{0},\tag{73}$$

where h is specific enthalpy h <sup>¼</sup> <sup>Ð</sup> ρ�1dp � � and **i**, **j**, and **k** are Cartesian unit vectors. The first term in the right-hand side of Eq. (72) is tidal acceleration, the second term is vertical gravitation, the third is acceleration of Coriolis, and the last is a viscous stress.

In the Cartesian coordinate system, the Burgers vortex (2) will be presented in the form

$$\begin{aligned} \mathbf{v\_x} &= -\mathbf{A}\mathbf{x} - \alpha \mathbf{r\_0}^2 \mathbf{y} \left[ \mathbf{1} - \exp\left(-\mathbf{r}^2/\mathbf{r\_0}^2\right) \right] / \mathbf{r^2}, \\ \mathbf{v\_y} &= -\mathbf{A}\mathbf{y} + \alpha \mathbf{r\_0}^2 \mathbf{x} \left[ \mathbf{1} - \exp\left(-\mathbf{r}^2/\mathbf{r}^2\right) \right] / \mathbf{r^2}, \\ \mathbf{v\_z} &= 2\mathbf{A}\mathbf{z}, \end{aligned} \tag{74}$$

where r<sup>2</sup> � x2 <sup>þ</sup> <sup>y</sup>2.

## **9. Motion of rigid particles in Burgers vortex**

Let us study the two-dimensional dynamics of dust rigid particles in a Burgers vortex. We will neglect the influence of rigid particles on dynamics of gas and the interaction of rigid particles among themselves.

As we consider centimeter- to meter-sized particles, then D considerably surpass the mean free path of gas molecules; therefore, the friction of rigid particles with gas will be described by Stokes drag force:

$$f = \beta(\mathbf{v} - \mathbf{u})\text{, where }\beta \equiv 18\text{ }\mathbb{M}/\mathbb{p}\text{\*}\text{D}^2,\tag{75}$$

**u** = (dX/dt, dY/dt) is velocity of a particle, and X and Y are particle coordinates.

In a dimensionless form, the equation of motion of particles in the accepted approach looks like

$$\mathbf{d}\mathbf{u}\_{\mathbf{x}}/\mathbf{dt} = \mathbf{2}\mathbf{u}\_{\mathbf{y}} + \gamma \left(\mathbf{v}\_{\mathbf{x}|\mathbf{r}=(\mathbf{X},\mathbf{Y})} - \mathbf{u}\_{\mathbf{x}}\right) - \partial\mathbf{h}/\partial\mathbf{x}\Big|\_{\mathbf{r}=(\mathbf{X},\mathbf{Y})},\tag{76}$$

$$\mathbf{d}\mathbf{u}\_{\mathbf{y}}/\mathbf{d}\mathbf{t} = \mathbf{3}\mathbf{y} - 2\mathbf{u}\_{\mathbf{x}} + \gamma \left(\mathbf{v}\_{\mathbf{y}}|\_{\mathbf{r}=(\mathbf{X},\mathbf{Y})} - \mathbf{u}\_{\mathbf{y}}\right) - \partial\mathbf{h}/\partial\mathbf{y}\Big|\_{\mathbf{r}=(\mathbf{X},\mathbf{Y})},\tag{77}$$

where γ is a dimensionless parameter

$$
\gamma = \mathfrak{h}/\mathfrak{\Omega}\_0 = \mathfrak{18}\mathfrak{\rho}\nu/\mathfrak{\rho} \* \mathrm{D}^2 \mathfrak{\Omega}\_0. \tag{78}
$$

The mass of the rigid particles captured by a vortex during this time is in the

2

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

> <sup>2</sup> � 4A2 <sup>þ</sup> <sup>Ω</sup><sup>0</sup> <sup>2</sup> � �z2

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

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

� <sup>1</sup> <sup>¼</sup> <sup>2</sup> ffiffi

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

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,

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

2 *=*2 � ��1*=*<sup>2</sup>

> 2 p ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi <sup>1</sup> <sup>þ</sup> <sup>4</sup>*A*<sup>2</sup>

*=*Ω<sup>2</sup> 0

q� � � 1, (89)

*=*2,

2

*:* (88)

, at the center

Σ ∗ , (87)

Mp ≈ πreff

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

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

h zð Þ¼ cs0

zK ffi cs0*=*2Ω0. Therefore the relative thickening

Δ*z ZK*

with the positive excess mass density (see **Figure 3**).

� *<sup>Z</sup>*<sup>0</sup> *ZK*

where cs0 is a sound speed at the vortex center (enthalpy, h0 ¼ cs0

z0 <sup>¼</sup> cs0 2A<sup>2</sup> <sup>þ</sup> <sup>Ω</sup><sup>0</sup>

order

which forms a planetesimal.

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

enthalpy from the z coordinate:

area of a whirlwind:

Burgers is thicker.

**37**

**11. Discussion and conclusion**

In Eqs. (76) and (77) a characteristic length is accepted: the size of a trunk of a vortex r0, for characteristic time and speed �1*=*Ω<sup>0</sup> and Ω0r0, respectively.

In the vortex trunk area (r2*=*r0 <sup>2</sup> <1), the profile of rotation has uniform character:

$$\mathbf{v\_x} = -A\mathbf{x} - \alpha \mathbf{y} + \mathbf{O}\left(\mathbf{r}^2/\mathbf{r\_0}^2\right), \mathbf{v\_y} = -A\mathbf{y} + \alpha \mathbf{x} + \mathbf{O}\left(\mathbf{r}^2/\mathbf{r\_0}^2\right). \tag{79}$$

where Α and ω are measured in unit Ω0. Therefore

$$
\partial \mathbf{h} / \partial \mathbf{x} = - \left( \mathbf{A}^2 - \alpha^2 - 2\alpha \right) \mathbf{x} - 2 \left( \mathbf{a} + \mathbf{1} \right) \mathbf{y}, \tag{80}
$$

$$
\partial \mathbf{h} / \partial \mathbf{y} = 2\mathbf{A} (\boldsymbol{\omega} + \mathbf{1}) \mathbf{x} - \left( \mathbf{3} + \mathbf{A}^2 - \boldsymbol{\omega}^2 - 2\mathbf{a} \right) \mathbf{y}. \tag{81}
$$

With the use of Eqs. (76)–(81), we receive the equations of motion of rigid particles in the field of a vortex trunk:

$$\begin{Bmatrix} \dot{X} \\ \dot{Y} \\ \dot{u}\_x \\ \dot{u}\_y \end{Bmatrix} = \begin{Bmatrix} 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1 \\ a & b & -\gamma & 2 \\ -b & a & -2 & -\gamma \end{Bmatrix} \begin{Bmatrix} X \\ Y \\ u\_x \\ u\_y \end{Bmatrix},\tag{82}$$

where

$$\mathfrak{a} = \mathbf{A}(\mathbf{A} - \boldsymbol{\eta}) - (\mathfrak{w} + \mathbf{1})\mathbf{2} + \mathbf{1}; \mathbf{b} = \mathbf{2}\mathbf{A}(\mathfrak{w} + \mathbf{1}) - \mathfrak{w}\mathfrak{o}.\tag{83}$$

From Eq. (82) it follows that the equilibrium position of rigid particles in a vortex trunk is its center X ¼ Y ¼ 0, where ux ¼ uy ¼ 0 and *u*\_ *<sup>x</sup>* ¼ *u*\_ *<sup>y</sup>* ¼ 0. Particles gradually come nearer to the center of the vortex by helicoidal trajectories.

For establishing the stability of this position of balance, it is necessary to require real parts of eigenvalues of a matrix in Eq. (82) to be zero or negative.

Eigenvalues are complex:

$$\Lambda\_{1,2,3,4} = -\gamma/2 \mp i \pm \sqrt{\left[a - \mathbf{1} + \mathbf{\gamma}^2/4 \pm i(b - \mathbf{\gamma})\right]}, \dots$$

which gives stability condition

$$(\mathbf{b} - \boldsymbol{\eta})^2 + \boldsymbol{\eta}^2 (\boldsymbol{a} - \mathbf{1}) \le \mathbf{0},\tag{84}$$

Taking into account Eq. (83), Eq. (84) leads to stability criterion γ> Α, which for viscosity, ν, in a dimensional form, gives

$$
\nu > \rho \ast \mathrm{AD}^2 / 18\rho \tag{85}
$$

Hence, the unique position of balance for rigid particles in a Burgers vortex is its center where all particles captured by a vortex will gather during the characteristic time:

τ � ωreff*=*A√βν*:* (86)

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

The mass of the rigid particles captured by a vortex during this time is in the order

$$\mathbf{M}\_{\rm p} \approx \pi \mathbf{r}\_{\rm eff} \mathbf{\bar{\Sigma}} \, \ast \,, \tag{87}$$

which forms a planetesimal.

duy*=*dt ¼ 3y � 2ux þ γ vyjr¼ð Þ X,Y � uy

<sup>γ</sup> <sup>¼</sup> <sup>β</sup>*=*Ω<sup>0</sup> <sup>¼</sup> <sup>18</sup>ρν*=*<sup>ρ</sup> <sup>∗</sup> <sup>D</sup><sup>2</sup>

vortex r0, for characteristic time and speed �1*=*Ω<sup>0</sup> and Ω0r0, respectively.

*=*r0

where Α and ω are measured in unit Ω0. Therefore

In Eqs. (76) and (77) a characteristic length is accepted: the size of a trunk of a

With the use of Eqs. (76)–(81), we receive the equations of motion of rigid

0 01 0 0 00 1 *a b* �*γ* 2 �*b a* �2 �*γ*

From Eq. (82) it follows that the equilibrium position of rigid particles in a vortex trunk is its center X ¼ Y ¼ 0, where ux ¼ uy ¼ 0 and *u*\_ *<sup>x</sup>* ¼ *u*\_ *<sup>y</sup>* ¼ 0. Particles

For establishing the stability of this position of balance, it is necessary to require

gradually come nearer to the center of the vortex by helicoidal trajectories.

real parts of eigenvalues of a matrix in Eq. (82) to be zero or negative.

<sup>Λ</sup>1,2,3,4 ¼ �γ*=*2∓<sup>ⅈ</sup> � <sup>√</sup> *<sup>a</sup>* � <sup>1</sup> <sup>þ</sup> <sup>γ</sup><sup>2</sup>

ð Þ b � γ

<sup>2</sup> <sup>þ</sup> <sup>γ</sup><sup>2</sup>

*ν*>ρ ∗ АD2

Hence, the unique position of balance for rigid particles in a Burgers vortex is its center where all particles captured by a vortex will gather during the characteristic time:

Taking into account Eq. (83), Eq. (84) leads to stability criterion γ> Α, which for

<sup>2</sup> � �, vy ¼ �Α<sup>y</sup> <sup>þ</sup> <sup>ω</sup><sup>x</sup> <sup>þ</sup> O r<sup>2</sup>

<sup>∂</sup>h*=*∂<sup>x</sup> ¼ � <sup>Α</sup><sup>2</sup> � <sup>ω</sup><sup>2</sup> � <sup>2</sup><sup>ω</sup> � �<sup>x</sup> � <sup>2</sup> Α ωð Þ <sup>þ</sup> <sup>1</sup> y, (80)

<sup>∂</sup>h*=*∂<sup>y</sup> <sup>¼</sup> <sup>2</sup>Α ωð Þ <sup>þ</sup> <sup>1</sup> <sup>x</sup> � <sup>3</sup> <sup>þ</sup> <sup>Α</sup><sup>2</sup> � <sup>ω</sup><sup>2</sup> � <sup>2</sup><sup>ω</sup> � �y*:* (81)

9 >>>=

8 >>><

>>>:

*X Y ux uy*

9 >>>=

>>>;

>>>;

*<sup>=</sup>*<sup>4</sup> � <sup>ⅈ</sup>ð Þ *<sup>b</sup>* � *<sup>γ</sup>* � �, *:*

ð Þ *a* � 1 ≤0, (84)

*=*18ρ (85)

τ � ωreff*=*A√βν*:* (86)

*a* ¼ A Að Þ� � γ ð Þ ω þ 1 2 þ 1; b ¼ 2Α ωð Þ� þ 1 γω*:* (83)

where γ is a dimensionless parameter

*Vortex Dynamics Theories and Applications*

In the vortex trunk area (r2*=*r0

vx ¼ �Α<sup>x</sup> � <sup>ω</sup><sup>y</sup> <sup>þ</sup> O r<sup>2</sup>

particles in the field of a vortex trunk:

Eigenvalues are complex:

which gives stability condition

viscosity, ν, in a dimensional form, gives

where

**36**

*X*\_ *Y*\_ *u*\_ *x u*\_ *y*

9 >>>= 8 >>><

>>>:

>>>; ¼

8 >>><

>>>:

� � � <sup>∂</sup>h*=*∂<sup>y</sup>

� �

<sup>2</sup> <1), the profile of rotation has uniform character:

*=*r0

Ω0*:* (78)

<sup>2</sup> � �*:* (79)

, (82)

<sup>r</sup>¼ð Þ X,Y , (77)
