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

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].

which describes the nonlinear Rossby waves in the atmosphere [2] and drift nonlinear waves in plasma [3]. Here ψxyzis a stream function: v=ez×∇ψ. In plasma physics ψis the electric potential, and constant v0is 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, 6, 7, 8, 9, 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≫λJperturbations, 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, 12, 13, 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, 22, 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

This is a vortex with a converging stream of substance to its center with gradient –Α,ωand r0as the circulation and the size of a trunk of a vortex. Rotation of a vortex trunk is almost uniform, then falls down under hyperbolic law, and at distance reff=4.5r0(effective radius of vortex) makes 1/3 of the maximum value of vθ(Figure 1). The asymptotic behavior of Burgers vortex in small and big distances from the vortex center represents the Rankin vortex [27, 28].

2. Magnitude of some parameters of circumstellar disks

A typical circumstellar disk is a few hundred AU (astronomical unit, 1 AU = 1.5 10¹³ 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

where γ=1.4is the gas adiabatic index, k is Boltzmann constant, mHis hydrogen atom mass.

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

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≈2Hρ.

In “α-model” of accretion disk [29], the expense of gas occurs with a speed dm/dt=3πνΣ, where νis the kinematic viscosity of gas, ν=αc_s H.

The dynamic time scale of a disk is

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

At cs∼1km/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: τd∼r/Δv∼R/A.E.102yr.

For a characteristic time τs∼Σ/αΩρ∗[31], where ρ∗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∗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]:

where vθis 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:

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, 0.01c_s, 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].

3. Model and basic equations for linear perturbations

Consider at first a gravitating pure gas disk of mass density ρr, which rotates with angular velocity Ωraround the z-axis. Explore 2D perturbations in plane of the disk, ignoring its vertical structure. Present any characteristic functions of the disk as f0r+frφt, where f0rdescribes the equilibrium state and frφtis a small but finite perturbation.

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

where Pis the pressure and

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

Perturbations of the disk in a rotating with angular velocity Ω0≡Ωr0cylindrical coordinate system (Figure 2) are described by 2D hydrodynamic equations¹:

where the velocity was presented in the form of

Φis the sum of perturbations of gravitational potential U and enthalpy

and the Poisson equation is

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

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

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:

In a uniformly rotating (Ω=const.) gravitating disk, no drifting stationary vortex solution can be obtained without specifying the function Bψ, because Eq. (18) can be represented as a Jacobean Jψcurlzv+2Ω/ρ=0,which satisfies the arbitrary circularly symmetric vortex disturbance around point O.

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:

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

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

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

The stream function ψxyof perturbed velocity (19) is expressed through perturbations Φby the following formula:

Imagine around a point O function ρxand Ωxin the form of

In this case V0≈r0Ω'0x.

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

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

where kR=Ω0/csis the Rossby wave number, κ02=2Ω02Ω0+r0Ω'0is the square of the epicyclical frequency, and

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

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

where ωJ2≡4πGρ0is the square of the Jeans frequency.

The order of magnitude of |H/U| can be estimated using the definition cs2=dP0/dρ0andΔΦ=k2Φ.

where k is wavenumber of perturbations

where kJ=ωJ/csis the Jeans wavenumber. Eq. (28) shows that the case H≫Udescribes the small-scale disturbances: matching k2/kJ2≫1,orλ≪λJ. In this case ψ=H/2Ω0≡h, and Eq. (25) takes the form

Limit H≪Ucorresponds to the large-scale perturbations −λ≫λJ. Then ψ=U/2Ω0≡ϕ, and Eq. (27) turns into

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, β=0,κ02=4Ω02, when the vortencity is constant and is equal to Γ/ρ0πa2where Γis the velocity circulation. We assume that the velocity circulation Γdiffers from zero only in a circle of radius a≪r0around point O. Using now the polar coordinates Rθ:x=Rcosθ,y=Rsinθ(Figure 2), the equation for disturbances (29) can be written as

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

where s≡2akR. This is a monopole vortex with the following perturbed velocity field:

Note that the vortices with positive and negative velocity circulation Γhave 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 2akR≈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=12. This vortex is a retrograde-circulating rarefaction around the center O condensation in the case of Γ<0and prograde-circulating rarefaction in the case of Γ>0(Figure 3). The decrease of density in the area R>aof 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]:

where γ=1−Ω2/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 e_z: e_z × Eq. (10), gives

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

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

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

where

The continuity equation now takes the form

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

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

where

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

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

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≪r0in the form of a vortex drifting in y-direction at a constant speed u. Introducing the wave variable

Eq. (46) can be rewritten in the form

or in the form of the Jacobean

where

On basis of Eq. (49)

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

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

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→0for large R. This condition implies

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

where J1and K1are Bessel and Macdonald functions, respectively, and g=kaand s=paare connected by “dispersion equation” which is transcendental

For long-scale perturbations

while for small-scale disturbances

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

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

The current lines are determined by dR/vr=Rdθ/vθthat gives

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

7. The contours of constant density

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

In the long-scale limit, Φ=Uandψ=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:

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:

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

where

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

where Rcand v0are constants, rotation of a disk in the small area (a≪Rc) can be considered as uniform, and βlwfor this model turns out to be equal to

where we used the relation r2≈Rc2+2RRccosθ, by placing the center O in R=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, 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, 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:

where x=R/a, h is the thickness of the gas disk, H1gxis the Struve function of the first order, and x1is the root of equation J1gx1=0. Similarly

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 Ω0at distance R0round 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

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=2,i.e.2≥q≥3/2.

The substance stream in chosen frame of reference, has X component of speed -iqΩ0y, centrifugal force is compensated by gravitation of the central star at distance R0, in other points their sum gives the tidal force j3Ω02y. The vertical component of gravitation ‐Ω02zis 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:

where h is specific enthalpy h=∫ρ−1dpand 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

where r2≡x2+y2.

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:

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

where γis a dimensionless parameter

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/Ω0and Ω0r0, respectively.

In the vortex trunk area (r2/r02<1), the profile of rotation has uniform character:

where Αand ωare measured in unit Ω0. Therefore

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

where

From Eq. (82) it follows that the equilibrium position of rigid particles in a vortex trunk is its center X=Y=0, where ux=uy=0and u̇x=u̇y=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:

which gives stability condition

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

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:

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

which forms a planetesimal.

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:

where cs0is a sound speed at the vortex center (enthalpy, h0=cs02, at the center of vortex is estimated by Clapeyron equation) and Ω₀ 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:

The question arises whether the disk thickness in area of vortex localization changed. On radius of R0the half-thickness of Keplerian disk from (4) is of order zK≅cs0/2Ω0. Therefore the relative thickening

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<0vortex 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 (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 0.5M⊙round a star of solar mass: M≈M⊙. Taking R0=20AUwe will obtain estimations Ω0≈8·10−9s−1and Σ∼1600g/cm2. For a typical protoplanetary disk at considering distance the vertical scale height is of order H≈108kmand sound speed cs≈HΩ0≈0.8km/s.

Let the maximum rotation speed of a vortex be ∼10 m/s at distance r0≈1010mfrom its center, and converging speed of a stream be vr=A·r0≈5m/s. Then we will have

The condition (85) is carried out with a large supply for protoplanetary disks. The molecular viscosity of gas, estimated by the formula ν∼λcs, 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 ν∼αcsH≈αH2Ω0. The dimensionless parameter α is constant value of an order α∼10−2. The scale of viscous length thus makes Lν≈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<Lshearcan have circular form.

Taking ρ∗/ρ≈1010in 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:

Therefore, during an order of ∼10⁶ 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.