Dynamics of Vortex Structures: From Planets to Black Hole Accretion Disks

Abstract

Thermo-vortices (bright spots, blobs, swirls) in cosmic fluids (planetary atmospheres, or even black hole accretion disks) are sometimes observed as clustered into quasi-symmetrical quasi-stationary groups but conceptualized in models as autonomous items. We demonstrate—using the (analytical) Sharp Boundaries Evolution Method and a generic model of a thermo-vorticial field in a rotating “thin” fluid layer in a spacetime that may be curved or flat—that these thermo-vortices may be not independent but represent interlinked parts of a single, coherent, multi-petal macro-structure. This alternative conceptualization may influence the designs of numerical models and image-reconstruction methods.

1. Introduction

Recently, a highly complex analysis of observations of the black hole Sagittarius A* (Sgr A*) yielded an impressive image (see Figure 1, panel A) that suggests the possible presence of three bright spots within its structure outside the shadow of a black hole (see Refs. [1,2,3,4,5,6]). Visually, this phenomenon is resemblant of the formations already observed in the atmospheres of planets (also shown in Figure 1, panels B–D). Indeed, regular in shape, long-lasting vortices and multi-vortex structures appear frequently in nature; some well-known examples are the Antarctic Polar Vortex, oceanic vortices, cyclones and anti-cyclones in atmospheres, cyclone storms on Jupiter poles, and analogous vortex structures in the atmospheres of outer planets. If further observational studies confirm that the bright spots in Figure 1A are not artifacts of the image-reconstruction algorithm but are indeed physical in nature (apparently associated with temperature anomalies/emissions in localized areas), then the list of known cosmic environments with multi-vortex structures would be expanded to include black hole accretion disk flows.

All vortex structures are naturally studied using frameworks and models that seem most appropriate for their native environments. Typically, however, the individual vortices are conceptualized as stand-alone items, sometimes interacting, sometimes not.

In this paper—while contemplating the image presented in Figure 1A and its visually bright spots (although the only significant real feature in the image is the shadow and its diameter, the non-uniformity of the brightness along the ring is not very significant and is subject to processing/reconstruction methods)—we analytically demonstrate that such spots may exist and represent connected “petals” of a single multi-petal thermo-vorticial macro-structure. Within such a structure, the vorticity of the “hydrodynamic” (i.e., collective) flow is interlinked with the temperature field of the medium. This vorticity, a Lagrangian characteristic of the hydrodynamic flow, is transported by the flow, thus creating spots with an elevated temperature.

The awareness of the existence of two interpretations of the structure and the linkage of thermo-vortices is important. The initial conceptualization—whether the thermo-vortices are treated from the start as stand-alone items or as inextricable parts of one macro-structure—may meaningfully influence the numerical models and image-reconstruction methods.

In our model, we make the following assumptions and impose the following restrictions: the flow of the disk fluid is assumed to be large-scale (the Reynolds number or its analogue is large, i.e., viscous effects are small and may be neglected); the flow is slow (relative to the sound speed); and the assumption of medium incompressibility is acceptable. To simplify the mathematical complexity of analytic solutions, our model considers the “thin”-layer case: (1) it is assumed that the accretion disk is relatively thin, i.e., the characteristic size of the structure is assumed to be significantly larger than the characteristic thickness of the disk (layer) at the location at which the structure is localized; (2) the motion of the medium is assumed to be quasi-two-dimensional, i.e., the component of the collective flow velocity in the direction perpendicular to the disk plane is suppressed or significantly smaller in magnitude than the characteristic flow velocity in the disk plane (in this context, a “thick” system may also, in principle, be described using the two-dimensional treatment if a vorticity patch is broken into columnar sub-patches, as in Refs. [9,10]); (3) the temperature decreases towards the center of the disk due to the assumption that the heating of the disk mainly occurs not from the hot central body but from the periphery where the capture and destruction of captured bodies occurs due to the tidal effect.

The medium (disk layer) in our model is presumed to rotate. Vorticity is non-zero where there is velocity shear. We describe the “average” vorticity, representative of some “band” within the disk, using the symbol $\omega$. (Obviously, this vorticity $\omega$ is not the same as the spin of the central body or the characteristic angular velocity $\mathrm{\Omega }$ of the disk’s rotation.) When $\mathrm{\Omega }$ is “sufficiently” high, the centrifugal effects (in the rotating frame) become prominent in the leading terms, and even dominate the gravitational effect of attraction to the central body.

To avoid any confusion, for the modelers accustomed to working with trajectories for particles, let us emphasize that we work with the field, not with individual particles (or their trajectories or orbits). Perhaps what may help the reader grasp this nuance better is the reminder that the velocity of the displacement of electrons in a usual house-wire is not the same thing as the speed of propagation of the electro-magnetic field perturbation along that same wire.

Additionally, we limit our consideration to the case when the magnetic field is weak and the plasma is non-relativistic. This means that in our treatment we neglect the effect of the magnetic field on the large-scale “hydrodynamics” of the flow, i.e., in the full tensor of the energy–momentum of the macroscopic “hydrodynamic” flow combined with the electromagnetic field, we neglect the magnetic part. However, this does not mean that this “weak” magnetic field has no impact on the electromagnetic radiation (for example, on the polarization of the synchrotron radiation). Indeed, as Ref. [11] indicates, EHT images in linear polarization have identified a coherent spiral pattern around the black hole, produced from ordered magnetic fields threading the emitting plasma.

Finally, our focus is on steady-state thermo-hydrodynamic structures. This is important to keep in mind for numerical modelers who typically initiate their models at time $t=0$ (and may start, for example, with the spherical Bondi accretion) and allow their models to evolve towards some steady state (hoping to minimize accumulated errors along the way). In contrast, our steady-state is derived analytically; we jump straight to the model of a flattened rotating medium (disk, layer) and examine its quasi-stationary dynamics.

The paper is organized as follows: Section 2 explains the model, Section 3 presents the results, and Section 4 concludes with the discussion. Appendices Appendix A and Appendix B explain the details of calculations; Appendix C elaborates the accretion disk model.

The derived insights may be useful for analyses of environments ranging from accretion disks (in an approximation of curved spacetime) to planetary atmospheres (in an approximation of flat spacetime): the design of numerical models and image reconstruction methods may be influenced by how the steady-state of the media is conceptualized—as multiple (separate) quasi-localized vortices or as a single (integrated) multi-petal structure.

2. Model

We use the quasi-2D fluid dynamics model, which describes large-scale vorticial motions of a “thin” rotating disk in a mildly curved spacetime.

Coordinates and Notations: In the model, we use the coordinate system where $x\equiv x_1$ and $y\equiv x_2$ are the axes in the horizontal plane and z points vertically up. We employ the following notations: $\mathrm{\Phi }$ is the potential of the force field; temperature $\theta \equiv {\theta }_0+{\tau }_s+\tau$, where ${\theta }_0$ is the (constant) baseline temperature of the disk, ${\tau }_s$ is the spatially inhomogeneous, axially symmetrical, time-independent part of the temperature distribution, and $\tau =\tau (t,x_j)$ is the temperature—a dynamical quantity related to the vortex structures in the fluid; $v_i$ signifies the components of the velocity field $\mathbf{v}$; ${\omega }_3$ is the z-component of vorticity $[\nabla \times \mathbf{v}]$; and ${ϵ}_{ijk}$ is the standard alternating tensor (${ϵ}_{123}=1$, ${ϵ}_{ijk}=0$ when any two indices are equal, ${ϵ}_{ijk}=+1$ for any even number of permutations of indices from ${ϵ}_{123}$, and ${ϵ}_{ijk}=-1$ for any odd number of permutations). Note also that ${ϵ}_{3jk}{\partial }_j{\tau }_s{\partial }_k\mathrm{\Phi }\equiv 0$ because both ${\tau }_s$ and $\mathrm{\Phi }$ are axially symmetric. Furthermore, we assume that the band of the disk between $r_1$ and $r_2>r_1$, where the vortex structure of interest is formed, is characterized by an approximately constant angular velocity $\mathrm{\Omega }=(0,0,\mathrm{\Omega })$.

“Thin” Disk: In the simplest model—in which the vortex structures are realized in a ”thin” flat sheet with $h\ll L$, $V^2\ll s^2\ll c^2$, $Re\gg 1$—the z-component of velocity, w, vanishes and may be dropped in all formulas (see Ref. [10], developing this particular model, or classical Refs. [12,13,14]). Here, h is the characteristic local thickness of the disk, which is slowly dependent on the radial coordinate; L is the characteristic spatial length over which the macroscopic flow characteristics change substantially; s is the so-called “isothermal sound speed”; V is the characteristic fluid velocity; c is the speed of light; dimensionless parameter $Re$ is the Reynolds number (or its analogs) such that $Re\gg 1$.

The System of Equations: The set of equations for the z-component of vorticity ${\omega }_3$ and for variations in temperature $\tau$ becomes:

$$\begin{array}{c}\hfill {\partial }_j{v}_j=0,D_t{\omega }_3=\beta {ϵ}_{3jk}{\partial }_j\tau {\partial }_k\mathrm{\Phi },D_t\tau =-v_i{\partial }_i{\tau }_s,\end{array}$$

(1)

where indices $j,k=1,2$. Operator $D_t\equiv {\partial }_t+v^k{\partial }_k$ is the total derivative. The condition of incompressibility $div\mathbf{v}=0$ does not mean that the density of the medium is constant. This condition simply means that the density evolves according to the equation $({\partial }_t+v^I{\partial }_i)\rho =0$. Taking $\rho ={\rho }_0+{(\partial \rho /{\partial }_T)}_p(T-T_0)+{(\partial \rho /\partial p)}_T(p-p_0)\simeq {\rho }_0(1-\beta (T-T_0))$, after some simple manipulation, the evolution equation for temperature perturbations takes the form of the third equation in Equation (1). For the equation of state, we will assume, in accordance with the assumption $div\mathbf{v}\to 0$, that the density essentially depends only on the temperature and not on the pressure. Thus, we write $\rho \simeq {\rho }_0(1-\beta (T-T_0))$, where subscript 0 denotes the reference values. The coefficient of thermal expansion $\beta =-{\rho }^{-1}{(\partial \rho /\partial p)}_T$ is presumed to be constant in the layer. Since quantity $\beta (T-T_0)$ is generally small (relative to 1), one may neglect the density variations and hence replace $\rho$ with the constant value ${\rho }_0$ in all terms of the dynamic momentum equation except for the “buoyancy” term.

We will presume that a collective fluid motion forms in the system, for which the relationship between the flow velocity and temperature variation is given by

$$\begin{array}{c}\hfill v_i=a{ϵ}_{3ij}{\partial }_j\tau \equiv a{ϵ}_{ij}{\partial }_j\tau .\end{array}$$

(2)

Here, tensor ${ϵ}_{ij}$ is the antisymmetric unit, second-order tensor, for which ${ϵ}_{12}=-{ϵ}_{21}$ and the diagonal components are zero; and a is some parameter assuring the correct dimension for the relationship between the considered quantities. In this case, vorticity ${\omega }_3=-a\Delta \tau +2{\mathrm{\Omega }}_3$, where $\Delta ={\partial }_1^2+{\partial }_2^2$. Then, notably, the presence (or absence) of the component with constant ${\mathrm{\Omega }}_3$ does not impact the core set of Equation (1).

As mentioned, the vorticial field is intertwined with the thermal field. Thus, in addition to the set of Equation (1), any model must include equations for the temperature field. We set the ansatz for the temperature profile as $\theta \equiv {\theta }_0+{\tau }_s+\tau$, where ${\theta }_0$ is the (constant) baseline temperature of the disk, ${\tau }_s$ is the spatially inhomogeneous, axially symmetrical, time-independent part of the temperature distribution, and $\tau =\tau (t,x_j)$ is the temperature—a dynamical quantity related to the vortex structures in the fluid. We chose the stationary temperature deviation ${\tau }_s$ in a form that allows for a self-consistent analytical solution. This choice does not contradict experimental observations (see, for example, Ref. [15]). Therefore, we will model (using notation $r=\left|\mathbf{x}\right|$ and $\mathbf{x}=(x_1,x_2)$, and the Taylor series expansion) the background distribution of the time-independent temperature deviation ${\tau }_s$ in the simplest form:

$$\begin{array}{c}\hfill {\tau }_s=\frac{{\tau }_1}{2}{r}^2,{\tau }_1>0,\end{array}$$

(3)

where ${\tau }_1$ is the parameter characterizing the “rapidity” of the increase in the basal temperature of the medium as the distance r from the disk rotation axis grows. (Generally speaking, the question of what shape the temperature profile takes within a black hole’s accretion disk is an open one. Astronomical measurements remain challenging, despite significant progress. See, for example, Refs. [15,16].) When the leading contribution to potential $\mathrm{\Phi }$ comes from the centrifugal effects, we can express $\mathrm{\Phi }=-K^2{\left|\mathbf{x}\right|}^2-(1/2){[\mathrm{\Omega },\mathbf{x}]}^2\simeq -(1/2){[\mathrm{\Omega },\mathbf{x}]}^2$. Here, the square brackets denote the cross product of two vectors. Parameter $K^2$ is determined by the expression $K^2=G{M}_{BH}/r^3=c^2{r}_g/\left(2r^3\right)$, where $K\sim r^{-3/2}$ is the so-called Kepler parameter (in the Newtonian approximation of gravity), which is interpreted as the angular velocity of a test particle on a circular orbit at distance r: G is the gravitational constant; c is the speed of light; $M_{BH}$ is the black hole “mass”. Thus, when angular velocity $\mathrm{\Omega }$ is presumed to remain constant within the band of r where the thermo-vorticial macro-structure forms, the set of the interlinked nonlinear evolution equations becomes:

$$\begin{array}{c}\hfill a{\partial }_t\Delta \tau +a^2{ϵ}_{ik}{\partial }_i\tau {\partial }_k\Delta \tau =\beta {\mathrm{\Omega }}^2{ϵ}_{ik}{x}_i{\partial }_k\tau ,\end{array}$$

(4)

$$\begin{array}{c}\hfill {\partial }_t\tau +a{ϵ}_{ik}{\partial }_i\tau {\partial }_k\tau =a{\tau }_1{ϵ}_{ik}{x}_i{\partial }_k\tau ,\end{array}$$

(5)

where, obviously, quantity $v_k=a{ϵ}_{ik}{\partial }_i\tau$ is the k-th component of the flow velocity.

Equations (4) and (5) have a transparent physical meaning. Their left sides describe the transport of dynamic (field) quantities: vorticity ($\sim \Delta \tau$) and temperature perturbation ($\tau$). The right sides of these equations describe “sources” that generate the vortices and the temperature perturbations. In other words, the vortices are generated by the source described by the right part of Equation (4), which is effective (non-zero) only when there exists an inhomogeneous gravity-like force field $\mathrm{\Phi }\sim {\mathrm{\Omega }}^2$ with which temperature perturbation $\tau$ interacts via the equation of state (when compressibility $\beta \ne 0$), while the quantity $\Delta \tau$—which characterizes the vortex field in the fluid—is transported by the self-induced flow (the left part of Equation (4)). On the other hand, in Equation (5), temperature perturbation $\tau$ is transported by the self-generated flow ($v_i\ne 0$); temperature perturbation $\tau (t,\mathbf{x})$ is generated by the source (the right part of Equation (5)), which is non-zero only when there exists a spatial- and time-independent temperature gradient (i.e., when ${\tau }_1\ne 0$). The processes are interlinked because the “source” of one dynamic quantity depends on a complex combination of other dynamic quantities. Equations (4) and (5) show that when the stratification of temperature is absent (${\tau }_1=0$) and there is no disk rotation ($\mathrm{\Omega }=0$, i.e., centrifugal force is zero), then Equations (4) and (5) degenerate into the traditional equations for vortex evolution and transport in a two-dimensional ideal fluid. Comparing Equations (4) and (5), we conclude that both the first and second equations describe the evolution of the same physical entity. As follows from the definition of hydrodynamical velocity, the dimension of coefficient a is $\left[a\right]={\theta }^{-1}{L}^2{T}^{-1}\equiv {\left[temperature\right]}^{-1}\times {\left[length\right]}^2\times {\left[time\right]}^{-1}$.

Lagrangian Quantity (“Q-Vorticity”): By combining Equations (4) and (5), i.e., after multiplying the second equation by $\beta {\mathrm{\Omega }}^2/a{\tau }_1$ and subtracting it from the first, we obtain:

$$\begin{array}{c}\hfill ({\partial }_t+a{ϵ}_{ik}{\partial }_i\tau {\partial }_k)(\Delta \tau -R^{-2}\tau )=0.\end{array}$$

(6)

Here, parameter $R^{-2}=a^{-2}(\beta /{\tau }_1){\mathrm{\Omega }}^2$. Its dimension is ${\left({\theta }^1{L}^{-2}{T}^1\right)}^2\times {\theta }^{-1}\times {\theta }^{-1}{L}^2\times T^{-2}=L^{-2}$, showing that R is a space scale factor. If the question is posed to examine the evolution of the temperature field, then the nonlinear differential equation Equation (6) is solved numerically, specifying the initial distribution of the temperature field, as well as the spatial boundary conditions.

On the other hand, we may define a new quantity

$$\begin{array}{c}\hfill q\equiv a(\Delta \tau -R^{-2}\tau ),\end{array}$$

(7)

which we will call “q-vorticity” because this quantity is linked to the streamfunction via definitions. This quantity is transported by the flow according to Equation (6), i.e., as a Lagrangian quantity. Thus, the integral of any function of q in the $xy$-plane is always conserved.

Temperature: Temperature $\tau$ is found via the Green function representation. It is presented as follows:

$$\begin{array}{c}\hfill \tau (\mathbf{x},t)=\frac{1}{a}\int d\mathbf{x}\prime G(\mathbf{x},{\mathbf{x}}^{\prime })q({\mathbf{x}}^{\prime },t).\end{array}$$

(8)

Here, the Green function satisfies the following equation:

$$\begin{array}{c}\hfill (\Delta -R^{-2})G=\delta (\mathbf{x},{\mathbf{x}}^{\prime }),\end{array}$$

(9)

where $\delta$ is the Dirac delta-function. The collective “hydrodynamical” velocity of the flow (orthogonal to the temperature gradient) is calculated using Equation (2).

Modeling Steady-State: So, what is the initial vorticity distribution $q(\mathbf{x},t)$ and how can it be modeled? Recall that we are not talking about the evolution of the system from some state of “nothing”, but are focusing on the (dynamical) steady-state to which the system eventually evolves.

Two methods are useful for modeling such a steady-state of $q(\mathbf{x},t)$.

For a strongly localized vortex—see, for example, Ref. [17]—quantity $q(\mathbf{x},t)$ can be parameterized by a function in the form $F(\mathbf{x}-{\mathbf{x}}_0\left(t\right))$, i.e., the one with the center at coordinate $\mathbf{x}={\mathbf{x}}_0\left(t\right)$, which satisfies the equation of transport ${\dot{x}}_{0k}\left(t\right)+{ϵ}_{ik}{\partial }_i\psi =0$. In other words, in such cases, the center of the vortex moves according to ${\dot{\mathbf{x}}}_0\left(t\right)-\mathbf{v}\left[{\mathbf{x}}_0\left(t\right)\right]=0$. (Here, the symbol “dot” signifies the derivative with respect to time.)

Alternatively—as we will consider below—the distribution of $q(\mathbf{x},t)$ can be described in the form of a macro-structure with “petals” (with constant $q(\mathbf{x},t)=q_0$ inside) whose moving boundaries evolve according to Equation (6).

Extended (i.e., not narrowly localized) stationary vortex structures—the ones rotating with constant angular velocity $\omega (\equiv {\omega }_3)$—are simpler to consider in a rotating coordinate system where the structures appear immovable, i.e., when their rotation direction is co-aligned with the z-axis and the derivative with respect to time becomes ${\partial }_t=-\omega {ϵ}_{ik}{x}_i{\partial }_k$. (The procedure is laid out, for example, in Ref. [18].) Indeed, when $f=f(\rho ,\varphi -\omega t$), then calculations relying on the properties of Jacobeans produce the following: ${\partial }_{t}f=-\omega {\partial }_{\varphi }f=-\omega \partial (f,\rho )/\partial (\varphi ,\rho )=-\omega (\partial (f,\rho )/\partial (x_1,x_2))(\partial (x_1,x_2)/\partial (\varphi ,\rho ))=-\omega \left({ϵ}_{ik}{\rho }^{-1}{x}_i{\partial }_{k}f\right)(-\rho )$. Then, Equation (5) may be rewritten as follows:

$$\begin{array}{c}\hfill {ϵ}_{ik}{\partial }_i\left(-\frac{\omega }{2}{\left|\mathbf{x}\right|}^2+a\tau \right){\partial }_k\left(\tau +\frac{{\tau }_1}{2}{\left|\mathbf{x}\right|}^2\right)=0,\end{array}$$

(10)

which is satisfied by the ansatz

$$\begin{array}{c}\hfill \tau +\frac{{\tau }_1}{2}{\left|\mathbf{x}\right|}^2=F(-\frac{\omega }{2}{\left|\mathbf{x}\right|}^2+a\tau ).\end{array}$$

(11)

The explicit expression of function F can be found from the obvious fact that temperature-driven flow perturbations must vanish when temperature perturbations vanish themselves. The suitable expression is as follows: function $F\left(u\right)=-{\tau }_{1}u/\omega$. Then, temperature fluctuations are expressed as follows:

$$\begin{array}{c}\hfill \tau =-\frac{{\tau }_1}{\omega }a\tau .\end{array}$$

(12)

It follows from here that parameter $a=-\omega /{\tau }_1$. Combining this with Equation (4), where ${\partial }_t=-\omega {ϵ}_{ik}{x}_i{\partial }_k$ is taken into account, we obtain the second evolution equation—the one which describes the rotation of a stationary vortex structure with angular velocity $\omega$ caused by the self-induced field of hydrodynamical velocity:

$$\begin{array}{c}\hfill {ϵ}_{ik}(-\omega x_i+a{\partial }_i\tau ){\partial }_k\left(\Delta \tau -R^{-2}\tau \right)=0.\end{array}$$

(13)

Here, $R^2={\left({\tau }_1\beta \right)}^{-1}{(\omega /\mathrm{\Omega })}^2\equiv {\lambda }_{\theta }^2{(\omega /\mathrm{\Omega })}^2$, where the thermal length scale ${\lambda }_{\theta }={\left({\tau }_1\beta \right)}^{-1/2}$ characterizes the background regime of this model. Obviously, an extended (multi-petal) thermo-vortex structure exists when parameter ${\tau }_1$, characterizing the “rapidity” of the change in temperature (and stream function) with the distance from the disk rotation axis of ${\tau }_1>0$.

The “Blob” in The Method of Contour Dynamics / Sharp Boundaries Evolution Method (SBEM): The term “thermo-vorticial blob” defines, in the plane ($x_1,x_2$), a domain D filled with constant vorticity $q_0$ (with the dimensions $\left[q_0\right]={\left[time\right]}^{-1}$) and bounded by a closed contour $\partial D$.

The contour can be described in the parametric form, where parameter s is the contour’s arc length (starting from some initial point on the contour), such that $0<s<L$, where L is the entire length of the contour. The shape of the blob boundary is given by the following equations:

$$\begin{array}{c}\hfill x\left(s\right)={\nu }_{1}cos\varphi -{\nu }_{2}sin\varphi ,y\left(s\right)={\nu }_{1}sin\varphi +{\nu }_{2}cos\varphi ,\end{array}$$

(14)

where the quantities ${\nu }_1$, ${\nu }_2$, and $\varphi$ are functions of s. Next, we introduce dimensionless variables and parameters: $\sigma =L/l,s\to Ls,\alpha =(\omega /q_0){(l/R)}^3,\eta =\sigma /4K\left(m\right)$. Here, $K\left(m\right)$ is the complete elliptic integral of the first kind. Omitting the details of the analysis, we will only point out that the functions

$$\begin{array}{c}\hfill {\nu }_1=\frac{1}{8\alpha }{\kappa }^{\prime },{\nu }_2=-1+\frac{1}{16\alpha }{\kappa }_0^2-\frac{1}{16\alpha }{\kappa }^2\end{array}$$

(15)

are linked with the curvature of the contour $\kappa$ via the following non-linear differential equation:

$$\begin{array}{c}\hfill {\kappa }^{″}+(8\alpha -\frac{1}{2}{\kappa }_0^2)\kappa +\frac{1}{2}{\kappa }^3-8\alpha =0.\end{array}$$

(16)

Here, prime signifies a derivative with respect to dimensionless argument $\sigma s$. The angular coordinate $\varphi \left(s\right)$ of the tracing point on the contour (Figure 2) is calculated from the following obvious expression:

$$\begin{array}{c}\hfill \varphi \left(s\right)-\varphi \left(0\right)={\int }_0^{\sigma s}d\xi \kappa \left(\xi \right)\end{array}$$

(17)

which follows from the definition of curvature $\kappa$.

Equation (16) resembles Duffing’s equation:

$$\begin{array}{c}\hfill y^{″}+(1-2m)y+2m{y}^3=0,\end{array}$$

(18)

which describes oscillations in a non-linear oscillator (see Refs. [19,20,21,22], for example). Equation (A34) is solved analytically in terms of Jacobi elliptic functions $y=cn(\theta ,m)$ with period $4K\left(m\right)$. The non-zero free term $8\alpha$ complicates the straightforward solving of Equation (16).

To bring the equation into a canonical form without the free term, we can make the following manipulations: make a “shift” in $\kappa$, “flip” the result “upside down”, make a change in the magnitude, make one more “shift” in the result, and finally, make a change in the magnitude again. In other words, the solution has the following form:

$$\begin{array}{c}\hfill \kappa \left(s\right)=A+\frac{B}{1-ϵz\left(\lambda s\right)}\end{array}$$

(19)

where $z\left(\lambda s\right)$ is the Jacobi elliptic cosinus and $\lambda =n4K\left(m\right)$ for n-petal configuration: $n=2,3,\cdots$. Parameter $\lambda$ provides the periodicity of the solution ($z\left(0\right)=z\left(\lambda \right)$), i.e., the closing of the contour. Therefore, we change the argument, transitioning from $\sigma s$ to $\theta =\lambda s$. The initial conditions are introduced as $\kappa \left(0\right)={\kappa }_0$ and ${\kappa }_{,s}\left(0\right)=0$.

The solution contains six free parameters: $\alpha$, $\eta =\sigma /4K\left(m\right)$, A, B, m, $ϵ$. However, one parameter remains free; this fixes the values of all other parameters. We select $ϵ$ as the controlling parameter. Parameter $ϵ$ determines the magnitude of the initial perturbation because of $\kappa \left(0\right)=A+B/(1-ϵ)$. Obviously, for the unperturbed state, when $ϵ=0$, we have $A+B=1,\sigma =2\pi ,m=0,\varphi \left(0\right)=\pi /2,\eta =1$.

Omitting the details of calculations, we found

$$\begin{array}{c}\hfill m={ϵ}^2\mu ,\\ \hfill A=\frac{(-1+2(-1+{ϵ}^2)\mu )({ϵ}^2+\mu -2{ϵ}^3\mu -{ϵ}^4\mu +2ϵ(1+\mu ))}{{ϵ}^2+\mu -{ϵ}^4\mu },\\ \hfill B=\frac{2(1-{ϵ}^2)(1+\mu -{ϵ}^2\mu )({ϵ}^2+\mu -2{ϵ}^3\mu -{ϵ}^4\mu +2ϵ(1+\mu ))}{{ϵ}^2+\mu -{ϵ}^4\mu },\\ \hfill \alpha =\frac{{({ϵ}^2+\mu -2{ϵ}^3\mu -{ϵ}^4\mu +2ϵ(1+\mu ))}^3}{8{({ϵ}^2+\mu -{ϵ}^4\mu )}^2},\\ \hfill \eta =-\frac{n(-{ϵ}^2-\mu +{ϵ}^4\mu )}{{(1-{ϵ}^2)}^{1/2}{(1+\mu -{ϵ}^2\mu )}^{1/2}({ϵ}^2+\mu -2{ϵ}^3\mu -{ϵ}^4\mu +2ϵ(1+\mu ))},\\ \hfill \sigma =4K\left({ϵ}^2\mu \right)\eta .\end{array}$$

(20)

Parameter $\mu (n,ϵ)$ is fixed by the condition that the solution of equation

$$\begin{array}{c}\hfill \frac{1}{\sigma }{\varphi }_{,s}=\kappa \left(s\right)\to \frac{d\varphi \left(s\right)}{ds}=4K\left({ϵ}^2\mu \right)\eta \left(A+\frac{B}{1-ϵcn(n4K\left({ϵ}^2\mu \right)s,{ϵ}^2\mu )}\right)\end{array}$$

(21)

has to satisfy the boundary conditions $\varphi \left(0\right)=\pi /2$ and $\varphi \left(1\right)=5\pi /2$. Here, $cn(\xi ,m)$ is the Jacobi elliptic cosinus, $K\left(m\right)$ is the complete elliptic integral of the first kind, $4K\left(0\right)=2\pi$, and n is the number of petals in the vortex structure. In Equation (19), parameters $A,B$, and others, are taken from Equation (20); dimensionless parameter s changes from $s=0$ to $s=1$.

The calculation, which we do not present in this section, provides the leading term in the expansion of $\mu =\mu (n,ϵ)$ in a series with respect to $ϵ$ (which does not vanish when $ϵ\to 0$): the expression $\mu \to n^2-1$.

3. Results

To analyze the thermo-vorticial “blobs”, we employed the so-called Sharp Boundaries Evolution Method (SBEM). Within the blob, $q(\mathbf{x},t)=q_0$ is constant; outside the blob, $q(\mathbf{x},t)=0$. This method originated in the 1960s–1970s and was further advanced in subsequent years. Despite the seemingly radical simplification, the method produces a reasonably good description and estimates for large-scale vortex flows/structures ([9,10]). Indeed, for the motion of an incompressible fluid in the $xy$-plane, the velocity vector $\mathbf{v}$ lies in the same plane, and the vorticity vector has only one non-zero component, $q_z$. For large-scale vortex structures, it is intuitively apparent that there exist zones (which are sometimes rather spacious) where the vorticity magnitude reaches its maxima (“hilltops”) when the sign of vorticity is positive and its minima (“chasms”) when the sign of vorticity is negative. There are also zones where vorticity is zero or close to zero (“flats”), transitional zones between hilltops and flats (“upper slopes”), and transitional zones between flats and chasms (“lower slopes”). Since typically the fluid velocity and thermodynamical quantities are analyzed in terms of the integral characteristics derived from vorticity, when the slopes are steep (i.e., the Reynolds number $Re$ or its equivalent is enormous), these transitional zones may be treated as sharp (precipitous, vertical) moving boundaries which evolve in accordance with the laws of hydrodynamic flows. In this paper—to not complicate the calculations—we consider only the case with the hilltops, upper slopes, and flats (not including the chasms and lower slopes).

One Coherent Multi-Petal Structure: We find that, because of the intertwined nature of the thermal and vorticial fields in the fluid medium, what often looks like a group of multiple individual “circular” vortices (as depicted in Figure 1, for example) is in fact one coherent macro-structure with a complex multi-petal shape. In other words, the localized “circular” vortices are not independent; they influence each other, redistributing conservable quantities—such as vorticity $q(\mathbf{x},t)$ and energy, which are linked to temperature $\tau (\mathbf{x},t)$—locally, within and between each other, while preserving the integrals over the entire multi-petal system.

Figure 3 illustrates the contours of thermo-vortices with three and five petals.

Describing the shape of such a multi-petal structure mathematically is not a trivial exercise. We demonstrate how we characterize the shape using several key parameters: the number n of petals (counting the “fingers”), the level of non-linearity $ϵ$ (describing whether the “fingers” are long or short; measuring the gap between the peak and trough); and parameter $\mu (n,ϵ)$ (describing whether the “fingers” are thick or thin, and sharply pointed or not). The latter is achieved by employing (instead of “smooth” sinus/cosinus functions) the Jacobi elliptic sinus/cosinus, which permits the inclusion of higher spectral harmonics into the description of the shape’s “fullness”.

In Figure 3, both shapes have the same peak/trough nonlinearity, i.e., they possess the same parameter $ϵ=-0.28$. But because they have a different number of petals n, their values of $\mu (n,ϵ)$ are different. Figure 4 illustrates how $\mu (n=3,ϵ)$ depends on $ϵ$ and how parameter $ϵ$ (and therefore $\mu (n,ϵ)$) impacts the contour shape.

Parameter $\mu =\mu (n,ϵ)$ is not an independent quantity. It can be found via the imposition of one more condition: the requirement that the contour must close, i.e., that phase difference $\varphi \left(1\right)-\varphi \left(0\right)=2\pi$, which follows from Equation (21). For moderate $-0.49\le ϵ\le 0$ and $2\le n\le 8$, function $\mu (n,ϵ)$ is well approximated by the simple expression $\mu (n,ϵ)\simeq (n^2-1)/{(1+c{ϵ}^2(n^2-1))}^d$, where $c\simeq 1.56$ and $d\simeq 0.98$.

Figure 5 illustrates how $\mu (n,ϵ)$ behaves for varying numbers of petals n and nonlinearity (peak/trough) parameter $ϵ$.

Once the shape of the contour of the macro-structure can be described, the properties of the system can be analyzed.

Number of Petals and Other Key Parameters: Generally speaking, the number of petals n evolves as the system evolves because of the complex linkage between the parameters in the system. Physically, multi-petal vortex structures arise when thermal energy is introduced into the system, leading to the formation of a hot (${\tau }_0\ne 0$) domain of finite ($0<l<\infty$) size with non-zero vorticity ($Q\ne 0$) inside. Mathematically, when three quantities Q, l, and ${\tau }_0$ are specified as the initial conditions, they “fix” three geometrical parameters: n, $ϵ$, and $\mu$. Here, the conserved quantity is as follows:

$$\begin{array}{c}\hfill Q=q_0{l}^2\sigma \frac{1}{2}{I}_n(ϵ,\mu ),\mathrm{where}I(ϵ,n)={\int }_0^{1}ds\left(({\nu }_1^2\left(s\right)+{\nu }_2^2\left(s\right))\kappa \left(s\right)\right).\end{array}$$

(22)

Recall that here, l is the characteristic size; $\sigma \equiv L/l$, where L is the contour length; parameter s is the contour’s arc length.

Quantities l and ${\tau }_0$ may be measured in observations. Thus, their links to other parameters in the system are informative:

$$\begin{array}{c}\hfill l={\left(\frac{2}{I_n}\right)}^{1/5}{\alpha }^{1/5}{\sigma }^{-1/5}{\left(\frac{Q{R}^3}{\omega }\right)}^{1/5},{\tau }_0={\left(\frac{2}{I_n}\right)}^{3/5}{\alpha }^{-2/5}{\sigma }^{-3/5}{\tau }_1{\left(\frac{Q{R}^{4/3}}{\omega }\right)}^{3/5}.\end{array}$$

(23)

All coefficients are functions of $ϵ,n$, and $\mu$. The dimensions of the quantities $\left[Q\right]=lengt{h}^2\times tim{e}^{-1}$, $\left[R\right]=length$, $\left[\omega \right]=tim{e}^{-1}$ and $\left[{\tau }_1\right]=temperature\times lengt{h}^{-2}$ assure the correct dimensions of the written expressions for l and ${\tau }_0$.

Temperature Distribution: Determination of the contour is the first step of the analysis. Once the steady-state contour for the thermo-vorticial blob is established, the temperature distribution $\tau =\tau (t,x_j)$ inside the blob may be determined. The expression for temperature distribution in terms of dimensionless variables is as follows:

$$\begin{array}{c}\hfill \tau (x,y)=\left(\frac{q_0{\tau }_1{R}^2}{2\pi \omega }\right)\left(\beta \sigma \right){\int }_0^{1}ds\left(\frac{1}{\beta |\widehat{z}-z|}-\frac{1}{2}{K}_1\left(\beta \right|\widehat{z}-z\left|\right)\right)\Re \left(\frac{exp\left[i\varphi \left(s\right)\right]({\widehat{z}}^{*}-z^{*})}{i|\widehat{z}-z|}\right).\end{array}$$

(24)

Figure 6 shows—for the same study—the vorticity distribution $q(x,y)$ and the temperature profile (in the $xz$-plane for $y=0$; obviously, temperature is distributed similarly within the planes slicing the other petals). The right panel also shows how the temperature profile varies depending on the geometrical parameter $\beta =l/R$. The greater $\beta$ is, the “tighter” the thermal structure within the petal—the petal peak is higher, the sides are narrower, and also, near the center of macro-rotation, the central thermo-vortex appears.

4. Discussion and Conclusions

The fact that rotating cosmic fluids contain vortices has been understood for a while. Vortices have been discussed in the context of planetary atmospheres (at least since the discovery of the Jupiter’s Great Red Spot), in the context of vortices and magnetic flux tubes in the accretion disks of black holes [23], galactic center hot spots [24], and so on. When the observed vortices are multiple, coherent, and long-lasting, they are also often clustered into orderly quasi-symmetric arrangements. These multiple vortices are typically interpreted as possibly interacting, but nonetheless as stand-alone items.

In our paper, we demonstrated that an additional physical interpretation of what these vortices represent may exist. What one sees in astronomical images as distinct spots may be not multiple individual quasi-circular-shaped vortices, which possibly interact and drift to form a quasi-symmetrical quasi-stationary macro-arrangement, but instead these distinct spots (thermo-vortices) may be the observable parts of one structure whose shape is a complex contour with multiple “petals”. Either alternative is theoretically possible, but the latter one is typically not considered.

To bring attention to this multi-petal interpretation, we presented and analyzed a model that can be illustrative and adaptable to various environments. Specifically, in order for our model to be of interest for studies of the accretion disks of black holes, we used the metric that works for the mildly curved spacetime (naturally, this model works in flat spacetime as is the case for planets). We had to limit our considerations to the model of a rotating “thin” layer because otherwise the mathematical calculations would become unwieldy. Our focus was on the description of the steady-state, large-scale, thermo-vorticial motions, not on the evolution from the state of “nothing”. We demonstrated that the steady-state may indeed look like a multi-petal structure and that the shape of individual petals may be highly nonlinear (the petals may look like “fat fingers”, or be elongated or pointy). Various parameters (relative dimensions of the system, thermal gradient, initial vorticity, etc.) and their linkages determine the outcome. We also demonstrated that the petals indeed have an elevated temperature and would appear as bright spots if observed from afar.

The physical explanation for why a system would evolve towards a steady-state with N petals rather than M petals is intuitively simple. The contour of the macro-structure bounds the inner domain of higher vorticity. Naturally, the contour is closed. Therefore, if subjected to some perturbation, the contour’s steady-state turns into a “standing wave” (like a standing wave on the surface of a swimming pool), with the number of crests (petals) defined by the properties of the system. The shape of the crests, however, is not sinus-wave-like but is complex (finger-like) because the system is highly nonlinear. Obviously, no steady-state is eternal, but we did not study the problem of the transition of quasi-steady-states.

Several notable multi-vortex arrangements, which are likely to be multi-petal structures, can be mentioned. As seen in Figure 1 (second row; panels C and D ), in the span of one year, the Antarctic ozone hole evolved from a quasi-circular into a two-petal structure. Based on the common knowledge of what the ozone hole represented, we can conclude with certainty that the two vortices observed in 2002 were not separate entities; they were indeed two petals of one shape-changing macro-structure. The arrangement of vortices on Jupiter’s South Pole (Figure 1B) is also a candidate multi-petal macro-structure. Specialists in geophysical fluid dynamics may spot an analogy between the study of this paper and the Rossby waves (see, e.g., Ref. [25]): the interface on the thermocline $\eta$ (or, equivalently, the streamfunction) corresponds to the temperature $\tau$ (which is proportional to the streamfunction due to the presumed Equation (2)), the pressure gradient responsible for the geostrophic flow corresponds to the gradient of the centrifugal potential $\mathrm{\Phi }$, and the potential vorticity corresponds to the q-vorticity. Not surprisingly, the global nature of planetary solitary waves corresponds to multi-petal structures and explains images like those in Figure 1B–D. One difference, however, is that the Rossby waves occur at the sharp boundaries between regions with different temperatures; in our case, the regions are of different vorticities.

As for the question of whether our mathematical treatment is valid for a “thick” system—such as the atmosphere of Jupiter or a “thick” double-layered accretion disk near a rotating black hole—the answer is yes. “Thick” systems may be, in principle, described using the same mathematical apparatus when the vorticity patch is broken into columnar sub-patches. (Refs. [9,10] point to the use of this technique.)

For a more detailed treatment—not included in this paper—it is necessary to consider the nuances of mathematical models for hydrodynamic flows (see, for example, Refs. [26,27]), collective plasma effects [28], relativism [29,30,31,32,33,34,35,36,37,38,39,40,41,42,43], and relativistic hydrodynamics [44].

For a strongly relativistic system—such as an accretion disk in the environment of a rapidly rotating supermassive object or a supermassive black hole, see Refs. [45,46,47]—the conditions (in covariant description) of adiabaticity and incompressibility remain the same, while the equation of fluid motion (in spacetime with a metric defined by the square of the four-interval $d{s}^2=m_{ik}d{q}^{i}d{q}^k$ in the equatorial plane of a rotating black hole when the coordinate $q^2\equiv \theta =\pi /2$) has the following form in the leading approximation (these details of our analysis are not listed in this paper):

$$\begin{array}{c}\hfill -\frac{w}{c^2}D{v}_{\beta }={\partial }_{\beta }p+2\frac{w}{c^2}{[\mathbf{v}\times \mathrm{\Omega }]}_{\beta }+w{\partial }_{\beta }ln\sqrt{g_{00}}+\dots \end{array}$$

(25)

Here, D is the operator of the total derivative with respect to the “synhronized time”; the enthalpy w is defined as $w/c^2=\gamma \rho +(1/c^2)(\rho e_1+p)$, where $\gamma$ is the Lorentz factor, p is the pressure, and $e_1$ is the non-relativistic part of the internal energy); c is the light speed; $v_{\beta }$ is the covariant $\beta$-component of the 3D velocity. Naturally, the basic elements of the model momentum in Equation (A84) are the pressure gradient (the first term), the “Coriolis force” (the second term), and the gravity and centrifugal forces, which are packed in the third term of the momentum equation. The key difference compared to the classic hydrodynamic treatment is the replacement of the classic gravity/centrifugal acceleration $-\nabla \mathrm{\Phi }+{\mathrm{\Omega }}^2\mathbf{r}$ with $-\nabla ln\sqrt{m_{00}}$, and the replacement of the classic averaged angular velocity $\mathrm{\Omega }$ of the accretion disk rotation with the relativistic parameter $\mathrm{\Omega }=(c/2)\gamma u^{0}curl(-g_{00}\mathbf{g})$. If the thermo-vortices are at some “distance” from the event horizon and conditions $x=r/r_g\gg 1,\mathrm{\Omega }x<1$ are satisfied, then, leaving the leading terms in the expansion ${\partial }_{\beta }ln\sqrt{m_{00}}$, we can obtain that ${\partial }_{\beta }ln\sqrt{m_{00}}\simeq -x{\mathrm{\Omega }}^2+1/2x^2+1/2x^3+(2a+\mathrm{\Omega })\mathrm{\Omega }/2x^2$. Here, the terms of higher orders of smallness are omitted. The first term describes the centrifugal effect, the second describes the contribution of the force of attraction in the Newtonian approximation of gravity, the third term describes the Schwarzschild model of a non-rotating black hole, the last takes into account the rotation of both the black hole and the disk. For $x\gg 1,\mathrm{\Omega }x<1$, but $\mathrm{\Omega }{x}^{3/2}>1/2$, the principal contribution provides the first term $\sim {\mathrm{\Omega }}^2$, i.e., the centrifugal effect, independent of the exact structure of the spacetime near the rotating, or not rotating, black hole.

To conclude, the illustrative model that we presented provides insights that are useful for a broad range of physical applications, ranging from planetary atmospheres to black hole accretion disks. Whether the observed vortices in cosmic fluids are conceptualized as stand-alone objects or inextricable parts of one macro-structure may meaningfully influence the numerical models employed in consideration of the phenomena and image reconstruction methods.