1. Introduction
Vorticity is a measure of the rotation of the velocity field at any point in a fluid. It is defined as the curl of the fluid velocity field $\boldsymbol {v}$:
Vortex sizes in a fluid can range from thousands of light-years (galaxies) to a few hundred metres (tornadoes) in nature and are often observed to self-organize into equilibrium configurations (Nitsche Reference Nitsche2006). The presence of these ordered vortex structures in fluid and plasma is quite important in understanding the formation and evolution of galaxies, accretion discs, stars, etc. (Brahic Reference Brahic1982; Acosta-Pulido et al. Reference Acosta-Pulido, Perez-Fournon, Calvani and Wilson1990; Abramowicz et al. Reference Abramowicz, Lanza, Spiegel and Szuszkiewicz1992; Shapiro Reference Shapiro1996; Klahr & Bodenheimer Reference Klahr and Bodenheimer2003; Porter, Jones & Ryu Reference Porter, Jones and Ryu2015; Jelic-Cizmek et al. Reference Jelic-Cizmek, Lepori, Adamek and Durrer2018). One particularly important class of equilibria in an ideal fluid can be identified with the Beltrami flow and expressed as
where $\alpha$ is an arbitrary function. When $\alpha$ is taken as a constant, it is also known as Trkalian flow.
An analogue of this condition can be derived in the context of force-free single-fluid magnetohydrodynamics when the flow velocity $\boldsymbol {v}$ is replaced by magnetic field $\boldsymbol {B}$ (Woltjer Reference Woltjer1958):
where $\alpha$ is a scalar field and must satisfy $\boldsymbol {B}\boldsymbol {\cdot }\boldsymbol {\nabla }\alpha =0$. This state, first discussed by Woltjer and Taylor, has been successful in modelling fusion and astrophysical plasmas (Woltjer Reference Woltjer1958; Taylor Reference Taylor1974). It can be derived by the minimization of the magnetic field energy
with magnetic helicity ($\boldsymbol {A}$ being the magnetic vector potential)
as a global constraint which leads to the ‘relaxed state’ with spatially homogeneous $\alpha$, also known as constant-$\alpha$-Beltrami field.
Later, this state was extended to multi-species plasmas where the energy now consists of both kinetic and magnetic parts (Mahajan & Yoshida Reference Mahajan and Yoshida1998; Steinhauer & Ishida Reference Steinhauer and Ishida1998; Yoshida & Mahajan Reference Yoshida and Mahajan2002):
with a generalized helicity, not the magnetic helicity,
playing the role of constraint in the minimization, where the new canonical momentum $\boldsymbol {P}=\boldsymbol {A}+mc/q \,\boldsymbol {v}$ and its curl $\boldsymbol {\varOmega }=\boldsymbol {\nabla }\times \boldsymbol {P}=\boldsymbol {B}+mc/q \,\boldsymbol {\nabla }\times \boldsymbol {v}$. Here, plasma density $\rho =mn$, with $m$ the mass of species, $n$ the number density, $q$ the charge, $c$ the speed of light and $i$ the individual plasma species. The collinearity condition, now defined as the generalized Beltrami condition, can be succinctly written as
where $\mu _i$ is the Lagrange multiplier. The generalized Beltrami condition implies an alignment of generalized vorticity and flow field in a multi-species charged fluid. To fully describe such equilibrium states, the Beltrami condition must be supplemented by the appropriate Bernoulli condition which indicates homogeneity in energy distribution in the plasma. These equilibrium states are called Beltrami–Bernoulli states, and are usually characterized by the number of independent single Beltrami systems needed to construct them.
In this paper, we explore the possibility of the formation of Beltrami–Bernoulli states in a weakly rotating self-gravitating neutral (uncharged) fluid. The significance of rotation within the context of frame-dependent effects such as Coriolis force in vortical fluid dynamics has been explored in the stellar structure, turbulence, zonal flows, dynamos, etc. (Singh & Singh Reference Singh and Singh1984; Hopfinger & Van Heijst Reference Hopfinger and Van Heijst1993; Shukla & Stenflo Reference Shukla and Stenflo2003; Itoh et al. Reference Itoh, Itoh, Diamond and Yoshizawa2006; González, Costa & Santini Reference González, Costa and Santini2010; Shatashvili & Yoshida Reference Shatashvili, Yoshida, Weiland and Lazzaro2011; González Reference González2014). Here, we consider the effect of fluid rotation in the context of the weak-field limit of general relativity, where the rotation of a self-gravitating fluid can twist the background space–time surrounding it, also known as frame dragging. This frame-dragging phenomenon can be identified with a magnetic-type gravitational field, i.e. gravitomagnetic field, and its governing equations can be derived by taking the weak-field, slow-velocity limit of Einstein's equation. As moving charges create a magnetic field, the gravitomagnetic field exists due to the mass currents in the rotating fluid. The role of gravitomagnetic field in various astrophysical phenomena such as jet collimation, pulsar beam precession, gyroscope precession, vorticity generation, etc., has been explored in detail (Bardeen & Petterson Reference Bardeen and Petterson1975; Nelson & Papaloizou Reference Nelson and Papaloizou2000; Lei, Zhang & Gao Reference Lei, Zhang and Gao2012; McKinney, Tchekhovskoy & Blandford Reference McKinney, Tchekhovskoy and Blandford2013; Nealon, Price & Nixon Reference Nealon, Price and Nixon2015; Krishnan et al. Reference Krishnan, Bailes, Van Straten, Wex, Freire, Keane, Tauris, Rosado, Bhat and Flynn2020; Bhattacharjee & Stark Reference Bhattacharjee and Stark2021). By defining generalized vorticity in the rotating fluid as a combination of flow vorticity and gravitomagnetic field, we obtain equilibrium flow configurations which are similar to a unique class of Beltrami–Bernoulli states known as double curl Beltrami states.
We present a brief overview of the Einstein–Maxwell equation based on the analogue of the electromagnetic Maxwell equation. Then, we construct the generalized vortical dynamics of a rotating fluid followed by an analysis of the equilibrium solution of the vorticity transport equation. We compare our results with the plasma equilibrium states and delineate features that are unique to uncharged self-gravitating rotating fluid. Next, we present an analysis of helicity and energy, which is helpful in understanding the physical meaning of the Lagrange multiplier. Finally, we discuss the limiting cases of these states, possible implications and future work.
2. Einstein–Maxwell equation
To study the gravitational dynamics of a weakly rotating self-gravitating fluid, we need to explore the linearized limit of the Einstein equation. When the space–time metric is almost Minskowskian, $g_{\mu \nu }=\eta _{\mu \nu }+\bar {h}_{\mu \nu }$, and terms of $\mathcal {O}(c^{-4})$ or higher are neglected, we can write the Einstein equation
as a set of linearized equations almost identical to the electromagnetic Maxwell equation as (Braginsky, Caves & Thorne Reference Braginsky, Caves and Thorne1977; Thorne Reference Thorne, Fairbank, Deaver, Everitt and Michelson1988; Manfredi Reference Manfredi2015)
where $T_{\mu \nu }$ is the stress energy tensor, $\boldsymbol {E}_g=-\boldsymbol {\nabla } \phi$ is the Newtonian gravitational field, $\phi$ is the gravitational potential and $\boldsymbol {B}_g=\boldsymbol {\nabla }\times \boldsymbol {A}_g$ is the gravitomagnetic field, with $\boldsymbol {A}_g$ being the corresponding vector potential. Also, $G$ is the gravitational constant and $\rho$ is the matter density. It should be noted that the right-hand side of (2.3) does not have any $\partial \boldsymbol {B}_g/\partial t$ to this order (Thorne Reference Thorne, Fairbank, Deaver, Everitt and Michelson1988).
3. Vortical fluid dynamics
In the weak-field, slow-velocity limit of general relativity, the Euler hydrodynamics equation for a rotating ideal fluid can be written as (Thorne, Price & MacDonald Reference Thorne, Price and MacDonald1986)
where ${\rm d}/{\rm d}t=(\partial /\partial t+\boldsymbol {v}\boldsymbol {\cdot }\boldsymbol {\nabla })$, $\boldsymbol {v}$ is the fluid velocity and $p$ is the fluid pressure.
The continuity equation for the corresponding fluid is
Now using the vector identity $(\boldsymbol {v}\boldsymbol {\cdot }\boldsymbol {\nabla }) \boldsymbol {v}=\boldsymbol {\nabla }(v^2/2)-\boldsymbol {v}\times (\boldsymbol {\nabla }\times \boldsymbol {v})$ and the expressions $E_g=-\boldsymbol {\nabla } \phi$ and $\boldsymbol {B}_g=\boldsymbol {\nabla }\times \boldsymbol {A}_g$, we can rewrite (3.1) as
where we have assumed a barotropic equation of state for pressure, i.e. $p(\rho )$. Since the vector potential $\boldsymbol{A}_g$ does not depend on time, we can rewrite (3.3) as
where we identify the quantity $\boldsymbol {P}_g = (\boldsymbol {v}+\boldsymbol {A}_g/c)$ as a new canonical momentum and $\varPhi$ contains all the potentials for the gradient forces.
Now, we take the curl of (3.4) and obtain the vorticity transport equation:
where we identify $\boldsymbol {\varOmega }_g=\boldsymbol {\nabla }\times \boldsymbol {P}_g= \boldsymbol {\nabla }\times \boldsymbol {v}+ \boldsymbol {B}_g/c$ as the generalized vorticity.
It should be noted here that (3.5) does not contain any source terms, which implies that if the vorticity is zero at any time, it remains so for all times in an ideal barotropic fluid.
4. Equilibrium state
In this paper, we are interested in large-scale equilibrium structures in a self-gravitating and weakly rotating fluid. The stationary solution of (3.5) can be written as
where we have defined $\boldsymbol {\omega }_g=\boldsymbol {B}_g/c$ and satisfies the requirement of vanishing divergence of generalized vorticity and time-independent continuity equation, i.e. $\boldsymbol {\nabla }\boldsymbol {\cdot }(\rho \boldsymbol {v})=0$.
To fully solve the equilibrium state, we need to supplement the Beltrami condition with the time-independent gravitomagnetic Ampère law rewritten in terms of the new quantity $\boldsymbol {\omega }_g$, which has the following form:
The separation constant $\mu$ in (4.1) can be identified as the Lagrange multiplier when the Beltrami condition is derived via the variational principle.
For (4.1) to be defined as the stationary solution of (3.5), we need to impose the Bernoulli constraint which is an expression of the balance of all remaining potential forces, i.e. $\boldsymbol {\nabla }\varPhi =0$.
Combining (4.1) and (4.2), we obtain the following equation:
where we have normalized $|\boldsymbol {\nabla }|$ to the inverse of skin depth $\tilde {\lambda }_J=\tilde\alpha \lambda _J$ with Jean's length $\lambda _J=c_{s0}/\omega _J$, Jean's frequency $\omega _J=(4{\rm \pi} G \rho _0\hat {\rho })^{1/2}$ and $\tilde\alpha =c/c_{s0}$. Here we have defined sound speed $c_{s0}$ in terms of some ambient mass density $\rho _0$ and $\hat {\rho }$ is the density envelope which we take to be a constant of order unity for the rest of this paper. Equation (4.3) is known as the double curl Beltrami equation and has been studied thoroughly in the context of Hall magnetohydrodynamics (Mahajan & Yoshida Reference Mahajan and Yoshida1998; Mahajan et al. Reference Mahajan, Miklaszewski, Nikol'skaya and Shatashvili2001; Ohsaki et al. Reference Ohsaki, Shatashvili, Yoshida and Mahajan2001).
Next, (4.3) is written as
where
It should also be noted that (4.4) is the combination of two Beltrami fields $\mathbb {G}_+$ and $\mathbb {G}_-$, i.e.
with the final solution
where the constant amplitudes $C_{\pm }$ are determined from initial conditions and characterize the double curl Beltrami states along with their corresponding eigenvalues $\lambda _{\pm }$. The explicit solution for the Beltrami condition in (4.6) is provided by the Chandrashekar–Kendall function in cylindrical coordinates whereas it takes the form of the Arnold–Beltrami–Childress solution in Cartesian coordinates (Chandrasekhar & Kendall Reference Chandrasekhar and Kendall1957).
One can obtain the solution for gravitomagnetic field by using (4.1) which can be written as follows:
Finally, it should be emphasized here that the coupling between gravitomagnetic and flow fields has enabled us to uncover a far richer equilibrium structure in the fluid compared with the traditional Trkalian flow.
4.1. Comparison with the double curl Beltrami states in plasmas
Though double Beltrami states emerge in both plasma and self-gravitating rotating fluid, the physical characteristics of these states might not be the same in both types of systems. In this section, we study possible differences between the two systems. For a single-species dynamic charged fluid with constant density in an appropriate neutralizing background, the generalized Beltrami condition can be written as
which, combined with the Ampère law, gives us the plasma counterpart of (4.3) (Mahajan Reference Mahajan2008):
with the following roots:
where species skin depth $\lambda _s=c/\omega _p$ and plasma frequency $\omega _p=\sqrt {4{\rm \pi} nq^2/m}$. The roots are real for $(\mu /\lambda _s)^2>4$ but form a complex conjugate pair when $(\mu /\lambda _s)^2<4$.
Apart from a factor of 4, there is a sign difference between the left-hand sides of (4.3) and (4.10) and the origin of this can be attributed to the fact that gravity is always attractive as reflected in (4.2). Contrary to the plasma equilibrium states, the roots $\lambda _{\pm }$ in (4.5) for double curl Beltrami states in self-gravitating fluid are always real. Moreover, the presence of an inherent length scale, i.e. Jean's length, has introduced a singular perturbation term $\boldsymbol {\nabla }\times \boldsymbol {\nabla }\times \boldsymbol {v}$ in (4.3). This implies equilibrium states in self-gravitating fluid are also endowed with two length scales in a gravitomagnetic field. This can have major consequences for the formation of large- and small-scale flow configurations in the fluid as demonstrated in charged fluid (Mahajan et al. Reference Mahajan, Miklaszewski, Nikol'skaya and Shatashvili2001; Kagan & Mahajan Reference Kagan and Mahajan2010).
Next, if we set $\mu =0$ and reverse the normalization of the gradients, (4.3) can be rewritten as
which can be compared to the corresponding limiting case of (4.10) in plasma:
which are nothing but the $\boldsymbol {\varOmega }=0$ solution of the vorticity transport equation for both plasma and fluid.
Though (4.12) and (4.13) have a similar structure, the physics exhibited by the two systems is completely opposite. Equation (4.13) is the superconducting limit in a plasma with a skin depth $\lambda _s$ beyond which the magnetic flux is completely expelled from the interior of the plasma; a complete antithesis to the behaviour of gravitomagnetic flux in a neutral fluid. Another way to interpret (4.12) and (4.13) is that electric current is restricted to the skin depth in a diamagnetic plasma equilibrium, whereas the mass current is not confined to Jean's length in a fluid equilibrium.
5. Helicity and energy
From our analysis in the previous section, we notice that the values of Lagrange multiplier $\mu$ determine the characteristics of equilibrium states in a fluid. First, we take the following definition of helicity as one of the invariants of the system:
which is a measure of different topological features of vortical field lines such as knottedness and twists (Moffatt Reference Moffatt1969).
The physical interpretation of the Lagrange multiplier can then be obtained by computing the helicity from (5.1):
yielding an expression for $\mu$ in terms of two invariants of the motion:
where $\langle \rangle =\int {\rm d}^3\,x$. Therefore, the Lagrange multiplier is a measure of generalized helicity as a fraction of total energy. It should be noted that, gravity being an attractive force, the energy density of the gravitomagnetic field has a negative sign in front of it (Sebens Reference Sebens2020).
6. Discussion
In light of our results, we notice that the equilibrium flow configurations of a rotating fluid in a stationary gravitomagnetic field allow for small-scale structures due to the singular perturbation term $\boldsymbol {\nabla }\times \boldsymbol {\nabla }\times \boldsymbol {v}$. These small-scale structures can be distinguished by comparing the ratio between Jean's length and system size. Moreover, two invariants of the motion, i.e. generalized helicity $H$ and energy $E$, emerge as the determinants of different classes of flow configurations in the fluid. The interaction and evolution of structures at different length scales can provide new insights into turbulence in accretion discs, differential rotation patterns in stars and jet formations in various astrophysical objects (Tobias, Dagon & Marston Reference Tobias, Dagon and Marston2011).
One can obtain the traditional Beltrami flow if the system length $L\gg \tilde {\lambda }_J$ and (4.3) reduces to $\boldsymbol {\nabla }\times \boldsymbol {v}=-(\mu ^{-1})\boldsymbol {v}$. Similar to the double Beltrami equilibria in a plasma, the double curl Beltrami states in a fluid can terminate and relax into single curl equilibria. Furthermore, such termination can result in catastrophic events. These events can have potential effects in astrophysics such as stability of a rotating star, formation of large-scale structure in galaxies, excitation of different wave modes, collimation of ionizing radiation in an accretion disc, etc.
The next step in this scheme will be to explore the gravitomagnetic effects in plasma equilibria. In that case, one has to appropriately incorporate the coupling between gravity and electromagnetic fields in Maxwell's equation which is beyond the scope of this paper. Finally, the consequence of double curl Beltrami flow in weakly rotating stars and accretion discs will be explored in future work.
Acknowledgements
The author thanks D.J. Stark for helpful discussions.
Editor Dmitri Uzdensky thanks the referees for their advice in evaluating this article.
Funding
C.B.'s work is supported by a New York Institute of Technology ISRC grant.
Declaration of interests
The author reports no conflict of interest.