The Impact of Ambipolar Diffusion on the Rossby Wave Instability in a Protoplanetary Disk

Recent observational and simulation studies have revealed that ambipolar diffusion is an important phenomenon in the outer regions of a protoplanetary disk (PPD). However, numerous simulation studies have found that ambipolar diffusion suppresses the turbulence caused by the magnetorotational instability (MRI) in these regions of a PPD. The aim of this study is to investigate the impact of ambipolar diffusion on the Rossby wave instability (RWI) at large radii of a PPD. To accomplish this, we examine the occurrence of the RWI in a PPD threaded by the magnetic field in the presence of ambipolar diffusion. Additionally, we scale the ambipolar diffusivity with respect to both the toroidal magnetic field and an important parameter known as the Elsässer number. We obtain the growth rate of unstable RWI modes in the outer regions of a PPD using linear perturbation analysis. In our nonaxisymmetric perturbation analysis, we find that the amplitude of the toroidal field oscillates in various modes for small values of the Elsässer numbers. For small Elsässer numbers, the growth rate of unstable modes associated with the RWI decreases. In other words, ambipolar diffusion suppresses the perturbation caused by the RWI. This effect is similar to the effect of ambipolar diffusion on the MRI. In contrast to the MRI, where there is a range around 1 for the Elsässer numbers, the existence of ambipolar diffusion supports the occurrence of RWI. Finally, we compare our findings with those of simulation studies to emphasize the importance of RWI in the outer regions of a PPD.


Introduction
In recent decades, there is no doubt that the magnetic field plays a crucial role in accretion disks.Protoplanetary disks (PPDs) are classified as weakly ionized disks because most regions are too cold to undergo sufficient thermal ionization (Bai & Stone 2011).In most regions of a PPD, especially at the outer regions, nonideal magnetohydrodynamic (MHD) effects should be considered instead of the ideal MHD effects, such as ambipolar diffusion, Hall, and ohmic diffusion (Bai & Stone 2011 also see Armitage 2011Armitage , 2019)).The nonideal MHD effects are also significant in the inner region of the disk larger than 0.1 au, where thermal ionization becomes weak.Recent simulation works demonstrate the significant importance of ambipolar diffusion in the outer regions of a PPD (Hawley & Stone 1998;Desch 2004;Kunz & Balbus 2004;Bai & Stone 2011;Simon et al. 2013aSimon et al. , 2013b;;Bai & Stone 2013;Simon & Armitage 2014;Gressel et al. 2015;Cui & Bai 2021).While it is believed that other nonideal MHD effects, such as Hall and ohmic diffusion, play significant roles in the inner regions of a PPD, ambipolar diffusion prevails over them in the outer regions (e.g., Simon et al. 2013b;Simon & Armitage 2014;Gholipour 2020).The ambipolar diffusivity determines the Elsässer number, and in the studies of a PPD, it is expressed with respect to the magnetic field, gas density, angular velocity, and the Elsässer number.
Approximately three decades ago, the magnetorotational instability (MRI) was introduced as a suitable mechanism for the transport process by Balbus & Hawley (1991; see also Balbus & Hawley 1998).Since then, numerous attempts have been made by various authors to assess it under different conditions.For example, the nonlinear evolution of the MRI under ideal MHD conditions has been examined through the use of both local numerical simulations (Hawley et al. 1995;Stone et al. 1996;Miller & Stone 2000) and global numerical simulations (Armitage 1998;Hawley 2000Hawley , 2001;;Fromang & Nelson 2006;Parkin & Bicknell 2013;Zhu & Stone 2018).The results of these studies reveal that the MRI generates strong MHD turbulence and facilitates efficient outward transport of angular momentum, with a rate that is consistent with observations.The problem here is that the nonideal MHD effects reflect the incomplete coupling between the disk material and the magnetic field, thereby significantly influencing the growth and saturation of the MRI (Armitage 2011).
Thanks to the results of observational studies conducted by the Atacama Large Millimeter/submillimeter Array (ALMA) project, nonaxisymmetric dust distributions have recently been detected in PPDs (van der Marel et al. 2013;Simon & Armitage 2014;Long et al. 2019;van der Marel et al. 2021).Additionally, recent simulation studies have revealed that MRI is suppressed by ambipolar diffusion, particularly in PPDs (weakly ionized disks) characterized by small Elsässer numbers (Simon et al. 2013b).The RWI is a suitable alternative mechanism to the MRI in weakly ionized disks, demonstrating greater consistency in such environments.The RWI was initially investigated in the context of a two-dimensional (2D) accretion disk (Lovelace et al. 1999;Li et al. 2000).Indeed, the RWI is closely linked to the Papaloizou and Pringle instability, where the wave becomes trapped between the inner and outer radii of a disk (Papaloizou & Pringle 1984).These studies have demonstrated that disks characterized by a steep radial structure, such as those with surface density enhancements or gaps, are indeed unstable to nonaxisymmetric perturbations.For instance, Lovelace et al. (1999)  of the RWI in a thin, nonmagnetized Keplerian disk using a 2D approach.They solved the linearized equations and obtained a set of ordinary differential equations (ODEs).They employed a local approximation to derive the dispersion relation.Subsequently, numerous studies have been conducted to explore the role of the RWI in accretion disks, both analytically and through simulation works (e.g., Balbus & Hawley 1991;Li et al. 2000;Varnière & Tagger 2006;Lyra et al. 2008Lyra et al. , 2009;;Yu & Li 2009;Umurhan 2010;Meheut et al. 2012Meheut et al. , 2013;;Lyra et al. 2015;Lin 2012Lin , 2014;;Lovelace & Hohlfeld 2013;Vincent et al. 2013;Gholipour & Nejad-Asghar 2014;;Lovelace & Romanova 2014;Miranda et al. 2016;Ono et al. 2016;Matilsky et al. 2018;Huang & Yu 2022;Liu & Bai 2023).In this context, three-dimensional (3D) investigations of the RWI were conducted through both linear analysis by Meheut et al. (2012) and nonlinear numerical simulations by Meheut et al. (2010).They discovered that both 3D calculations and 2D calculations yield comparable results (Lin 2012;Richard et al. 2013).Accordingly, it is reasonable for us to utilize 2D assumptions when considering the RWI for the sake of simplicity (e.g., Liu & Bai 2023).
Despite the considerable efforts dedicated to the expansion and development of the RWI in this field, the role of nonideal MHD, including ambipolar diffusion, has not received significant attention thus far.Therefore, in this work, our main focus is to examine the impact of ambipolar diffusion on the occurrence of the RWI.To achieve this objective, we delve into the formulation of the problem in Section 2, which includes the examination of a stationary disk and the incorporation of ambipolar diffusivity within a PPD.In Section 3, we thoroughly investigate the perturbation of the disk.Moving on to Section 4, we derive a dispersion relation and subsequently analyze the obtained results within the same section.Finally, in Section 5, we provide a comprehensive conclusion and engage in a discussion of the results.

Formulation of the Problem
We consider a magnetized thin accretion disk in the cylindrical coordinate system (r, f, z).We begin by formulating the problem with the vertically integrated main equations (2D equations) of a PPD in the nonideal MHD regime where ambipolar diffusion dominates in the outer regions of a PPD.The basic equations of the system consist of the continuity equation, momentum equation, induction equation, and the polytropic equation.The equation of continuity is The equation of momentum conservation is given by The Gauss equation is 3 The induction equation becomes 0 .5 Here, Σ, P, V, Φ, T, and B are the surface density, gas pressure, velocity vector, gravitational potential due to the central object, viscous tensor, and magnetic field vector, respectively.B is the unit vector for the magnetic field direction.η A is the ambipolar diffusivity, and the specific entropy is defined as "S = P/Σ γ " where γ is the polytropic index.

The Disk Background
To study the stationary state, we consider a PPD under axisymmetric and steady-state conditions (∂/∂f ≡ ∂/∂t ≡ 0).We also assume that the vertical thickness of the disk is much smaller than the radial distance (i.e., r ?H), allowing us to consider the PPD in 2D.In this study, we only consider the rf component of the viscous tensor (T rf = αP, where α is a constant; Shakura & Sunyaev 1973;Pringle 1981).Additionally, we use the Newtonian gravitational potential (Φ = GM * /r, where M * is the mass of the central object and G is the gravitational constant) and neglect the self-gravity of the disk.The angular velocity in a magnetized PPD can be determined by examining the radial component of the momentum equation.The equation is given by Here, we assume that the magnetic field is toroidal at large radii of the PPD.Based on the work of Yu & Li (2009), we propose a power-law relationship between the toroidal magnetic field and radius, represented as B f = B 0 r − n , where B 0 and n are constants.Consequently, we have In the outer regions of a PPD, the temperature is too cold to consider the gradient of gas pressure at large radii.By substituting n = 1 into Equation (6), we obtain This equation represents a nearly Keplerian angular frequency, where * W = GM r 3 .To facilitate further analysis in subsequent sections, it is helpful to introduce the square of the radial epicyclic frequency, denoted as Next, we consider a bump with a fixed center at radius r 0 .The equilibrium surface density profile can be approximated by a Gaussian function: Here, F and Δ represent the height and width of the bump, respectively.In this study, we set F = 1.4 and Δ = 0.05r 0 , with 0.5 r/r 0 1.5.Typically, the disk aspect ratio (H/r = c s /v f , where c s is the sound speed) is assumed to be in the range of 0.05-0.1.Additionally, Σ b denotes the surface density of the background disk, which can be defined as Here, Σ 0 represents the value of Σ at r 0 , and κ characterizes the slope (typically between 1/2 and 3/4 in the standard alpha disk model; see also Li et al. (2000) and Huang & Yu (2022)).
Lastly, we will examine the β p parameter in a PPD, which is known as the ratio of gas pressure to magnetic pressure (i.e., b For simplicity in the following sections, we will introduce β as the inverse of β p using the equation In the study of PPDs, β p has generally been assumed to have a value in the range of approximately 10 2 -10 5 (e.g., Simon et al. 2013aSimon et al. , 2015;;Khaibrakhmanov 2024).However, some studies indicate that it is possible to overcome the magnetic pressure with the gas pressure in the outer regions of a PPD (e.g., Li et al. 2016).This is due to the observational evidence of a relatively strong toroidal magnetic field in these regions.
Additionally, the outer regions of a PPD are too cold, leading to thermal pressure of a similar magnitude to the magnetic pressure (e.g., Armitage 2011;Li et al. 2016;Vlemmings et al. 2019;Khaibrakhmanov 2024).To address these issues, we consider β to be in the range of 10 −3 -10 in the outer regions.

Ambipolar Diffusivity in a PPD
As previously seen in Equation (4), the profile of ambipolar diffusivity needs to be taken into account in the set of equations.In this regard, we scale this parameter with respect to the Elsässer number (i.e., Λ) as follows (e.g., Simon et al. 2013b): It is better to rewrite the Equation (4) as follows: where η 0 is defined by It is useful to consider the following term, which will be helpful for the calculations in Section 3: This term is very straightforward around the bump region (i.e., r ≈ r 0 ) and can be written as where we have 3/2 ν < 2. In Section 3, we will consider the global linear perturbation on the original equations.

Perturbation of the Disk
In this section, we consider the global linear perturbation on the main equations, which have been described in Section 2. The form of the perturbation is ( , where δf (r) is the amplitude of perturbation, m is the azimuthal mode number, and ω = ω r + iω i is the mode frequency.The amplitude of perturbation of the velocity and the magnetic field are δV = (δv, δu) and δB = (δB r , δB f ) in the 2D approach.Although we are more interested in the case of n = 1 in this study, for future works, we solve the equations for the general case of the initial magnetic field.Applying the perturbation on the continuity equation leads to 18 The radial component of the perturbed momentum equation is The azimuthal component of the perturbed momentum equation becomes The Gauss equation results in The perturbed induction equation becomes ).Thus, we consider this case and investigate other cases in future works.

Case of n = 1
The case of "n = 1" leads to the Keplerian disk, which was discussed in Section 2.1.Also, this case is compatible with the presence of ambipolar diffusion at large radii of a PPD.In this case, Equation (15) becomes  Equations ( 23)-( 28) form a set of ODEs that can be numerically solved using appropriate boundary conditions.These equations represent two-point boundary eigenvalue problems that can be readily solved using the relaxation method (Press et al. 1992).In this technique, an approximate finite-difference equation is employed on a mesh of points to replace the ODEs (Huang & Yu 2022).

Small Ambipolar Elsässer Number
As mentioned earlier, the magnetorotational instability MRI is suppressed in the regime of ambipolar diffusion, particularly for small Elsässer numbers (i.e., Λ = 1).Equation ( 27) is coupled to the other equations through its right-hand side.However, in the case of small Elsässer numbers (Λ = 1), this equation can be approximately decoupled.It is preferable to make the Equation (27) dimensionless, considering the initial magnetic field at the bump region as B f (r 0 ) = B 0 /r 0 in the following form: where we have B 1 = δB f (r)/B f (r 0 ) and x = r/r 0 .The solution of Equation ( 29) is as follows: where c 1 and c 2 are constant, and we have ( ) If we set m = 0 (axisymmetric stability) and ν = 2 in Equation (31), the result is However, c 2 should be 0 because the amplitude of the magnetic field cannot be infinite at large radii of a PPD.Also, c 1 is equal to "1" because we have "B 1 = 1" at x = 1.It is better to consider another typical case (i.e., m = 1 and ν = 1.8).In this case, we have The perturbation amplitude of the magnetic field is shown in Figures 1 and 2 for small values of the Elsässer number.As observed, the perturbation amplitude of the toroidal magnetic field exhibits a helical behavior in the imaginary plane, causing it to oscillate as the parameter "m" increases.This behavior may be significant in the study of turbulent stress in the outer regions of a PPD.6 The Astrophysical Journal, 965:81 (13pp), 2024 April 10 Gholipour

Dispersion Relation
For the corotation mode (| ω| 2 = Ω 2 ), Lovelace et al. (1999) obtained a dispersion relation by incorporating the r dependence ( ) ~ik r exp r into the main equations when k r r ? 1.To ensure the validity of the WKB approximation in this scenario, the wavelength must be smaller than the length scale of the background change.Since the disk without an initial Gaussian bump is stable against the RWI, it is the Gaussian bump itself that can cause the RWI.The width of the Gaussian bump must correspond to the length scale of the background change.Additionally, we have set the width of the Gaussian bump to be approximately equal to the disk-scale height H (see Section 2.1).Therefore, the condition for the WKB approximation to be valid is λ = H and λ = r (i.e., σ → ∞).
Following the work of Lovelace et al. (1999), we consider the r dependence ( ) ~ik r exp r for the perturbations in Equations ( 23)-( 28  where a 4 , a 3 , a 2 , a 1 , and a 0 are constants (see the Appendix).This dispersion relation is a fourth-order equation with four roots.The imaginary part of ω * associated with * ( ) w < Im 0 and * ( ) w > Im 0indicates the stable and unstable modes.If the imaginary part of ω * is always negative, the disk is stable against perturbation.
On one hand, the outer regions of a PPD are sufficiently cold, resulting in trivial viscosity effects in these regions.On the other hand, the MRI, which is regarded as the source of alpha viscosity, is ineffective in these regions due to the presence of ambipolar diffusion.Therefore, we consider a typical case of the outer regions where the viscosity is very low (i.e., α → 0), highlighting the effect of ambipolar diffusion on the RWI in a PPD.At this stage, we ignore the effect of viscosity.We solve the dispersion relation using the Laguerre method (Press et al. 1992).We found that the dispersion relation has three roots that are always negative, along with a root associated with the presence of ambipolar diffusion.Since the results are sensitive to two parameters, β and Λ, we plot the results with respect to these parameters.
The results are shown in Figures 3-5.Figures 3-4 depict the relationship between the imaginary growth rate and wavenumber, while Figure 5 illustrates the growth rate in relation to the Elsässer number for different values of the ratio of magnetic pressure to gas pressure.
Figure 3 displays the imaginary part of the nondimensional growth rate, denoted as " * ( ) w Im ," plotted against the nondimensional wavenumber σ, for different values of the Elsässer number (β = 0.01, 0.1, and 1) and the azimuthal mode number (m = 1, 3, and 5).This is for the case where Λ = 0.1 and x = 0.8.
It can be observed that as σ tends toward infinity, the growth rate approaches a small constant value for m = 1, 3, and 5.The magnitude of this constant value depends on the values of β, so decreasing β leads to a lower growth rate.
Figure 4 is similar to Figure 3 but for Λ = 1.It is evident that the growth rate increases as the values of β increase while maintaining constant Elsässer numbers.This observation suggests that a relatively strong toroidal magnetic field promotes the occurrence of the RWI, whereas ambipolar diffusion tends to suppress the RWI.Furthermore, the increase of "m" leads to an increase in the highest growth rate for σ < 500.For σ > 500, the change of "m" has trivial effects on the growth rate.From these two figures, we can conclude that the growth rate has a linear relationship with both Λ and β as σ approaches infinity.Additionally, the following relationship between the imaginary part of the nondimensional growth rate and Λ and β can be obtained: p It is now useful to consider the above equation in the dimensional form as follows: where a » c 0.5 c s 2 , a constant coefficient in the isothermal disk.Clearly, this equation shows that ambipolar diffusion suppresses the RWI, while an increase in magnetic pressure relative to gas pressure supports the occurrence of the RWI.
Figure 5 illustrates the impact of the Elsässer number and the ratio of magnetic pressure to gas pressure on the growth rate of the RWI.However, the values of β differ from Figures 3 and 4. The graphs demonstrate that, when σ is large, the growth rate increases linearly with respect to Λ as β increases.Furthermore, when β 1, the growth rate increases linearly with respect to β as Λ increases.It is worth noting that, at β = 10, the growth rate reaches a maximum point at Λ = 6 and then rapidly decreases.

Conclusion and Discussion
In this study, we investigated the influence of ambipolar diffusion on the RWI in the outer regions of a PPD at large radii.To accomplish this, we examined the linear evolution of the RWI in a PPD with a threaded magnetic field, taking into account the impact of ambipolar diffusion.Additionally, we parameterized the ambipolar diffusivity using the Elsässer number.We calculated the perturbation amplitude of the toroidal magnetic field over the bump regions, particularly for small values of the Elsässer number.We observed that the amplitude of the toroidal magnetic field oscillates as the value of "m" increases.
Next, we derived the dispersion relation for the growth rate of the RWI with the WKB approximation.Our findings demonstrate that the growth rate of the RWI decreases as the Elsässer number decreases.This is consistent with the results of simulation studies on the effect of ambipolar diffusion on the growth of the magnetorotational instability MRI, as it has been observed that the MRI is suppressed by ambipolar diffusion when the Elsässer number decreases.
Now is an appropriate time to compare simulation and observational works.Simon et al. (2013b) utilized local numerical simulations to survey the strength and nature of MHD turbulence in the outer regions of PPDs.These regions are characterized by the dominance of ambipolar diffusion as the primary nonideal MHD effect.They found that the turbulence caused by the MRI disappears when Λ < 1 and " vs. the nondimensional wavenumber σ (the nondimensional wavenumber) for various values of Elsässer number (Λ = 0.1) for the case of (β = 0.01, 0.1, 1) and x = 0.8.becomes progressively stronger as the surrounding ambipolar diffusion decreases.Nearly ideal MHD behavior is restored when Λ > 10 3 .In the intermediate regime (10 < Λ < 10 3 ), ambipolar diffusion leads to a substantial increase in both the MRI dynamo cycle and the characteristic scale of the magnetic field structure.In other words, in the absence of a vertical magnetic field, the MRI is entirely deactivated in a specific area within the outer region of the PPD (Simon & Armitage 2014).
Although the structure and occurrence of the RWI compared to the MRI in ambipolar-dominated regions are very different, the findings of this study show that ambipolar diffusion has similar effects on both instabilities in these regions.However, there is a difference: in the presence of ambipolar diffusion, the RWI can arise in these regions, but this depends significantly on the value of the initial magnetic pressure to gas pressure in the intermediate regime (10 < Λ < 10 3 ).Furthermore, it was " vs. the nondimensional wavenumber σ (the nondimensional wavenumber) for various values of Elsässer number (Λ = 1) for the case of (β = 0.01, 0.1, 1) and x = 0.8.found that the amplitude of the toroidal field strength, which oscillates over time due to the dynamo, serves as a parameter controlling the stress levels (Simon et al. 2011).The larger the amplitude of oscillation (while avoiding suppression of the MRI due to ambipolar diffusion), the stronger the turbulent stresses.Regarding Section 3.2 and Figures 1 and 2, the perturbation amplitude of the toroidal field strength of the RWI oscillates at high values of "m."As stated previously, small values of the Elsässer number correspond to large values of ambipolar diffusivity and vice versa.Thus, in the outer regions of a PPD, the RWI generates turbulent stress in the presence of ambipolar diffusion.
ALMA observations have now detected the trapping of dust particles in the outer regions of PPDs (van der Marel et al. 2013; " vs. the Elsässer number for various values of β, i.e., β = 0.1, 1, 10 when σ = 10 5 and x = 0.8.Fukagawa et al. 2013;Flock et al. 2015;Pinilla et al. 2015;van der Marel et al. 2015;Ruge et al. 2016;Zhang et al. 2016;Villenave et al. 2020).In dust traps, particles are expected to experience easier growth due to the locally enhanced dust-to-gas mass ratio (Booth & Clarke 2016;Surville et al. 2016).However, the details of this coagulation process remain poorly understood, primarily due to the largely unknown properties of the dust.In the outer regions of a PPD, the ionization rate may decrease enough to suppress MRI turbulence.However, the occurrence of the RWI leads to the production of strong vortices, which are significant candidates for dust trapping (Lovelace et al. 1999;Lyra et al. 2009).In this regard, we can indicate that recent observational studies have revealed a preference for identifying ring/cavity and arc substructures at larger radii (∼tens of astronomical units; Huang et al. 2018Huang et al. , 2018;;Long et al. 2018;Pinilla et al. 2018;van der Marel et al. 2019;Andrews 2020).The summary of the results can be outlined as follows: 1. Small values of the Elsässer number suppress the occurrence of the RWI in the outer regions of a PPD, while values around 1 amplify the occurrence of the RWI in the outer regions of a PPD. 2. The perturbation amplitude of the toroidal magnetic field exhibits a helical behavior in the imaginary plane, resulting in oscillations as the value of "m" increases.This behavior is significant in studying turbulent stress in the outer regions of a PPD. 3.An increase in magnetic pressure relative to gas pressure supports the occurrence of the RWI.This support is significant for Elsässer number values around 1 but insignificant for small values.4. The growth rate increases as the values of the Elsässer number increase.However, when magnetic pressure dominates over gas pressure (β = 10), the growth rate increases with higher values of the Elsässer number until it reaches Λ = 6.After that point, the growth rate decreases as the Elsässer number continues to increase.Therefore, it can be concluded that the optimal range for angular momentum transport in the outer regions of a PPD due to the RWI is when Λ is around 1, and β is maximized.
Finally, it is strongly recommended to consider the influence of ambipolar diffusion on the RWI through the utilization of nonlinear numerical simulations.and The Gauss equation: The induction equation: The polytropic equation: The equations can be converted to a compact form if we define the following constants: After some calculations, Equation (A16) can be linearized as follows: The dispersion relation in Equation (A17) reduces to a thirdorder equation for a nonmagnetized disk, i.e., η ≡ μ ≡ β = 0 because in this case, we have a 0 = 0.

ORCID iDs
Mahmoud Gholipour https:/ /orcid.org/0000-0003-2599-0082 investigated the occurrence Original content from this work may be used under the terms of the Creative Commons Attribution 4.0 licence.Any further distribution of this work must maintain attribution to the author(s) and the title of the work, journal citation and DOI.
Some observational and numerical studies indicate that n = 1 in the outer regions of a PPD (e.g.,Yu & Li 2009;Gholipour 2020

Figure 1 .Figure 2 .
Figure1.The amplitude of the perturbed magnetic field between x = 0.5 and x = 1.5 for the case of Λ = 1 and ν = 1.5.

)
The calculations are somewhat detailed, so we review them in the Appendix.It is very useful to consider nondimensional variables instead of dimensional variables.Thus, we have the region of the bump.Notably, the case of γ = 1 leads to ϒ 0 = 1, Ξ 0 = Π 0 and *