Thermal Effects of Ambipolar Diffusion during the Gravitational Collapse of a Radiative Cooling Filament

In this study, we consider the effects of ambipolar diffusion during the gravitational collapse of a radiative cooling filamentary molecular cloud. Two separate configurations of magnetic field, i.e., axial and toroidal, are considered in the presence of the ambipolar diffusion for a radiative cooling filament. These configurations lead to two different formulations of the problem. The filament is radiatively cooled and heated by ambipolar diffusion in both cases of magnetic field configurations. The self-similar method is used to solve the obtained equations in each case. We found that the adiabatic exponent and ambipolar diffusivity play very important roles during the gravitational collapse of a cooling filament. The results show that the ambipolar heating significantly increases the temperature in the middle regions of a cooling filament. Furthermore, we found that the ambipolar diffusion has very important effects during the collapse, so that its heating effect is dominant over its dynamical effect in the middle regions of a cooling filament. The obtained results also address some regions where the rate of star formation is more or less compared to the observational reports.


Introduction
Recent observations by the ALMA project and the Herschel space telescope reveal that stars are often formed in regions associated with filamentary molecular clouds (e.g., André et al. 2010André et al. , 2014;;Henning et al. 2010;Molinari et al. 2010;Arzoumanian et al. 2011;Schneider et al. 2012;Bonne et al. 2020).In quiescent regions where turbulence has little effect, it is believed that the clouds are supported by thermal pressure and the magnetic field, as opposed to gravity.In this context, the cooling and heating processes as well as magnetic field diffusion play crucial roles in initiating the collapse phase.These processes also have significance during the gravitational collapse of filamentary clouds.The magnetic fields associated with the filamentary clouds are often either perpendicular or parallel to the clouds (e.g., Palmeirim et al. 2013;Soler et al. 2013;Planck Collaboration 2016;Fissel et al. 2019;Seifried et al. 2020;Li et al. 2022;Priestley & Whitworth 2022).For example, Li et al. (2022) conducted a stability analysis of two regions, OMC-3 and OMC-4, in the massive and extensive molecular cloud complex of Orion A using observational data.They found that the filamentary molecular cloud and dense clumps are magnetically supercritical in OMC-3, whereas they are often magnetically subcritical in OMC-4.They concluded that the regions associated with OMC-3 are likely in the gravitational collapse phase, and the dynamic effects of the magnetic field are significant in the dense gas structures of OMC-3.
Our understanding of the magnetic field has significantly improved due to recent advancements in observational and simulation work in molecular clouds (e.g., Körtgen & Banerjee 2015;Wurster et al. 2017Wurster et al. , 2018)).For instance, it has been established that there is a relationship between density and the magnetic field (e.g., Crutcher et al. 2010;Mouschovias & Tassis 2010;Crutcher 2012;Tritsis et al. 2015).However, determining how different processes, such as cooling and heating, affect the dynamics of gas during gravitational collapse remains challenging.While analytical works have their limitations, the self-similar solution method has proven to be a powerful technique for studying collapsing filaments.Many authors have explored gravitational collapse using similarity solutions (e.g., Larson 1969;Penston 1969;Whitworth & Summers 1985;Shu et al. 1987;Inutsuka & Miyama 1992;Kawachi & Hanawa 1998;Hennebelle 2003;Tilley & Pudritz 2003;Shadmehri 2005;Holden et al. 2009;Khesali et al. 2014;Aghili & Kokabi 2017;Gholipour 2018).However, the dynamical and thermal effects of ambipolar diffusion (AD) on a radiative-cooling filamentary cloud have not been fully considered in previous studies of self-similar collapse.Therefore, our aim is to investigate the dynamical and thermal effects of AD during the self-similar collapse of a cooling filament.We present the general formulation in Section 2, followed by the standard assumptions in Section 3. The self-similar solutions are provided in Section 4, and the results are discussed in Section 5. Finally, we conclude our findings in Section 6.

General Formulation
Here, we consider a magnetized cooling filament that undergoes collapse in the presence of AD.The basic equations are as follows: The continuity equation is given by where ρ is the gas density and V is the velocity vector.The equation of motion is as follows

2
where P is the gas pressure, Φ is the gravitational potential, and B is the magnetic field vector.The Poisson's equation is where G is the gravitational constant.The Gauss' Law for magnetism is given by: The induction equation in the presence of AD becomes A where η A is the ambipolar diffusivity.The energy equation is where γ is the polytropic exponent (adiabatic index), and Ω = ρΛ − Γ is the net cooling function (with Λ as the cooling rate and Γ as the heating function).Finally, the equation of state is where T is the temperature, R g is the gas constant and μ B is the effective molar mass.Furthermore, σ is defined as the ratio of the gas constant to the effective molar mass (i.e., σ = R g /μ B ) for simplicity in Section 4.

Standard Assumptions
We use the standard cylindrical coordinates (r, f, z) with the origin at the center of the filament.Furthermore, the filament is assumed to be axisymmetric and very long, aligned along the zaxis.Therefore, we can assume ∂/∂f ≡ ∂/∂z ≡ 0.
As stated before, the magnetic fields associated with the filamentary clouds are often either perpendicular or parallel to the clouds (e.g., Palmeirim et al. 2013;Soler et al. 2013;Seifried et al. 2020;Priestley & Whitworth 2022).Furthermore, some observational reports indicate the existence of a toroidal magnetic field in the filament in some star-forming regions, such as Orion A and Orion B (Tahani et al. 2019).Accordingly, we will consider two cases of magnetic field orientation as follows: 1. Case I.This corresponds to a purely toroidal magnetic field, where the magnetic field is perpendicular to the cloud (B f ≠ 0 and B z = 0).2. Case II.This corresponds to a purely axial magnetic field, where the magnetic field is parallel to the cloud (B f = 0 and B z ≠ 0).
In this situation, the physical variables only depend on the radius and time.Based on these assumptions, we can express the gravitational potential as the mass per unit length M: 8 Thereupon, the continuity equation can be rewritten as: The momentum equation for Case I is: For Case II, the momentum equation is: The induction equation for Case I becomes: For Case II, the induction equation is: Regarding Equations (1) and (6), we can write: Finally, we have: As can be seen, the momentum equation and the induction equation differ between the two cases.Although the equations have been simplified by the standard assumptions, it is still necessary to understand the behaviors of the magnetic diffusivity as well as the net cooling function in the next section.

Cooling and Heating
If we assume that the net cooling function is zero (i.e., Ω = 0 or Γ = ρΛ) in Equation (6), the energy equation can be transformed into the polytropic equation ( r ).It is widely accepted that low-energy cosmic rays (CRs) play a significant role in collisional ionization and heating within the interstellar medium (e.g., Zweibel 2017;Padovani et al. 2020).In fact, the temperatures of molecular clouds are determined by the balance between radiative cooling and heating mechanisms (such as CR heating and ambipolar drift).
The isothermal molecular cloud is a result of this balance at the pre-collapse stage, where it is in equilibrium.However, the question that arises here is: Under what conditions does the collapse become isothermal during gas contraction?This is because, even though the net cooling function may be nonzero during gas contraction, the gravitational compression of gas also causes heating.This question can be asked in another way: Is it necessary to differentiate between an isothermal cloud (in a static state) and an isothermal collapse (in a dynamic state) within a cooling radiative filament?In fact, one of our aims in the next sections of this study is to find the answer to this question.
Cosmic-ray heating is an external source, while cooling is an intrinsic characteristic of the cloud.Meanwhile, ambipolar drift heating possesses an inherent and dynamic nature, causing its value to change during gravitational collapse.It is without a doubt that the ambipolar drift (AD) effect is an appropriate candidate to initiate the gravitational collapse in a molecular cloud.The cloud is supported by both thermal pressure and the magnetic field, which act against the gravitational collapse, thus maintaining a state of force balance.However, the AD effect, also known as ion-neutral friction, leads to the diffusion of the magnetic field.As a result, it overcomes the gravitational support provided by the magnetic field and thermal pressure.Consequently, the cloud enters a phase of gravitational collapse, with the magnetic field being dominated by the induction equation that includes the AD effect.In this case, the ambipolar diffusivity appears as an important parameter in this equation.In a weakly ionized molecular cloud, the ambipolar diffusivity can be written as (e.g., Gholipour 2018;Gholipour et al. 2022) where ρ i is the ion density and γ A is the drag coefficient that is a result of the frictional force between the ions and the neutrals (e.g., Gholipour et al. 2022).The relation between the ion density and the gas density is ρ i ≅ òρ n where ò is a constant coefficient as ò = 3 × 10 −16 g 1/2 .cm −3/2 and n is equal to 1/2.Thus, we can write (e.g., Gholipour 2018) where C 0 = 1/4πγ A ò is a constant.Accordingly, the derivative of the ambipolar diffusivity is Now let us consider the heating by ambipolar diffusion (e.g., where v d is the drift velocity of the ions relative to the neutrals given by Thus, we can write Consequently, we can rewrite the ambipolar heating for each case as follows where Γ f and Γ z correspond to the ambipolar heating for Case I and Case II, respectively.Generally, the cooling function is given by Whitworth & Jaffa (2018) is the magnitude of the velocity divergence, and Λ 0 , ν, α, and δ are constants that should be substituted from the observational reports.This function was obtained by curve fitting the observed data on the CO emission of the molecular clouds.However, the challenging issue is determining the values of ν, α, and δ.At low densities and low column densities, when a CO excited level is collisionally excited it usually de-excites through spontaneous emission (so the level populations are not thermalized) and the photons usually escape without being absorbed (so it is optically thin).In this limit, we have ν ≈ 1, α ≈ 3/2, and δ ≈ 0.
At high densities and high column densities, when a CO molecule is collisionally excited, it usually de-excites collisionally as well (so the level populations are thermalized) and any emitted photons usually get absorbed and re-emitted before they eventually escape (so it is optically thick).In this limit, we have ν ≈ − 1, α ≈ 4, and δ ≈ 1.In this study, we aim to consider the outer regions of the molecular clouds where the density is low and the cloud is optically thin.Therefore, the cooling function is given by where α is equal to 3/2.Consequently, the net cooing function is as follows where it is assumed that the AD heating is dominant over the heating of cosmic rays during the gravitational collapse process.

AD Timescale
The AD heating has been considered an important heating mechanism in molecular clouds by many authors (e.g., Scalo 1977;Zweibel & Josafatsson 1983;Elmegreen 1985;Padoan et al. 2000;Nejad-Asghar 2011;Li et al. 2012;Elyasi & Nejad-Asghar 2016).It is useful to compare the dynamical timescale with the ambipolar diffusion timescale.The dynamical contraction timescale of a filamentary molecular cloud can be assumed as a multiple of the freefall timescale, given by 3.8 10 yr 10 cm .26 The characteristic timescale of magnetic flux loss is given by 3 10 yr 10 cm 30 G 0.1 pc , where R is the radial size of the filament.Since the collapse timescale is shorter, the flux would be getting concentrated by the collapse, despite AD.However, it is helpful to consider these timescales from a different point of view.Following the work of Zweibel & Brandenburg (1997), the AD Reynolds number R AD can be written as where t f is the dynamical crossing time and t AD is the AD timescale.Li et al. (2012) considered the effect of AD heating and found that the AD heating rate is significant in the range of R AD > 1. Simulation and analytical works indicate that the values of R AD are often >1 at large radii of a cloud (e.g., Li et al. 2006).

Self-similar Method
The self-similar method represents physical variables that can be as dimensionless functions of a self-similar variable.To achieve this, we introduce an independent self-similar variable, denoted as x ≡ Art a (where a is a dimensionless index; for example Lou & Xing 2016;Gholipour 2018).The MHD selfsimilar transformation follows the self-consistent form:  where the numerators on the right-hand side of the above relations are nondimensional functions.Additionally, the nondimensional ambipolar diffusivity is denoted as η(x) = κ/ñ 3/2 where k p = G C 2 0 represents the strength of η(x) (see also Appendix).Moreover, we have Substituting Equation (29) into Equation (8) gives Furthermore, the induction equation in Case I (Equation ( 12)) leads to For Case II, the induction equation (Equation ( 13)) converts to Finally, Equation (14) becomes where the indices f and z (in Π f,z ) refer to Case I and Case II, respectively.Also, we have Equations (31)-( 38) form a set of ordinary differential equations (ODEs).In Case I, the set of equations consists of Equations (31), ( 34), (36), and (38) with Π f .In Case II, the set of the equations consists of the Equations (31), ( 35), (37), and (38) with Π z .These sets of equations can be solved numerically if we know the boundary equations at the outer regions of the filament.This issue is discussed in detail in Appendix.

Results and Discussion
The results are presented in Figures 1-8, which contain a total of 54 plots.Typically, we choose a range between 140 and 240 for x (see Appendix), corresponding to the middle radius and outer radius, respectively.We found that the problem is highly sensitive to the values of γ (the polytropic exponent) and κ (the strength of the nondimensional ambipolar diffusivity).Therefore, we have selected three values for κ = 0.003, 0.005, and 0.007 (see Appendix).Additionally, we have considered two cases for γ as follows: 1) γ < 1 and 2) γ > 1.
For the case of γ < 1, we found that, as γ approaches 0, collapse occurs with significant effects.Accordingly, we typically choose three values for γ: 0.01, 0.3, and 0.7.For the case of γ > 1, we typically choose three values for γ: 4/3, 3/2, and 5/3.Since our focus is on studying AD heating during the collapse, we examined the results for two scenarios: one where Γ AD ≠ 0 (including AD heating) and another where Γ AD = 0 (excluding AD heating).It is important to note that there are no self-similar solutions for γ > 1 and Γ AD ≠ 0. Therefore, we only analyzed the case of Γ AD = 0 for γ > 1.Each figure includes two panels, with the left panel representing the purely axial magnetic field and the right panel representing the purely toroidal magnetic field.Figure 1 shows the profile of the nondimensional temperature, i.e., τ(x) = p(x)/ñ(x), versus −x for certain values of κ and γ, without the AD heating (Γ AD = 0) and γ < 1.As can be seen in the cases of γ = 0.3 and γ = 0.7, τ(x) increases as |x| decreases from 240.This indicates that an increase in κ leads to a significant rise in the value of τ(x).For the case of γ = 0.01, increasing κ leads to a slight increase in τ(x).However, it can be considered nearly constant during the collapse, particularly for κ = 0.003.This demonstrates that the dynamical effects of AD can effectively control the temperature profile in the absence of AD heating.This observation holds true regardless of the direction of the magnetic field.Moreover, an increase in γ results in higher values of τ(x) and vice versa.This suggests that γ has a resistance effect against gravitational collapse in a cooling filament.Although the impact of the magnetic field direction is not very noticeable for γ < 1, the toroidal field has a greater effect on the temperature compared to the axial field.Figure 2 is the same as Figure 1, but for γ > 1.In contrast to the previous figure, the temperature significantly increases as γ increases.The dynamical effect of AD is also noticeable, as the temperature increases with increasing κ.However, the important point to highlight here is that when Γ AD = 0, the temperature increases dramatically, reaching a maximum point in the middle regions of the cloud.After that, it rapidly decreases until the collapse is halted.This is consistent with the findings of Nakamura (1998), Federrath & Banerjee (2015), and Toci & Galli (2015).In both cases of γ, it can be concluded that the dynamical behavior of the AD tends to make the collapse hotter, similar to the polytropic exponent in the range of 140 < x 240.
Figure 3 is the same as Figure 1, except for Γ AD ≠ 0. When comparing Figures 1-3, it is clear that AD heating significantly increases the temperature during the collapse.In the absence of AD heating, increasing γ and κ leads to an increase in temperature.However, in the presence of AD heating, decreasing γ and κ results in an increase in temperature.When the axial field's κ is increased, the curves with different κ values converge to a point in the presence of Γ AD ≠ 0. On the other hand, when the toroidal field's κ is decreased, the curves with different κ values diverge in this scenario.This observation suggests that AD heating distinguishes between the effects of Case I and Case II on the temperature profile as γ approaches zero.Consequently, AD heating prevents the collapse when γ > 1.The key point to note here is that the filaments heat up rapidly near |x| = 240, except when γ approaches zero.This is because the filament is assumed to have an isothermal configuration at |x| = 240 in order to establish the boundary conditions (see Appendix).As |x| decreases, the filament takes on a new configuration that depends on the parameters associated with γ and κ.
Figure 4 represents the profile of nondimensional radial velocity versus −x for some values of κ and γ in the case of Γ AD = 0 and γ < 1.There is a linear relationship between u r (x) and x so that the slope of the curve increases with the increase of κ and γ. Figure 4 is the same as Figure 5, but for γ > 1.Compared to the previous figure, the curve is outside of its linear shape so that the collapse occurs faster in the case of γ > 1 than in the case of γ < 1. Figure 6 is the same as Figure 4, but for Γ AD ≠ 0. As can be seen, the AD has a significant braking effect on the collapse.Regarding these figures, the fast and slow collapse can be discussed regarding the cases of Γ AD ≠ 0 and Γ AD = 0.
Figures 7, 8, and 9 depict the profiles of nondimensional magnetic fields in three cases: (γ < 1 and Γ AD = 0), (γ > 1 and Γ AD = 0), and (γ < 1 and Γ AD ≠ 0), respectively.As observed in Figure 7, the magnetic field increases as one moves toward the central regions of the cloud (i.e., as |x| decreases) in all cases.This issue is also valid for Figures 8  and 9.The profile of the magnetic field is highly sensitive to the values of κ, while the change in adiabatic exponent and the presence of the AD heating are not as significant.However, the rate of increase in Case I is greater than that in Case II.This finding may be significant in the study of the magnetic flux problem in the star formation (e.g., Bodenheimer 2011;Zhao et al. 2011), and in considering fossil magnetic fields in accretion disks.Now, it is useful to discuss the relation of our findings with the polytropic exponent.The adiabatic exponent (index) is a crucial parameter for considering gas contraction during the collapse phase.Essentially, it describes the resistance of the gas to compression.It is important to mention the processes that influence the values of polytropic exponents in molecular clouds.Generally, the polytropic exponent is a complex function of gas temperature, dust temperature, velocity, radiation intensity, and chemical composition (Spaans & Silk 2000).Therefore, a detailed analysis involving chemistry, thermal balance, and radiative transfer is necessary to explain the value of the polytropic exponent.In this regard, Toci & Galli (2015) conducted a survey on the observational properties of filamentary clouds in cylindrical symmetry and found that the polytropic exponent should fall within the range of 1/3 > γ > 2/3.These results are also in full agreement with the findings of Federrath & Banerjee (2015).They derived a theoretical prediction for the dependence of the star-formation rate on γ, by numerically integrating the Hopkins (2013) intermittency probability distribution function (PDF).Specifically, they found that the star-formation rate increases by a factor of 1.7 for γ = 0.7 compared to γ = 1.For γ = 5/3, the star-formation rate decreases by a factor of 3 compared to γ = 1.
Here, it is useful to compare our results with the results of simulation work done by Nakamura (1998), which is similar to our work but differs in the consideration of the magnetic field.In fact, he performed one-dimensional simulations on the collapse of a quiescent, nonmagnetized filamentary molecular cloud, taking into account the heating and cooling processes.He found that when the central density exceeds 10 4 -10 5 cm −3 and γ is ∼1.1, a shock wave is formed at r ∼ 0.05 pc.The shock wave separates the cloud into two parts, namely a dense spindle and a diffuse envelope.If we consider the toroidal field with κ = 0.005, γ = 4/3, and Γ AD = 0 (see also Figure 2), the collapse halts at x = 122.This point indicates a radius of approximately 0.05 pc (see Appendix), which is consistent with the findings of Nakamura (1998).

Summary and Conclusions
In this study, we examined the thermal and dynamic effects of ambipolar diffusion (AD) on the self-similar collapse of a filamentary molecular cloud undergoing radiative cooling.The filament cools due to radiation and is heated by AD heating, particularly at large radii where the dynamical effects of AD are significant.We also investigated two different structures of the magnetic field separately: Case I, which involves a purely toroidal field, and Case II, which involves a purely axial field.Accordingly, we formulated the problem in two separate regimes regarding the structure of the magnetic field.
To account for the cooling and heating processes, we used the energy equation.The cooling function was scaled by the density and temperature parameters from the work of Goldsmith & Langer (1978; see also Goldsmith 2001).The heating function was adjusted to reflect the thermal effect of AD.We also employed the self-similar technique to solve the equations.Two important parameters, κ (the nondimensional coefficient of ambipolar diffusivity) and γ (the polytropic exponent), play significant roles in the evolution of a collapsing (cooling) filamentary cloud.We considered the problem for three values of κ and two cases of γ (i.e., γ < 1 and γ > 1).In this regard, κ controls the two effects of AD, i.e., the thermal and dynamical effects, while γ indicates the dominant thermodynamics of the cloud.Additionally, we have determined the temperature profile during the gravitational collapse of a cooling filament for different configurations of the magnetic field, both with and without AD heating.
It is crucial to mention again that there are no self-similar solutions for γ > 1 and Γ AD ≠ 0. The reason for this can be justified in the following way: the temperature increases significantly during the collapse, reaching its maximum value when γ > 1 and Γ AD = 0. Afterward, the temperature rapidly reduces until the collapse halts.On the other hand, when AD heating was added to γ < 1, the temperature increased significantly.Therefore, we expect that when AD heating is added to γ > 1, the temperature will rapidly and significantly increase compared to γ > 1 and Γ AD = 0. Consequently, it can be inferred that the star-formation rate should decrease when γ > 1 and vice versa.
We found that gravitational collapse is approximately isothermal for low values of the polytropic exponent, i.e., γ = 0.01, in the presence of the purely dynamical effect of AD (Γ AD ≠ 0).However, in the absence of AD effects (κ = 0), the temperature decreases during gravitational collapse for low values of γ and increases for other values of γ.Last, it can be concluded that the AD effect leads to an increase in temperature, regardless of whether this effect belongs to the dynamical or heating effect.The other results can be summarized as follows: 1.The regions associated with γ < 1 have more potential for star birth than the regions associated with γ > 1.This is because the temperature increases significantly to a maximum value at a certain radius during the collapse for γ > 1, and then decreases rapidly until the collapse halts.The specific radius at which this occurs depends on the ambipolar diffusivity.It can also be concluded that the smaller the ambipolar diffusivity value, the smaller the specific radius.2. The heating effect of AD is dominant over its dynamical effect on a cooling filament.3.In the absence of AD heating and at a larger radius than the specific radius discussed in item 1, gravitational collapse occurs at a faster rate in filaments associated with γ > 1 than in those associated with γ < 1.This is because the higher temperature is a consequence of the higher collapse velocity, i.e., there is less time for the gas to cool. 4. The comparison between Case I and Case II reveals that there is no significant difference in the absence of AD heating.However, in the presence of AD heating, the distinction between the two cases becomes apparent when γ approaches zero. 5.The presence of the AD effects leads to a reduction in the magnetic field values.It should be noted that the values of the magnetic field increase at a higher rate in Case I than in Case II.Furthermore, increasing the values of the polytropic exponent increases the values of the magnetic field.6. Regarding the inside-out collapse hypothesis, it suggests that the collapse of centrally concentrated objects occurs in an inside-out manner (Shu 1977).In starforming regions, we anticipate the clouds to have a γ value of less than 1 in the outer regions and a γ value greater than 1 in the central regions.This is because there are two reasons for this.First, the collapse occurs at a faster rate for γ > 1.Second, the values of γ increase with increasing temperature and density in the inner regions of a cloud.
Finally, two suggestions are given for further improvements to apply.First, it is recommended to include cloud rotation in this work.Second, it is suggested that this work also be performed through simulation.

r
Substituting Equation (31) into the second part of Equation (32) leads to If the physical variables of Equation (29) are replaced from the Equations (10)-(11), the final results are

Figure 1 .
Figure1.The profile of the nondimensional temperature, i.e., τ(x) = p(x)/ñ(x), vs. −x for some values of κ and γ < 1 when Γ AD = 0.The left and right panels present Case II (purely axial magnetic field) and Case I (purely toroidal magnetic field), respectively.Here, τ 0 corresponds to the temperature at the outer radius of the filament (|x| = 240).

Figure 4 .
Figure 4.The profile of nondimensional radial velocity, i.e., u(x), vs. −x for some values of κ and γ < 1 when Γ AD = 0. Here, u 0 is the value of u(x) at the outer radius of filament (|x| = 240).

Figure 5 .
Figure5.The profile of nondimensional radial velocity, i.e., u(x), vs. −x for some values of κ and γ > 1 when Γ AD = 0. Here, u 0 is the value of u(x) at the outer radius of the filament (|x| = 240).

Figure 7 .
Figure 7.The profile of nondimensional magnetic field vs. −x for some values of κ and γ < 1 when Γ AD = 0.

Figure 8 .
Figure 8.The profile of nondimensional magnetic field vs. −x for some values of κ and γ > 1 when Γ AD = 0.

Figure 9 .
Figure 9.The profile of nondimensional magnetic field vs. −x for some values of κ and γ < 1 when Γ AD ≠ 0.