Exploring the Impact of Self-Excited Alfvén Waves on Transonic Winds: Applications in Galactic Outflows

Abstract

The impact of cosmic rays is crucial to understand the energetic plasma outflows coming out from the Galactic centers against the strong gravitational potential well. Cosmic rays can interact with thermal plasma via streaming instabilities and produce hydromagnetic waves/fluctuations. During the propagation of cosmic rays it can effectively diffuse and advect through the thermal plasma which results the excitation of Alfvén waves. We are treating thermal plasma, cosmic rays and self-excited Alfvén waves as fluids and our model is referred as multi-fluid model. We investigate steady-state transonic solutions for four-fluid systems (with forward as well as backward propagating self-excited Alfvén waves) with certain boundary conditions at the base of the potential well. As a reference model, a four-fluid model with cosmic-ray diffusion, wave damping and cooling can be studied together and solution topology can be analyzed with different set of boundary conditions available at the base of the gravitational potential well. We compare cases with enhancing the backward propagating self-excited Alfvén waves pressure and examining the shifting of the transonic point near or far away from the base. In conclusion we argue that the variation of the back-ward propagating self-excited Alfvén waves significantly alters the transonic solutions at the base.

1. Introduction

Cosmic rays play a significant role in the interstellar environment and galaxy evolution. They are highly energetic charged particles, mostly protons and atomic nuclei, that originate from outside the solar system. These energetic particles follows a power law spectrum which extends over many orders of magnitude in energy. Cosmic rays usually originate from the energetic sources available in galaxies. Some of the energetic events like supernovae, active galactic nuclei and tidal disruption events may contribute to the flux of cosmic rays. Cosmic rays can travel from galactic to extragalactic sources with relativistic speeds and keep extremely high energies, often exceeding those produced by human made particle accelerators.

Cosmic ray energy density and the energy density of other components of the interstellar medium are of the same order of magnitude [1,2]. The equipartition of energy between the different components indicates there are significant interactions among these components of the interstellar medium. Cosmic rays can play a dynamical role in the structure and evolution of the interstellar as well as intergalactic medium. Cosmic rays can influence the formation of stars by affecting the interstellar medium and molecular clouds. Cosmic Rays can effectively ionize and heat the interstellar environment and hugely impacting the formation of galaxies.

The propagation of cosmic rays in highly conducting plasma has been discussed throughout the years. Propagation and interactions of cosmic rays in the magnetized thermal plasma remain tremendously important in interstellar as well as intergalactic environments. Cosmic rays interact with thermal plasma through magnetic fluctuations. Streaming of cosmic rays through plasma can generate waves such as Alfvén waves. When a particle’s gyro-radius is comparable to the wavelength of a wave ($r_g\sim \mathsf{\lambda }$), particle scattering occurs [3,4,5,6,7,8,9]. In the presence of both forward and backward propagating self-excited Alfvén waves, cosmic rays also diffuse in momentum space via stochastic acceleration (i.e., second order Fermi acceleration). Many studies have been discussed on the hydrodynamical perspective of cosmic rays and it can significantly alter different modified structures and instabilities in the cosmic ray plasma system [10,11,12,13,14,15,16,17,18,19].

The nature of the Alfvén waves propagation, particularly in the nonuniform plasma background, is dependent on the relationship between its wavelength $\mathsf{\lambda }$ and the characteristic scale of the background plasma L that can be put into the treatise as WKBJ or non-WKBJ models. For the consideration of small-size wavelength with comparison to the characteristic length $\mathsf{\lambda }\ll L$ and unidirectional flow, the Alfvén wave transport is ideally described by the WKBJ approximation. Regarded as the simplest model that was formulated and initially developed in the 1960s and 1970s, the main characteristics of the Alfvén wave WKBJ model features high frequency and small wavelength that propagates in slowly varying media with the absence of coupling interaction and mixing between different waves modes [20,21,22,23,24,25,26,27]. The studies by Alazraki [22], Belcher [23], and Belcher and Davis [24], were among the first to incorporate the WKBJ approach for the Alfvén waves description to develop wave-driven plasma magnetohydrodynamics (MHD) one-fluid model in the context to study the dynamics of the solar wind. One of these studies highlighted that the existence of the outward propagating waves within the plasma can generate a considerable amount of pressure and energy to accelerate solar wind towards high velocities over extensive distances within the interplanetary medium. Comprehensive studies carried out by Hollweg include the development of the two-fluid plasma Alfvén wave driven model that explores the non-linear damping heating effects on the proton species, with the implications of the model tend to reproduce solar wind properties at 1 AU [25,26,28,29]. On the other hand, his analyses of the WKBJ expansion for Alfvén waves highlights non-convergent behaviour that imposes serious limitations in accurately describing wave propagation and energy transfer processes. McKenzie and Völk [30] formally presented the WKBJ Alfvén wave equation to further develop the MHD models within the framework to study the cosmic-ray modified shocks. Furthermore, McKenzie employed the MHD models in WKBJ limit to investigate the interactions between Alfvén waves and various charged ion species in the solar wind model which specifically incorporates the wave action conservation, thereby highlighting the significance of the Alfvén wave pressure in binding the plasma system, thus enabling the model to provide insights of the solar wind acceleration and streaming instability [31]. But the WKBJ approximation fails to deal with cases in terms of the extreme fluctuations in the inhomogeneous plasma background and wavelength size comparable or greater than the characteristic length $\mathsf{\lambda }\ge L$ found in the realistic astrophysical scenarios such as the inner corona of the Sun and the interplanetary regions. Hence, non-WKBJ approach is required to capture the behavior of the alfvén waves propagation in these turbulent conditions. In contrast to their counterparts, the common traits of the non-WKBJ Alfvén waves model are mainly associated with low frequency and large wavelength, irregular amplitude variations, simultaneity of both inward and outward propagation directions due to reflection, mixing and coupling interactions between different waves modes [27,32,33,34,35,36]. The basic framework of the non-WKBJ Alfvén waves model was forwarded by Heinemann and Olbert [32] to study Alfvén waves near sun’s corona. The study mainly analyzed the coupling interactions and wave mixing between inward and outward propagating Alfvénic fluctuations. The basic phenomena of the reflection and transmission are crucial in forming coupling between different waves modes that develops analogous to the standing wave interference pattern. By this aspect the wave mixing influences the interactions of longer wavelengths in the inhomogeneity plasma background [27]. Perhaps the modifications of the Heinemann and Olbert model by the inclusion of the incompressible turbulence MHD effects, appear to have been motivated to address the observation related to solar wind fluctuations [33,34,35]. Developed by Zhou and Matthews [34,35] and Marsch and Tu [33], the framework consists of wave mixing turbulence equations for the Alfvénic waves fluctuation mainly expressed in terms of Elsässer fluctuation variables $Z^{\pm }$ which recover the Heinemann–Olbert model as a limiting case under purely poloidal fields. For convenience, the Velli sign convention is adopted, by which + is associated with outward(forward) and − for the inward(backward) waves, respectively [37]. So in the incompressible MHD turbulent framework, the interaction between forward and backward waves is governed by mixing tensor term that depends on the gradient of the background flow [27]. However, the coupling interactions between Alfvén waves and magneto-acoustic waves are usually treated to be negligible in both WKB and non-WKB models due to the assumption that the magneto-acoustic modes tend to die out due to decaying and compression by the Landau damping effect [5,27,32]. In the context of the hydrodynamic four-fluid model proposed by Ko to examine the cosmic ray-driven plasma system utilized in this study, turbulence conditions are not considered, leading to the automatic disregard of the effects of mixing and the coupling interactions between self-propagating forward and backward waves. Consequently, the four-fluid model is treated in the scope of WKBJ approximations.

Galactic outflows are energetic powerful winds originating from the center of the galaxies under the influence of strong gravitational potential and extend towards very large distances (i.e., up to several Kpc scales). Thermal gas is not solely playing a role to modify these energetic parsec winds but several other components available in the interstellar medium also playing a major role (i.e., magnetic field and cosmic ray). The velocity of these outflows can start from several hundred to few thousands km/s. These outflows can also be observed at very high red shift galaxies as well. Cosmic ray driven wind mechanism was first discussed by [38]. Furthermore [39,40,41], wrote down the proper formulation of cosmic ray driven winds with different variations available in the ISM components (i.e., including CR streaming and diffusion). Consequently, some major studies published includes [42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64] showed that continuous gas emissions in the form of outflows can hugely impact the evolution and fate of the galaxies. The binding of different components which serve for energy exchange mechanisms for galactic winds are poorly understood. Energy and momentum transferred by these energetic winds, tidal disruption events by black holes, and supernova explosions seems to be connected to get the overall feedback and impact on galaxies and it need more theoretical, observational and simulations scrutiny. The feed back coming from tremendous energetic sources are yet to be model. To understand the basics of cosmic ray driven galactic winds reader is referred to the recent reviews [42,60,65] for details.

2. Four-Fluid Cosmic Ray Plasma Model

A four-fluid model is developed by Ko [66], in which thermal plasma, cosmic rays and two opposite propagating self-excited Alfvén waves were included for the first time. Each component of the cosmic ray plasma system can be treated as fluid represented by its energy flux and pressure. The mass-density of the cosmic ray plasma system is solely contributed by the thermal plasma, whereas the waves and cosmic rays are characterized as massless fluids. The model is governed by total mass and momentum equations, as well as the energy exchange equations for different components. In flux-tube formulation, the steady state of the model governed by:

$$B\Delta ={\psi }_B=\sqrt{{\mu }_0}{\psi }_B^{\prime }\Delta ,$$

(1)

$$\rho U\Delta ={\psi }_m={\psi }_m^{\prime }\Delta ,$$

(2)

$$\rho U{\displaystyle \frac{dU}{d\xi }}=-{\displaystyle \frac{d}{d\xi }}\left(P_g+P_c+P_w^{+}+P_w^{-}\right)-\rho {\displaystyle \frac{GM}{{(\xi +a)}^2}},$$

(3)

$${\displaystyle \frac{1}{\Delta }}{\displaystyle \frac{d{F}_g\Delta }{d\xi }}=U{\displaystyle \frac{d{P}_g}{d\xi }}-\Gamma +L_w^{+}+L_w^{-},$$

(4)

$${\displaystyle \frac{1}{\Delta }}{\displaystyle \frac{d{F}_c\Delta }{d\xi }}=\left[U+(e_{+}-e_{-})V_A\right]{\displaystyle \frac{d{P}_c}{d\xi }}+{\displaystyle \frac{P_c}{\tau }},$$

(5)

$${\displaystyle \frac{1}{\Delta }}{\displaystyle \frac{d{F}_w^{\pm }\Delta }{d\xi }}=U{\displaystyle \frac{d{P}_w^{\pm }}{d\xi }}\mp e_{\pm }{V}_A{\displaystyle \frac{d{P}_c}{d\xi }}-{\displaystyle \frac{P_c}{2\tau }}-L_w^{\pm },$$

(6)

where the energy fluxes are

$$F_g=\left(E_g+P_g\right)U={\displaystyle \frac{{\gamma }_g{P}_g}{\left({\gamma }_g-1\right)}}U,$$

(7)

$$\begin{array}{cc}\hfill F_c& =\left(E_c+P_c\right)\left[U+(e_{+}-e_{-})V_A\right]-\kappa {\displaystyle \frac{d{E}_c}{d\xi }}\hfill \\ \hfill & ={\displaystyle \frac{{\gamma }_c{P}_c}{\left({\gamma }_c-1\right)}}\left[U+(e_{+}-e_{-})V_A\right]-{\displaystyle \frac{\kappa }{\left({\gamma }_c-1\right)}}{\displaystyle \frac{d{P}_c}{d\xi }},\hfill \end{array}$$

(8)

$$F_w^{\pm }=E_w^{\pm }\left(U\pm V_A\right)+P_w^{\pm }U=P_w^{\pm }\left(3U\pm 2V_A\right).$$

(9)

$$e_{\pm }={\displaystyle \frac{P_w^{\pm }}{P_w^{+}+P_w-}}$$

(10)

$${\displaystyle \frac{1}{\tau }}=16\alpha {\displaystyle \frac{V_A^2}{c^2}}{\displaystyle \frac{P_w^{+}{P}_w^{-}}{P_w^{-}+P_w^{+}}}$$

(11)

$$\kappa ={\displaystyle \frac{c^2}{3\alpha (P_w^{+}+P_w^{-})}}$$

(12)

$$\Gamma ={\delta }_c{\rho }^2\sqrt{P_g/\rho }$$

(13)

$$L_w^{\pm }={\delta }^{\pm }{P}_w^{\pm 2}\sqrt{P_g/\rho B^2}$$

(14)

Equations (1) and (2) come from zero divergence of magnetic field and mass continuity equation, and ${\psi }_B$ and ${\psi }_m$ represents the magnetic and mass flux flow rates, respectively. $\Delta$ is the cross-sectional area of the flux-tube, $\xi$ is the coordinate along the flux-tube which is the normal of the galactic disk in the present model. In Equation (3) $\rho$ and U are the density and flow speed of the plasma, $P_g$, $P_c$ and $P_w^{\pm }$ are the pressures of thermal plasma, cosmic ray and self-excited Alfvén waves (± denote forward/backward propagating waves). $g_e=-GM/{(\xi +a)}^2$ is the external gravitational potential. Equation (4) represents the thermal plasma energy. The terms $\mp e_{\pm }{V}_{A}d{P}_c/d\xi$ in Equations (5) and (6) are wave excitation by cosmic ray streaming instability. In Equation (8) $\kappa$ is the diffusion coefficient of cosmic rays along the magnetic field line (see Equation (12)), $P_c/\tau$ represents stochastic acceleration or second order Fermi acceleration (see Equation (11)). $\alpha P_w^{\pm }$ in Equations (11) and (12) is the scattering frequency of cosmic rays by the forward and backward waves, $e_{\pm }$ in Equation (10) are ratio of the scattering frequencies. where c represents the speed of light. ${\gamma }_g$ (adiabatic index of thermal plasma) and ${\gamma }_c$ (adiabatic index of cosmic rays) can be regarded as closure parameters. Equation (13) represents Bremsstrahlung cooling with $\left({\delta }_c\right)$ is the cooling strength and in Equation (14) denotes non-linear Landau wave damping with $\left({\delta }^{\pm }\right)$ is the wave damping strength.

The coupled system is also presented in [54,57,58]. The Alfvén flow speed is given by

$$V_A={\displaystyle \frac{B}{\sqrt{{\mu }_0\rho }}}=M_A^{-1}U={\displaystyle \frac{{\psi }_B^{\prime }\sqrt{\rho }U}{{\psi }_m^{\prime }}}=\tilde{\psi }\sqrt{\rho },$$

(15)

where $M_A$ is the Alfvén Mach number. Here $\tilde{\psi }={\psi }_B^{\prime }/{\psi }_m^{\prime }$, ${\psi }_m^{\prime }={\psi }_m/\Delta =\rho U$ and ${\psi }_B^{\prime }={\psi }_B/\sqrt{{\mu }_0}\Delta =B/\sqrt{{\mu }_0}$ (Equations (1) and (2)).

With Equations (4) and (6), the momentum equation (Equation (3)) can be written as:

$$\begin{array}{cc}\hfill & \left(1-M_{\mathrm{eff}}^{-2}\right)U{\displaystyle \frac{dU}{d\xi }}={\displaystyle \frac{a_{\mathrm{eff}}^2}{\Delta }}{\displaystyle \frac{d\Delta }{d\xi }}-g_e-{\displaystyle \frac{1}{\rho }}{\displaystyle \frac{d{P}_c}{d\xi }}\left[{\displaystyle \frac{e_{+}(1+\frac{1}{2}{M}_A^{-1})}{(1+M_A^{-1})}}+{\displaystyle \frac{e_{-}(1-\frac{1}{2}{M}_A^{-1})}{(1-M_A^{-1})}}\right]\hfill \\ \hfill & +{\displaystyle \frac{1}{\rho U}}\left[{\displaystyle \frac{P_c}{2(1-M_A^{-2})\tau }}-({\gamma }_g-1)\left(\Gamma +L_w^{+}+L_w^{-}\right)\right.\left.+{\displaystyle \frac{L_w^{+}}{2(1+M_A^{-1})}}+{\displaystyle \frac{L_w^{-}}{2(1-M_A^{-1})}}\right].\hfill \end{array}$$

(16)

this is called the wind or outflow equation. The effective Mach number can be defined as

$$M_{\mathrm{eff}}={\displaystyle \frac{U}{a_{\mathrm{eff}}}},$$

(17)

here the effective sound speed is

$$a_{\mathrm{eff}}^2=a_g^2+a_w^{+2}{\displaystyle \frac{\left(1+\frac{1}{3}{M}_A^{-1}\right)}{\left(1+M_A^{-1}\right)}}+a_w^{-2}{\displaystyle \frac{\left(1-\frac{1}{3}{M}_A^{-1}\right)}{\left(1-M_A^{-1}\right)}}.$$

(18)

where

$$a_g^2={\displaystyle \frac{{\gamma }_g{P}_g}{\rho }},a_w^{+2}={\displaystyle \frac{3P_w^{+}}{2\rho }},a_w^{-2}={\displaystyle \frac{3P_w^{-}}{2\rho }}.$$

(19)

the energy integral will no longer hold due to the inclusion of the cooling and wave damping effects along the flow. We have total seven equations and we need seven initial conditions to solve the ODEs system numerically. We adopt the same set of equations available in Ko et al. [54]. In the given procedure of solving the boundary condition of the system of coupled ODEs, one of the important criterion used to identify a physical admissible solution is the convergence towards equilibrium [67]. That is, the morphological solutions in terms velocity and pressure profiles must reach uniform asymptotic states at both upstream and downstream region $x\to \pm \infty$ to attain spatial boundary equilibrium at these respective extreme regimes [67]. While the Friedmen and Rotenburg [68] extends the linearized MHD models under time-dependent conditions to analyze the behavior of the hydromagnetic fluid stability in the stationary equilibrium, the four fluid description is based in the framework of Ko [67] under steady state assumptions. Since it is restricted to time independent conditions, it lacks the capability to capture the perturbed behaviors and therefore it cannot offers insgihts about the stability of the plasma system especially at the equilibrium conditions [67,68].

3. Results

Cosmic ray driven galactic outflows has become very interesting under the influence of cooling and wave damping effects altogether along the flow. Several interesting mechanisms include strong and weak coupling of waves to the thermal plasma as well as re-coupled stage has been a new era to investigate. In this paper we are trying to encounter the different ingredients available and their energy exchange mechanisms in coupled/re-coupled stage to the whole on galactic winds under the influence of strong gravitational potential well.

3.1. Cosmic Rays and Waves in the Coupled Stage

Initially, the cosmic rays is strongly tied and coupled with the thermal plasma by the presence of both forward and backward propagating self-excited Alfvén waves. The early strong coupling stage tends to exist around until $10<\xi <15$ (depending on backward wave pressures) as illustrated by the velocity (Figure 1a,c) and pressure profiles (Figure 1b,d). Strong wave damping and cooling heavily influence the dynamics of the galactic wind flow in the four-fluid cosmic ray plasma system. The velocity gradient of the wind outflow remains positive throughout subsonic regime, whereas the effective sound speed falls down respectively in this regime. Upon examining the profiles the cases of the backward wave pressure at the base $P_w^{-}=0.1$ (solid line) and $P_w^{-}=0.2$ (dashed line) depicted by velocity and pressure profiles (Figure 1a,b), the outflow wind speed and sound speed exhibit monotonic increasing and decreasing behavior, respectively. Meanwhile, both forward and backward Alfvén wave pressure decreases dramatically ${\displaystyle \frac{d{P}_w^{\pm }}{d\xi }}<0$ as this can be inferred by the effects of stochastic acceleration ($P_c/\tau$) and wave damping ($L_w^{\pm }$). Hence, forward and backward waves quickly diminish at the early stage (around $\xi \approx 1$). There are also noticeable declines in other pressure components, such as thermal plasma, mainly caused by bremsstrahlung cooling, and cosmic rays, which initially reach a minimum where the backward Alfvén wave pressure vanishes completely. It is interesting to point out that although stochastic acceleration is present, it is still unable to thwart the cosmic-rays deceleration at the early stage, as the backward wave withered quickly. Therefore, the decline of the pressure components provides the support of the increasing and decreasing of the outflow wind speed and effective sound speed respectively. After the backward wave dies out completely $P_w^{-}=0$, and with only forward wave $P_w^{+}$ surviving, the plasma system is reduced to three fluid system. There seems to be the suspicion of the complex and counter interactions between gravitational field, cooling, wave damping and the energy exchange mechanisms that may cause the roughly constant flow in part of the subsonic regime. The only prominent energy exchange mechanism occurring in the three-fluid would be cosmic-ray streaming instability and work done by plasma against pressure gradients. At around $\xi >10$ there is a rise of cosmic-ray pressure, but continuously downfall of thermal and forward wave pressure. At the same time, wind speed gradually begins to rise again due to negative decreasing impact of the thermal and forward wave pressure. An interesting and important point to mention here is that with the increasing behavior of the wind flow speed in the coupled stage, the density continuously dilutes $\left(\rho \right)$ by which it causes to weaken the cooling effects given by $(\Gamma \propto {\displaystyle \frac{{\rho }^2}{\sqrt{p}}})$. Furthermore, with the continuous decreasing of the forward wave pressure leads the diminishing of the damping effect and strengthening the diffusive flux that clearly suggests the cosmic-rays is slowly detached from the plasma and gradually accelerating near the end of the coupled stage. Finally, when the forward wave dies out completely ($\xi \simeq 15$ for solid line and $\xi \simeq 12$ for dashed line) around the transonic point of the wind flow speed that marks the end of the coupled stage. Upon comparisons with the results from larger backward wave pressure at the base $P_w^{-}=0.3$ (solid line) and $P_w^{-}=0.4$ (dashed line) in Figure 1c,d (velocity and pressure profiles), generally the coupled regime exhibit similar features of the subsonic flow speed with the exception of few noticeable drastic changes in the dynamics of the cosmic ray plasma system. Firstly, quasi-steady state of the wind flow disappears with the increasing of the backward wave pressure in the system. There is the major shift of the transonic point towards the base of the gravitational potential well. This highlights clearly that besides in addition of increasing mass flux [58] shown from previous studies, elevating backward wave pressure in the system also play significant role in bringing the transonic point closer towards the gravitational potential well. Naturally, all the major shifts in pressure profile behavior especially the diminishing of both forward and backward wave pressure occurs at early stage. With increasing the backward wave pressure in the system sets the greater strengthening effect on the wave damping mechanism that contributes in driving slight increase of the thermal pressure at the beginning of the coupled stage before it starts to fall dramatically due to the diminishing of wave damping and cooling effects. It is worthwhile to mention during the transition of the plasma system towards three-fluid system (comprises of cosmic-rays, thermal plasma and forward Alfvén wave), the wind outflow model with its increasing subsonic speed draws somewhat similar behavior with previous comparison studies of the three-fluid model by Ko et al. [54,57] based on the constrains set by the certain conditions. The previous three fluid-model with the assumption of negligible diffusion coefficient $\kappa =0$ shows the possibility of the wind model to undergo acceleration in subsonic regime $({\displaystyle \frac{dU}{d\xi }}>0)$, due to the dominance of the gravitational field over the cooling $(g_e>\Gamma )$ and the decreasing behavior of thermal pressure $({\displaystyle \frac{d{P}_{th}}{d\xi }}<0)$ which specifically corresponds with the $a1$ case condition [54]. But in our case the diffusion coefficient is non-zero or rather it varies due to the explicit dependency on the waves, so it might not be necessary that the $a1$ condition in [54] might be the criteria for our wind outflow model as there could be the account of extra factors or terms which are not yet explored.

3.2. Cosmic Rays and Waves in the De-Coupled Stage

During the decline of both forward and backward Alfvén waves due to the influence of energy exchange mechanisms, wave damping, and the cooling effects on the thermal plasma, sets the motion for the cosmic-rays and Alfvén waves to gradually become decoupled from the plasma system. With the Alfvén waves, the diffusion coefficient term $\kappa$ is activated and playing its role along the flow, enabling the diffusive flux mechanism to influence the cosmic-rays transport in the coupled stage. Initially, when the backward waves die out, the wind outflow transits towards three-fluid system where the cosmic-rays is being weakly coupled to thermal plasma by the forward wave. Afterwards when the forward wave die out, marking the absence of both forward and backward waves i.e., $P_w^{\pm }=0$ in the plasma system. Ultimately, the diffusion coefficient becomes infinite due to the model $\kappa \propto {\displaystyle \frac{1}{P_w^{+}+P_w^{-}}}$ (see Equation (12)), resulting in the cosmic rays to become completely detached from thermal plasma, thereby forming the decoupled stage. Naturally, the wind outflow transits towards a decoupled two-fluid system, thereby forming a quasi-thermal outflow characterized by the thermal plasma and cosmic-rays individually [54]. The natural consequence by the vanishing of both waves and the diffusion coefficient $\kappa$ to become infinite sets the cosmic ray pressure to be constant and decoupled from the thermal plasma to form the quasi-thermal state of the wind outflow. In this case, the work done on thermal plasma and the cooling mechanism solely govern the dynamics of the outflow during the decoupled regime. As observed in the velocity and pressure profiles for the cases $P_w^{-}=0.1$ (solid line) and $P_w^{-}=0.2$ (dashed line) represented by Figure 1a,b, the decoupled regime begins near the transonic point ($\xi =15$ for solid line and $\xi =12$ for dashed line) and eventually reaching to steady state system at the far end. The presence of diffusive flux and the work done by plasma against cosmic-rays in the decoupled stage sustains the cosmic rays transport and acceleration as indicated by the rise of cosmic ray pressure (${\displaystyle \frac{d{P}_c}{d\xi }}>0$) until it reaches to uniform state at very far distance from the base. Wind outflow switches to supersonic speed, where the speed gradient manages to maintain its increasing behavior (${\displaystyle \frac{dU}{d\xi }}>0$) that without surprise reduces the density and for the effective sound speed ($a_{\mathrm{eff}}$) still continues to exhibit monotonically decreasing behavior. Consequently, at far distance, the wind flow reaches to constant speed asymptotically, whereas the (${\mathrm{a}}_{\mathrm{eff}}$) disappears completely posing clear indication of the plasma wind outflow carrying with minimal density reaches to steady state condition. Despite the absence of wave damping effect, the thermal pressure still persists to decline (${\displaystyle \frac{d{P}_{th}}{d\xi }}<0$), to the extent that it greatly reduces the temperature gradient (${\displaystyle \frac{dT}{d\xi }}<0$) of the wind outflow and diminishing the cooling mechanisms. Furthermore, the gravitational effects has become weaker on the wind outflow due to its strength diminishing with the ever increasing distance from the base of the gravitational potential well. Without the influence of thermal pressure, cosmic-rays and gravity, the wind simply cruises with constant speed (see Equation (3) with R.H.S close to zero). With the increase of the backward wave pressure profile (i.e., $P_w^{-}=0.2$) in the four-fluid system, it is apparently witnessed that the decoupled stage in the wind outflow model occurs at the earlier stage due to the shift of transonic point closer to the base of the gravitational potential well. However, the dynamics of the wind outflows becomes interesting upon the further increase of the backward wave pressure at the base $P_w^{-}=0.3$ (solid line) and $P_w^{-}=0.4$ in terms of velocity and pressure plots illustrated in Figure 1c and Figure 1d, respectively. The outflow models displays abrupt transitions characterized by non-monotonic variations in both the wind flow and the effective sound speed, with the latter remains present. In this dramatic change of behavior, it is observed that the wind flow velocity reach at its maximum point, which corresponds to the minimum point reached by the effective sound speed point during its downfall. Hence, these points mark the transition from the decoupled stage to the re-coupled stage.

3.3. Cosmic Rays and Waves in the Re-Coupled Stage

As discussed above, the consequence of raising the backward wave pressure at the base in the four-fluid system, i.e., $P_w^{-}=0.3$ (solid line) and $P_w^{-}=0.4$ (dashed line) shown in Figure 1c,d in terms of velocity and pressure profiles, produces non-monotonically behavior of the outflow wind speed and its associated effective sound speed potentially forming the dramatic hump shape region in the velocity profile Figure 1c. The velocity gradient appears to rise during the decoupled stage, even though it experiences deceleration until it hits its peak. Conversely, the effective sound speed $\left({\mathrm{a}}_{\mathrm{eff}}\right)$ diminishes, approaching its lowest point. As shown in Figure 1c, the point at which velocity peak and the effective sound speed reaches minimum, typically occurs for the backward wave pressure profiles $P_w^{-}=0.3$ and $P_w^{-}=0.4$ at around $(\xi =48)$ and $(\xi =32)$ respectively. At this point, the backward wave pressure starts to reappear and begins to rise, as illustrated in Figure 1d, thus commencing the re-coupled stage for the wind outflow. The wind outflow evolves towards three fluid-system where the cosmic rays tend to recouple with the backward Alfvén wave in the plasma system and the backward wave survives throughout the progression of the wind outflow carrying it at far away distances. In relation with the behavior of the pressure components observed in Figure 1d, as we observe the increase of both cosmic-rays and backward Alfvén waves, their effects directly disfavors (favors) the wind outflow speed U (effective sound speed $\left({\mathrm{a}}_{\mathrm{eff}}\right)$) leading to their downfall ${\displaystyle \frac{dU}{d\xi }}<0$ (growth ${\displaystyle \frac{d{a}_{\mathrm{eff}}}{d\xi }}>0$). This ultimately develops the drastic hump behavior pattern for both the wind outflow speed and effective sound speed during the transition between decoupled towards re-coupled stage as observed in the velocity profile in the Figure 1c. Following the severe reduction of both density and temperature of the plasma system, the cooling mechanism becomes ineffective in accelerating the wind outflow during its supersonic regime as it is unable to counteract the opposite effect of the cosmic ray pressure. Hence, the negative impact by the rising of the cosmic ray pressure gradient due to work done by plasma flow against cosmic rays and waves and partly from streaming instability decelerates the wind outflow. As a result, the flow speed and density decreases and increases respectively. Meanwhile both cosmic-rays and backward waves pressure increases as well. So the correlation between velocity and pressure profiles clearly shows the exacerbate effect by cosmic-rays gradient on the wind outflow. This positive feedback drives the abrupt drop in the speed of the wind outflows and the rise of both cosmic-rays and backward wave pressures. The wind outflow enters into the re-coupled stage by the aid of backward waves that couples cosmic-rays back again to thermal plasma through the streaming instability. This again encourages the growth of backward wave pressure by positive cosmic-ray pressure gradient. In this manner, the backward waves starts to grow enough to play crucial role in the confinement of the cosmic-ray transport. As a consequence, we witness the rise of the cosmic-rays as illustrated in Figure 1d, i.e., for $P_w^{-}=0.3$ and $P_w^{-}=0.4$ that occurs at around $(\xi >48)$ and $(\xi >32)$ respectively. The wind outflow in the re-coupled stage again evolves towards three-fluid system. Interestingly, the presence of backward waves reactivates the wave damping mechanism again in the re-coupled regime. But unfortunately, the ongoing decreasing trend of the thermal pressure $({\displaystyle \frac{d{P}_{th}}{d\xi }}<0)$ reduces both thermal pressure and temperature to the extent that it makes almost both wave damping and cooling mechanisms to become ineffective. So taking account of the increasing cosmic-rays and thermal pressure and the presence of cooling mechanism we witness the fall of the supersonic wind outflow speed that clearly depicts the decelerating stage of the wind outflow at the beginning of the re-coupled stage. Once again we draw a comparison of our three-fluid outflow model in the decoupled stage with the respective counterpart of Ko et al. [54], more specifically with their case of supersonic outflows in the $b3$ condition. They have shown for the negligible diffusion coefficient, the wind outflow model display accelerating stage ${\displaystyle \frac{dU}{d\xi }}>0$ in the supersonic regime caused by the dominance of the cooling effect and the decreasing behavior of the thermal pressure ${\displaystyle \frac{d{P}_{th}}{d\xi }}<0$ and temperature gradient ${\displaystyle \frac{dT}{d\xi }}<0$. But the wind flow moving supersonic speed in our model displays completely opposite behavior as its speeds slows down or decelerate ${\displaystyle \frac{dU}{d\xi }}<0$ due to the positive cosmic ray pressure gradient and the ineffectiveness of the cooling mechanisms. Nevertheless both thermal pressure and temperature decline rampantly just similar to the $b3$ condition. Furthermore, the gravitational effects is also negligible in this range. Ultimately for the profiles $P_w^{-}=0.3$ and $P_w^{-}=0.4$ the wind outflow reaches to steady state condition at around $(\xi >60)$ and $(\xi >40)$ respectively. Notice with the comparisons between $P_w^{-}=0.3$ and $P_w^{-}=0.4$, we want to draw attention again that the effects of increasing backward wave pressure at the base shifts the re-coupled stage closer to the base of gravitational potential well and cause sudden hump behavior for both wind flow and effective sound speed.

4. Implications in Galactic Outflows

Now we draw attention towards the applications of our given morphological results in the context of galactic outflows. Generally speaking there are several proposed approaches or mechanisms to deal with the description of the galactic driven winds. Based on emission-driven sources, the outflows falls into these categories, such as thermal and radiation outflows and cosmic-ray driven outflows [54,57,60,65,69]. Thermal and radiation outflows are primarily energy-momentum driven winds originating from supernovae explosions, stellar winds, massive stars radiation effects on dust and charged grains, and accretion of gas by super-massive black holes. Speaking of pure thermal outflows, it usually relies on thermal pressure giving characteristics of high temperature (around ${10}^6$–${10}^7$ K) and low density [65,70]. The thermal pressure plays a prominent role, dominating the gravitational binding force, enabling it to carry the ejected gas at high velocities $\approx 1000$ km/s away from the galactic disk towards the circumgalactic medium [65]. Hence, these features of thermal outflow come in handy while ejecting the winds at highly energetic galactic environments such as active galactic nuclei and star-burst galaxies. But the thermal outflows pose serious limitations when taking account of cooling effects due to rapid losses of thermal energy, leading to the demise of thermal pressure. Henceforth, the thermal outflow offers a lack of support to carry the gas at large distances against the gravitational potential well of the galactic center, which eventually leads to the cessation of winds at a certain altitude before they fall inwards into the galactic disk [54,69]. On the other hand, the cosmic-ray driven outflows are attributed by the additional pressure gradients due to cosmic-rays propagation. Commonly arising from highly energetic astrophysical events such as shockwave produced at supernova events or tidal disruption at supermassive black holes that are of non-thermal origin, cosmic rays are deemed to propel towards extreme energies at around $GeV-TeV$. While carrying high energies, cosmic-rays can inject themselves into the plasma system, where it gets scattered by the formation of self-excited Alfvén waves to be coupled with thermal plasma and hence imparting energy-momentum transfer to the plasma system. Therefore, this develops the overall buoyancy effect by the combined pressure contributions due to thermal plasma, cosmic rays, and magnetic fields fluctuations to overwhelm the inward gravitational force and accumulate enough mass of the gas to be ejected from the galactic disk. With these given characteristics, the cosmic-ray driven with the presence of self-excited Alfvén waves outflows can drive large mass outflow rates and accompany the winds towards extended distances from the galactic center as compared to thermal outflows [48,54,57,58]. So the cosmic-ray driven outflows fit the suitable mechanisms of the galactic winds at normal galaxies i.e Milky Way Galaxy and dwarf galaxies primarily for these main reasons (i) sharing comparable equipartition with thermal gas and magnetic fields (ii) thermal plasma and magnetic fields alone are not strong enough to provide buoyant force and accelerate the gas that aids in carrying winds towards extended distances [1,2,54,60,65,69,71].

The presence of galactic cosmic rays serves crucial factor in shaping the galactic environment, especially its involvement in galactic winds and regulation of star formation rates. In exploring the dynamics of the galactic outflows, a hydrodynamic approach is often employed to investigate the mechanisms that are responsible in driving the winds and their correspondence with the known observational signatures. Bustard et al. [72] employed one-fluid model to study the steady-state wind solutions solely based on pure thermal outflows in the presence of cooling effects. Their key findings highlight the role of radiative losses in reducing the efficiency of wind-driven mass outflows. Moreover, in comparison between subsonic and transonic wind models, the former tend to have greater radiative losses that sharply decrease its temperature and overall efficiency which results the wind to cover at certain limited distances. Interestingly enough for the transonic wind model, the factors involving low mass ejection rates and low temperatures despite the presence radiative losses by cooling effect the conditions still remain favorable for efficient wind-driven expulsion of mass and energy. Breitschwerdt et al. [39,40] were among the pioneer to adopt the hydrodynamic approach that incorporated the cosmic-rays coupling with thermal plasma with the presence of self-excited Alfvén waves, to investigate the impact of cosmic-rays on the galactic wind models. In this treatment of three-fluid system, they explicitly showed that the combined pressures by both cosmic rays, waves and thermal plasma can carry the outflow rate up to $1M_{\odot }$ yr⁻¹. Depending of the formation of the cosmic rays coupling, it is shown that the wind flows are usually launched and driven at far distance about 1 kpc away from the galactic center where below this height the cosmic ray diffusion tends to be more prevalent. Perhaps the recent work by Ko et al. [54] attempts to unlock closer accurate description to the nature of the galactic outflows by adapting the extension of the cosmic-ray plasma hydrodynamic model with the inclusion of the pressure gradients of forward and backward self-excited Alfvén waves while taking account of diffusion, non-linear Landau wave damping and bremsstrahlung cooling mechanisms altogether. Hence, the framework is studied in the form of three-fluid or four-fluid framework that is able to produce myriad of possible allowable existence of wind driven solutions. Besides Bustard et al. [72] models, Ko’s models also extensively generated transonic solutions that either fall into class of two categories namely sub-supersonic and super-subsonic. It is realized among these two winds models, the sub-supersonic are able to carry and drive galactic winds at farther distances away from the galactic center and hence this matches accurately with the observed criteria of outflows.

Our studies of the four-fluid model, is simply a successful extension of Ko et al. [54] that provides comprehensive insights of the mechanism in terms of complexity interaction between the components and the inclusion of the cooling and wave damping effects that collectively effects the dynamics of the galactic winds. We mainly examine the morphological effects of the wind profile when we increase the backward wave pressure at the base of the potential well. First, we have the creation of three stages of the wind flow along the vertical distance above from the galactic center that are labeled as (i) coupled (ii) decoupled and (iii) re-coupled stages. The coupled stage always lies near to the galactic center that serves crucial point for the launch site of the galactic winds from the galactic center. In the case of streaming instability sets the stage for the coupling effect between cosmic-rays and thermal plasma with the presence of self excitation of Alfvén waves enabling the overall outward pressure contribution by these components to overcome the strong gravitational potential well. As a result the overall outward pressure helps to launch and accelerate the winds away from the galactic center. The simulation studies Ruszkowski et al. [61] have demonstrated for the cosmic rays to propagate at super-Alfvénic speed during streaming transport enhances the efficiency of driving the winds. But the cooling mechanism and wave damping have adversarial effect on the energy losses for both thermal plasma and Alfvén waves that overall can impact the wind flow. Recent simulation studies have shown despite the presence of radiative cooling, it still have minimal impact to the cosmic-rays by which itself can manage to carry and accelerate the outflow [73]. Next stage of the wind-flow is the decoupled stage where it lies at mid-far away from galactic center. The starting phase of the decoupled stages occurs near the transonic point in which the wind outflow transcends towards supersonic speeds. As the cooling getting less effective as density and temperature decreases, its effect on increasing flow speed in the supersonic regime is taken over by the opposite effect of the cosmic ray pressure gradient (i.e., work done by the flow against pressure gradients, partly from cosmic ray and partly from waves through streaming instability). The flow speed decreases and the density increases. The pressures of cosmic rays and waves increase accordingly. This exacerbates the effect of cosmic ray pressure gradient on the flow. This seemingly positive feedback drives the abrupt drop in flow speed and the rise of pressures of cosmic rays and backward propagating waves (note that for positive cosmic ray pressure gradient, the streaming instability encourages the growth of backward waves but discourages the forward waves). The flow then goes into the re-coupled stage.

The last stage of the wind flow is the re-coupled stage that lies farthest away from the galactic center.Here the backward waves are revived and remain always present in the outflows. With both Cosmic rays and backward waves pressure rising (see Figure 1d). Surprisingly, the wind outflows seem to decelerate in this farthest region, but nonetheless still continue to travel away from the galactic center, but the thermal plasma is severely weakened and becomes completely diminished as a result of continuous energy losses due to cooling mechanisms and decrease of temperature. So only the cosmic-rays and backward Alfvén waves are left in responsibility to transport the galactic winds at the farthest region towards the halo region or intergalactic medium. Second consequence of increasing backward Alfvén waves pressure in the outflow system brings the transonic point close to the base of the gravitational potential well (i.e., towards the galactic center). So it turns out that besides variation of mass outflow rates highlighted in previous studies [74], backward Alfvén waves also account important factor in the determining the location of the transonic point.

5. Conclusions

The four-fluid system is much more complicated than the three-fluid system. In the presence of both waves the stochastic acceleration cannot be ignored ($1/\tau \ne 0$) and it can impact the galactic outflows. Further, the two waves can be affected by the cosmic ray streaming instability in opposite ways (see Equation (6)). In conclusion, four-fluid outflow keeps an interesting effects with cooling, which has some intrigue features that are totally different from three-fluid (with forward propagating wave) outflows. There is a number of distinct features with variations in cooling and wave damping effects. On the left panel of Figure 1c, after the transition from subsonic flow to supersonic flow, the flow velocity (blue line) continue increases for a distance and then decreases rapidly, forming a hump in the profile and then decreases gradually to a constant value. This non-monotonicity is interesting within multiple fluid system with cooling and wave damping effects and need more studies for scrutiny.

On the right panel of Figure 1d, a rapid rise in cosmic ray pressure (red thick line) and backward-propagating wave pressure (black solid and dotted line) seems to be associated with the rapid drop in flow velocity of the hump. Due to the cooling, the thermal pressure (green thick line) decreases to zero at large distances. The backward-propagating wave survive to large distances because of the non-linear Landau damping become ineffective as thermal pressure approaches zero (see Equation (6)).

Lastly, we would like to indicate that in the four-fluid system, the wave pressures are essentially zero for $\xi$ =10∼30. As a result, in this range the cosmic ray diffusion coefficient is very large and cosmic rays are decoupled from the thermal plasma. Yet, the backward-propagating wave is revived around $\xi =40$ (probably due to cosmic ray streaming instability). Cosmic ray becomes re-coupled to the thermal plasma again. We can visually compare the cases with cooling, wave damping and with different initial backward wave pressures. In the Figure 2, we use $U=2/3$, $P_g=1$, $P_c=1$ and $P_w^{+}=0.5$ at the base of gravitational potential well. Green, blue, magenta, red and cyan denote cases with $P_w^{-}=$ 0, 0.1, 0.2, 0.3 and 0.4, respectively. We need to tune the cosmic ray diffusive flux ($\kappa {\displaystyle \frac{d{E}_c}{d\xi }}$) at the base of the gravitational potential well for every case to get a transonic solution.