Indian Institute of Science Education and Research, Pune, India

Corresponding author: P. Subramanian, Indian Institute of Science Education and Research, Sai Trinity Building, Garware Circle, Pashan, Pune, Maharashtra 411021, India. (p.subramanian@iiserpune.ac.in)

[1] The forces acting on solar Coronal Mass Ejections (CMEs) in the interplanetary medium have been evaluated so far in terms of an empirical drag coefficient C_{D} ∼ 1 that quantifies the role of the aerodynamic drag experienced by a typical CME due to its interaction with the ambient solar wind. We use a microphysical prescription for viscosity in the turbulent solar wind to obtain an analytical model for the drag coefficient C_{D}. This is the first physical characterization of the aerodynamic drag experienced by CMEs. We use this physically motivated prescription for C_{D} in a simple, 1D model for CME propagation to obtain velocity profiles and travel times that agree well with observations of deceleration experienced by fast CMEs.

[2] The aerodynamic drag experienced by CMEs as they traverse the interplanetary medium between the Sun and Earth is generally thought to arise due to the coupling of the CMEs to the ambient solar wind. CMEs which start out slow (with respect to the solar wind speed) near the Sun seem to accelerate en route to the Earth, while fast CMEs are decelerated, suggesting that the solar wind strongly mediates CME propagation [Gopalswamy et al., 2000; Manoharan, 2006] in the interplanetary medium. This fact has been invoked in several papers that derive a heuristic aerodynamic drag coefficient for CMEs [e.g., Byrne et al., 2010; Maloney and Gallagher, 2010; Vrsnak et al., 2010, 2012; Cargill, 2004]. In particular, Borgazzi et al. [2009] have used two different drag prescriptions to investigate CME slowdown using a simple 1D hydrodynamical model that lends itself to analytical solutions.

[3] While there has been a fair amount of progress in this direction, a physical understanding of the viscosity mechanism that leads to the drag on CMEs is still lacking. Many authors have focussed on the overall dynamics of the CME, using the drag coefficient C_{D} only as an empirical fitting parameter that remains constant throughout the propagation [e.g., Byrne et al., 2010; Maloney and Gallagher, 2010; Lara et al., 2011].

2. This Work

[4] Our primary focus here is therefore on computing the drag force on the CME using a physical prescription. We compute the viscosity of the ambient solar wind using a prescription relevant to collisionless plasmas. We use this solar wind viscosity prescription to compute the drag on the expanding CME employing approaches that are commonly used in fluid dynamics. We restrict ourselves to a simple 1D, hydrodynamical model in order to focus on the essential physics. We first recapitulate the basics of the drag force experienced by CMEs.

3. Viscous Drag on a CME

[5] We start with the oft-used 1D equation of motion for a CME that experiences only aerodynamic drag [e.g.,Borgazzi et al., 2009]:

mCMEVCMEdVCMEdR=12CDNimpACMEVCMEâˆ’Vsw2,

where m_{CME} is the CME mass, V_{CME} is the CME speed, C_{D}is the all-important dimensionless drag coefficient,N_{i} is the proton number density in the ambient solar wind, m_{p} is the proton mass, A_{CME} ≡ πR_{CME}^{2}is the cross-sectional area of the CME andV_{sw} is the solar wind speed. The term on the right hand side of equation (1) represents the drag force experienced by the CME. It may be noted that this form for the drag force is appropriate only for a solid body moving through a medium at high Reynolds numbers [e.g., Landau and Lifshitz, 1987]. Most authors adopt an empirical drag coefficient C_{D} that remains constant with heliocentric distance. The value for C_{D} is chosen to ensure that the computed velocity profile using an equation like equation (1) agrees with observations [e.g., Lara et al., 2011].

[6] It is well known, however, that the drag coefficient C_{D} is a function of the Reynolds number of the system under consideration. We start with data from Achenbach [1972] who gives the widely used characterization of C_{D} as a function of the Reynolds number for high Reynolds number flow past a solid sphere. This standard characterization can be found in most fluid dynamics texts [e.g., Landau and Lifshitz, 1987]. The drag coefficient C_{D}(Re) is a very slowly increasing function of the Reynolds number Re until Re ≈ 10^{5}, above which it exhibits a sharp drop; this is the so-called drag crisis. Beyond this sharp drop,C_{D} is an increasing function of Re; this spans the supercritical and the transcritical regimes. While these facts are well known, we are not aware of any analytical fits to Achenbach's [1972] data, especially for high Reynolds numbers. We therefore fit Achenbach's [1972] data for Re > 10^{6} to the following functional form:

CDRe=0.1478âˆ’42834Re+9.8Ã—10âˆ’9Re.

[7] In our case, the Reynolds number is a function of the CME velocity V_{CME}, the typical macroscopic lengthscale R_{CME} and the viscosity coefficient according to the usual formula

Reâ‰¡VCMERCMEÎ½=VCMERCMENimpÎ·,

where v (cm^{2} s^{−1}) is the coefficient of kinematic viscosity and η ≡ νN_{i}m_{p}. Our use of the CME radius R_{CME}for a typical macroscopic lengthscale implies that (as for a flux rope CME) the cross-sectional dimensions of the CME are ≈R_{CME}. In what follows, we will develop a physically motivated viscosity prescription for the solar wind and use it to determine the Reynolds number self-consistently. We will use the Reynolds number thus determined to calculateC_{D} and use it to solve the CME equation of motion (equation (1)).

4. Viscosity Prescription for a Collisionless Plasma

[8] It is well known that the solar wind is a collisionless plasma above ∼10 R_{⊙}, where the (Coulomb) collision mean free path is longer than macroscopic scale lengths. However, a fluid description is a generally well accepted one for the solar wind. This implies that the collisionless particles are confined via effective collisions with scattering centers of some sort, which lends validity to a fluid treatment. Clearly, viscosity in the solar wind is not due to interparticle collisions, as is the case in everyday experience.

4.1. Hybrid Viscosity

[9] We follow Subramanian et al. [1996, hereinafter SBK96], who developed a model for collisionless viscosity. They considered a one-dimensional, plane-parallel shear flow, with viscosity being provided by the flux of protons that originate from one of the layers and impinge on the other. Tangled magnetic fields are embedded in the flow, and the average coherence length (i.e., the length for which an average magnetic field line is expected to remain straight) of the “kinks” in the magnetic fields is taken to beλ_{coh}. The protons, whose gyroradii are negligible in comparison to other length scales, are envisaged to slide along the field lines, and change their direction (i.e., get scattered) either when they encounter another proton, or a kink in the tangled magnetic field. Since the protons are collisionless, proton-proton collisions are unlikely, and momentum transfer (and consequently viscosity) is dominated by proton-magnetic kink encounters. The magnetic kinks can also be envisaged as turbulent eddies that act as scattering centers; the only restriction in this scenario is that the scattering centers evolve over a timescale that is slower than that of the travel time of a typical proton, so that they appear stationary to it. Using this scenario,SBK96 arrive at the following simple and physically motivated formula for a “hybrid” coefficient of dynamic viscosity η_{hyb}; one that is neither due to magnetic field stresses alone, nor due to proton-proton (Coulomb) collisions alone:

Î·hybâ‰¡Î½hybNimp=215Î»Î»iiÎ·ffgcmâˆ’1sâˆ’1,

where λ denotes the effective mean free path, λ_{ii}is the (Coulomb) mean free path for proton-proton collisions andη_{ff} is the standard proton viscosity due to Coulomb collisions alone [Spitzer, 1962]. The viscosity η_{hyb} in a collisionless plasma is thus suppressed with respect to the standard Coulomb value η_{ff} by a factor (2/15) λ/λ_{ii}. For the sake of completeness, we reproduce the expressions for λ_{ii} and η_{ff}:

where v_{rms} ≡ (3kT_{i}/m_{p})^{1/2} is the rms thermal velocity of the protons, t_{ii}is the mean interval between proton-proton (Coulomb) collisions,N_{i} is the proton number density in cm^{−3}, T_{i} is the proton temperature in Kelvin and lnΛ is the Coulomb logarithm, which is taken to be equal to 20. Equations (4) and (5) comprise our operational definition for hybrid viscosity, which we will apply to the solar wind. It may be noted that the hybrid viscosity (equation (4)) can be equivalently written as

where v_{turb} is the velocity of the turbulent eddies of lengthscale l_{turb}. Furthermore, if l_{turb} is interpreted as the dissipation lengthscale in a turbulent cascade, and if we adopt the usual expression for energy dissipation rate in Kolmogorov turbulence ϵ ∼ v_{turb}^{3}/l_{turb}, the expression for turbulent viscosity can be rewritten as [Verma, 1996]

Î·turbâ‰ˆNimpÏµlturb41/3.

4.2. Solar Wind Viscosity

[10] We now use equations (4) and (5) to compute the operative viscosity in the ambient solar wind. As mentioned earlier, the viscosity would be determined primarily by collisions between protons and magnetic kinks, and therefore the effective mean free path λ ≈ λ_{coh}. The coherence length of the magnetic field irregularities (λ_{coh}) represents the shortest lengthscale over which the turbulent magnetic field is structured. It is fairly well established that density turbulence in the solar wind follows the Kolmogorov scaling, with an inner (dissipation) scale that is determined by proton cyclotron resonance [Coles and Harmon, 1989]. We assume that the magnetic field irregularities follow the density irregularities [e.g., Spangler, 2002], and are governed by the same inner scale. We therefore take λ_{coh} to be equal to the inner scale of solar wind density turbulence advocated by Coles and Harmon [1989]:

Î»â‰ˆÎ»coh=684Niâˆ’1/2km.

[11] The solar wind proton density N_{i} is assumed to be given by the model of LeBlanc et al. [1998]:

where R is the heliocentric distance in solar radii. We take the proton temperature to be T_{i} = 10^{5} K.

[12] Thus equation (4), with λ given by equation (9) and the density given by equation (10) defines the hybrid viscosity prescription for the ambient solar wind for our purposes. This enables us to obtain the viscosity of the ambient solar wind as a function of heliocentric distance.

[13]Eviatar and Wolf [1968] obtain an estimate of v ≈ 800 km^{2} s^{−1}at the Earth by considering the momentum transfer across the magnetopause due to fluctuations induced by the two-stream cyclotron instability. This is in excellent agreement with the value of 788 km^{2} s^{−1} we obtain for the kinematic viscosity coefficient from the hybrid viscosity model. Pérez-de-Tejada [2005] has derived a rough estimate of v ≈ 1000 km^{2} s^{−1} for the kinematic viscosity of the solar wind near the ionosheath of Venus (heliocentric distance 0.72 AU). The hybrid viscosity model yields a value of 600 km^{2} s^{−1} for the coefficient of dynamic viscosity at 0.72 AU.

5. Results

[14] We use the prescription for the viscosity given in § 4 to determine the Reynolds number (equation (3)), which in turn is used to determine C_{D} (equation (2)). The resulting expression for C_{D} is used in the equation of motion (equation (1)) to solve for the CME speed as a function of heliocentric distance.

[15] A representative result is shown in Figures 1 and 2. In this example, the CME radius R_{CME} is assumed to expand as R_{CME} = KR^{P}, where K is a constant of proportionality, which we determine by assuming that the radius of the CME is 1 R_{⊙} at a heliocentric distance of 2 R_{⊙}. The power law index p is assumed to be equal to 0.78 [Bothmer and Schwenn, 1998]. The solar wind speed V_{sw} is taken to be 375 km/s and the velocity of the CME at R = 5R_{⊙} is taken to be equal to 1048 km/s. This CME initial velocity is representative of a CME on May 17 2008 that was well observed by the STEREO spacecraft from the Sun to the Earth [Wood et al., 2009]. We chose this CME since it is one of the few fast ones that were well observed from the Sun to the Earth by the STEREO spacecraft during the minimum and ascending phases of cycle 24. The CME mass m_{CME} is taken to be 5 × 10^{14} g. This value for the CME mass is representative of CMEs during this part of the solar cycle [see Vourlidas et al., 2010].

[16] The solid line in Figure 1 is the solution for the velocity profile calculated from equation (1). The plus signs represent data for the CME of May 17 2008. In order to get the velocity-distance data for this CME, we started with the distance-time data for this CME [Wood et al., 2009, Figure 6 (top)]. The velocity-distance data points inFigure 1are derived by numerically differentiating the distance-time data. The large undulations in the velocity-distance data shown inFigure 1, especially for distances > 20 R_{⊙}, are due to fluctuations in the distance-time measurements, which are accentuated in the derived velocity. In particular, the significant dip in CME speed after ∼150R_{⊙} is almost certainly an unphysical artefact. It is well known that CMEs achieve an asymptotic speed by the time they reach the Earth, and often as soon as ∼100 R_{⊙} [e.g., Poomvises et al., 2010]. The dip in the CME speed beyond 150 R_{⊙} is probably due to the difficulty in tracking CME in HI data, which are well known to have poor signal to noise ratio at those distances. In view of this, the agreement between the solid line, which is the solution to equation (1)and the velocity-distance data for the May 17 2008 CME is evidently rather good. The Reynolds number for the CME (in units of 10^{8}) and the dimensionless constant C_{D}(which is now computed self-consistently, using the prescription for solar wind viscosity) are shown inFigure 2. The results shown in Figure 2 justify the function we adopt for C_{D} (equation (2)), which is valid only for Reynolds numbers ≫10^{4}. We note that Wood et al. [2009]obtain the velocity for the May 17 2008 not by piecewise numerical differentiation of the distance-time data (as we have done), but by assuming an empirical model that incorporates an initial acceleration phase and a subsequent deceleration phase that is followed by a constant velocity phase.

[17] We have also carried out this exercise for some of the fastest Earth-directed halo CMEs observed so far. Our results are summarized inTable 1. For each event, the parameters supplied to the model (equation (1)) are the solar wind speed V_{SW}, the CME mass m_{CME} and the CME initial velocity V_{i}. The initial velocity of each halo CME is determined from the LASCO CME catalog (http://cdaw.gsfc.nasa.gov/CME_list/). Since these are halo CMEs, their masses are hard to estimate, and we have used reasonable guesses, based on typical CME masses during the appropriate phase in the solar cycle [Vourlidas et al., 2010]. As before, the proton temperature is assumed to be 10^{5} K and the Leblanc et al. [1998] density model (equation (10)) is used for determining the solar wind viscosity and the drag coefficient C_{D}. The quantity V_{ICME}represents the speed of the relevant interplanetary CME detected in-situ by spacecraft near the Earth andTT_{ICME} is the time elapsed between the first detection of the halo CME in the LASCO FOV and the detection of the corresponding ICME at the Earth. The quantity V_{model} represents the predicted ICME velocity at the Earth and TT_{model}denotes the predicted Sun-Earth travel time. It is evident fromTable 1 that the model predictions for the ICME speed at the Earth and the total travel time agree quite well with the observations.

Table 1. A Comparison Between Observations and Model Results for Near-Earth ICME Speeds and Sun-Earth Travel Times for Some Fast Earth-Directed CMEs

Event

V_{SW} (km/s)

m_{CME} (g)

V_{i} (km/s)

V_{ICME} (km/s)

V_{model} (km/s)

TT_{ICME} (hours)

TT_{model} (hours)

20010409

450

5 × 10^{14}

1192

670

696

53.4

53.5

20031028

450

2 × 10^{15}

2459

1400

1350

24.8

25.6

20040726

400

10^{15}

1366

900

894

38.63

37

[18] We have used the first CME in Table 1 (20010409) as a representative event to ascertain the effects of varying some of the parameters on the model predictions. All other quantities remaining fixed, we find that a 10 % decrease (increase) in V_{sw} results in a 7.5 % decrease (increase) in V_{model} and a 7 % increase (decrease) in TT_{model}. Similarly, we find that a 50 % decrease (increase) in m_{CME} leads to a 13 % decrease (increase) in V_{model} and a 15 % increase (decrease) in TT_{model}. On the other hand, a 100 % increase (decrease) in T_{i} results in a 4 % increase (decrease) in V_{model} and a 4.6 % decrease (increase) in TT_{model}. Therefore, the results of our model are most (least) sensitive to the solar wind speed (proton temperature).

6. Summary and Conclusions

6.1. Summary

[19] To summarize, we have considered the motion of a CME under the influence of only a drag force (equation 1). The drag force is proportional to the square of the CME speed (relative to the background solar wind); this form is appropriate for a solid body moving through a background medium at high Reynolds numbers. The proportionality constant involves the important (dimensionless) drag coefficient C_{D}, which has been treated as an empirical fitting parameter in the literature so far.

[20] It is, however, well known that the drag coefficient C_{D} is in fact a function of the CME Reynolds number Re. We use standard data pertaining to the motion of a solid sphere at high Reynolds numbers to characterize C_{D} as a function of Re (equation (2)). It is necessary to know the coefficient of kinematic viscosity v in order to determine Re (equation (3)). We compute the operative viscosity in the (quiescent) collisionless solar wind using a prescription that considers protons colliding with magnetic scattering centers (equations (4) and (5)). We assume that the proton temperature in the solar wind is T_{i} = 10^{5}K, the proton number density is given by equation (10) and that the operative mean free path λ is the inner scale of the spectrum of density irregularities (equation (9)). This gives a simple, physically motivated prescription for the drag coefficient C_{D}, which we use in solving a simplified 1D equation of motion (equation (1)). Using observational estimates for the CME starting speed and reasonable guesses for the ambient solar wind speed and the CME mass, we find that our results for the CME speed profile and the Sun-Earth travel time agree quite well with observations (Figure 1 and Table 1).

6.2. Conclusions

[21] The agreement between our theoretical predictions and observations of CME deceleration is remarkable, especially in light of the several simplifying assumptions we have adopted. It confirms the essential validity of our physical prescription for the dimensionless viscous drag parameter C_{D}, which can be profitably used in semi-analytical treatments [e.g.,Vrsnak et al., 2012] and simulations of CME propagation.

[22] We next mention some caveats that accompany our conclusions. Firstly, this is a rather simplified, 1D hydrodynamic treatment that only addresses the essential physics of the CME-solar wind interaction. There is no attempt to include Lorentz force driving of CMEs; something that is known to be important up to around 30R_{⊙} [e.g., Subramanian and Vourlidas, 2007], at least for CMEs that are only moderately fast. Very fast CMEs, such as the ones considered in Table 1, probably experience Lorentz force driving and acceleration very early in their evolution. Upcoming instruments such as the ADITYA-I coronograph [Singh et al., 2011] could address this issue via high cadence images of the inner solar corona.

[23] Secondly, it is surprising that a drag law of the form of equation (1), which is in fact valid only for solid bodies, works so well. CMEs are often thought of as bubbles, which are technically defined as bodies that deform in response totangential stresses on their surfaces. The drag force for high Reynolds number flow past a bubble is in fact proportional to V_{CME}, and not to V_{CME}^{2} [Landau and Lifshitz, 1987; Merle et al., 2005]. With this form for the drag force, our preliminary results indicate that a typical fast CME slows down only by about 1 % by the time it reaches the Earth. It is therefore worth investigating why CMEs seem to behave like solid bodies as far as their interaction with the ambient solar wind is concerned. In a flux rope CME, the large-scale, ordered magnetic field of the flux rope could provide an explanation for this apparent “solid body” effect. Finally, we note that the drag coefficient derived from ideal MHD simulations [e.g.,Cargill, 2004] can often significantly exceed unity, in contrast to the values obtained in this work.

Acknowledgments

[24] We acknowledge several comments and suggestions from the anonymous referees that have helped improve this paper. PS acknowledges financial support from the RESPOND program of the Indian Space Research Agency. AL acknowledges partial support from DGAPA-UNAM IN112412-3 and CONACyT grants.

[25] The Editor thanks two anonymous reviewers for assisting in the evaluation of this paper.