Effects of Coulomb collisions on lower hybrid drift waves inside a laboratory reconnection current sheet

Magnetic reconnection converts magnetic energy into plasma thermal and flow energy via topological rearrangements of the magnetic field lines. Energy conversion processes during magnetic reconnection result in many free energy sources for waves and instabilities near the diffusion region, such as strong gradients of the magnetic field and plasma parameters. Among them, the lower hybrid drift wave (LHDW) has been widely observed near the diffusion region in both space (e.g., Refs. 1–7) and laboratory plasmas (e.g., Refs. 8–10). The free energy source of LHDWs is the cross field current.¹¹ The large density gradients near the separatrix can particularly be a free energy source by inducing a perpendicular current via a diamagnetic drift.

LHDWs have been a candidate for generating anomalous resistivity because it can interact differently with magnetized electrons and non-magnetized ions, resulting in momentum exchange between the two species (e.g., Refs. 7–9 and 12–16). For reconnection with a negligible guide field, the fast-growing, short-wavelength (⁠$kρe∼1$⁠; k is the magnitude of the wave vector k, $ρe$ is the electron gyroradius), quasi-electrostatic LHDW (ES-LHDW) is found to be localized at the edge of the current sheet⁸ due to the stabilization by the high plasma beta (β).¹⁷ On the other hand, the long-wavelength (⁠$kρeρi∼1$⁠; $ρi$ is the ion gyroradius), electromagnetic LHDW (EM-LHDW) that propagates obliquely to the magnetic field exists in the electron diffusion region.⁹ However, extensive efforts via numerical particle-in-cell (PIC) simulations^15,16 show that the EM-LHDW does not play an important role in fast reconnection and electron energization near the electron diffusion region during antiparallel reconnection.

Recent observations by the magnetospheric multiscale (MMS) mission show that the ES-LHDW can be generated inside or near the electron diffusion region,^5–7 when there is a sizable guide field. The ES-LHDW can drive electron heating and vortical flows⁶ near the electron diffusion region. Moreover, the ES-LHDW is capable of generating anomalous drag between electrons and ions.⁷ 

Motivated by these observations, Yoo et al.⁷ have developed a local, linear theoretical model that explains the dynamics of both ES- and EM-LHDWs in the presence of a guide field. This model is based on collisionless closures for the electron heat flux with the assumption of a gyrotropic electron pressure tensor. The results from the model agree with the activities of the ES- and EM-LHDWs inside a current sheet at the magnetopause.⁷ 

In laboratory experiments, such as the magnetic reconnection experiment (MRX), the effects of Coulomb collisions on magnetic reconnection and electron heating are not negligible. The classical Spitzer resistivity,¹⁸ for example, can balance the reconnection electric field in the collisional regime and can even account for 10%–20% of that in the collisionless regime.^19,20 This indicates that Coulomb collisions may also affect the dynamics of LHDWs in laboratory plasmas.

These collisional effects on LHDWs have not been considered previously, even though LHDWs in the reconnection current sheet have been extensively studied via theoretical analyses and numerical simulations (e.g., Refs. 11, 14, 21–23). This paper provides the first quantitative study of the effects of Coulomb collisions on LHDWs. Through this model, we can address how the dynamics of LHDWs in laboratory plasmas are different from those in collisionless plasmas and when collisional effects become important. To include the effects from collisions, we have advanced the previous models^7,24 by using closures of the electron heat flux, heat generated by collisions, and resistivity that can be used for plasmas with arbitrary collisionality.^25,26 For a self-consistent modeling of the heat flux and energy conservation, we also have allowed a first-order perturbation of the perpendicular electron temperature (⁠$Te1⊥$⁠), which was set to be zero in a previous model by Yoo et al.⁷ Unlike previous models, the zeroth-order electron temperature anisotropy is not allowed in the current model because the available closures were developed under the assumption of isotropic electron pressure at equilibrium. Except these changes, all other assumptions are the same: we used a kinetic equation for unmagnetized ions, fluid equations for electrons, and a gyrotropic pressure tensor for electrons.

This linear model can be used to quantify the effects of LHDWs on electron heating and reconnection dynamics in weakly collisional plasmas; with measured wave amplitudes and quasi-linear arguments, wave-associated anomalous terms and heat generated by collisions with ions can be directly estimated. It should be noted that the wave-associated heating power cannot be estimated by collisionless models.

In Sec. II, we explain the theoretical model for LHDWs in a local geometry. Then, in Sec. III, we numerically calculate dispersion relations of LHDWs for two cases. The biggest difference in the two cases is the value of electron beta, $βe$⁠. For the low-$βe$ case, which represents conditions near the electron diffusion region during reconnection with a strong guide field, the ES-LHDW is unstable. For the high-$βe$ case, which represents conditions in the same region but with a negligible guide field, the EM-LHDW has positive growth rates. In both cases, collisional effects on LHDWs with typical MRX parameters are not significant (⁠$≲20$ %). Finally, in Sec. IV, we discuss the results and propose future research.

Figure 1 shows the geometry of our local theoretical model for a LHDW inside a current sheet. Here, the subscript 0 indicates equilibrium quantities. We chose the ion rest frame, and electrons have velocity (⁠$ue0$⁠) on the x–z plane. The equilibrium magnetic field is along the z direction and the density gradient direction is along the y direction. In this model, there is neither equilibrium temperature gradient nor ion temperature anisotropy. The equilibrium electron temperature is also assumed to be isotropic, but anisotropy is allowed in the perturbed electron temperature. The wave vector (k) lies on the x–z plane due to our assumption of negligible k_y. Thus, our theoretical model is local and valid only when the wavelength of the LHDW is much smaller than the thickness of the current sheet in the y direction.²⁴ 

To balance the force associated with the pressure (density) gradient, there is an equilibrium electric field along the y direction. By using the ion and electron force balance equations, the equilibrium electric field E₀ can be expressed in terms of other plasma parameters. From the ion force balance along the y direction, we have

where n₀ is the equilibrium density, $Ti0$ is the equilibrium ion temperature, and $ε=(dn0/dy)/n0$ is the inverse of the density gradient scale. From the y component of the electron momentum equation, we have

where $ue0x$ is the x component of the equilibrium electron flow velocity and $Te0$ is the equilibrium electron temperature. Then, the equilibrium electric field is

The inverse of the gradient scale is given by

Note that Eqs. (3) and (4) are the same as those in the collisionless model in Yoo et al.,⁷ because the resistivity term is zero along the y direction.

All perturbed quantities have a normal mode decomposition proportional to $exp [i(k·x−ωt)]$ with the wave vector $k=(k⊥,0,k∥)$⁠. Here, the subscript 1 indicates perturbed quantities. For the dispersion relation, Maxwell's equations without the displacement current term are used,

The displacement current term is ignored because the phase velocity of the wave is much smaller than the speed of light.

Assuming the equilibrium ion distribution function to be locally Maxwellian, the perturbed ion current density (⁠$Ji1$⁠) is given by²⁴ 

where $mi$ is the ion mass, $vti=2Ti0/mi$ is the ion thermal speed, $ζ=ω/kvti$⁠, and $Z(ζ)$ is the plasma dispersion function. This is from a perturbed Vlasov equation for unmagnetized ions. This means that any dynamics slower than the ion cyclotron frequency have been ignored, including collisional effects on ion dynamics. In our regime of interest, the ion collision frequency is smaller than the ion cyclotron frequency. The perturbed ion temperature can be also obtained, which is

The perturbed electron current density $Je1$ is obtained from fluid equations. This is different from the classical formulation of LHDWs, where the kinetic (Vlasov) equation is used for electron dynamics (e.g., Refs. 17, 27, and 28). Since electrons are magnetized, a gyrotropic electron pressure tensor is assumed. In this case, the 3 + 1 fluid model (n, u, $p∥$⁠, and $p⊥$⁠; $p∥$ and $p⊥$ are the parallel and perpendicular pressure, respectively) is appropriate.²⁵ In this fluid model, off diagonal terms of the electron pressure tensor are ignored.

The first-order electron momentum equation is given by

where $Pe1$ is the perturbed electron pressure tensor and $Re1$ is the perturbed resistivity. The perturbed electron density $ne1$ is given by the electron continuity equation, which is

To close the momentum equation, we need closures for $Pe1$ and $Re1$⁠. For $Pe1$⁠, we only need closures for $pe1⊥$ and $pe1||$⁠, since we assume a gyrotropic pressure tensor as mentioned earlier. To obtain $pe1⊥$ and $pe1∥$⁠, we start from the following kinetic equation:

where $fe$ is the electron distribution function and $C(fe)$ is the collision operator. First, multiplying the kinetic equation with $me(vz−uez)2$ and integrating over the velocity space yield

where

Similarly, multiplying the kinetic equation with $me[(vx−uex)2+(vy−uey)2]/2$ and integrating over the velocity space yield

where

Linearizing Eq. (11) yields

By using $pe1||=ne1Te0+n0Te1||$ and Eq. (9), Eq. (19) can be written as

Similarly, linearizing Eq. (15) yields

We now need fluid closures for $qe1||, qe1⊥, Ce1||$⁠, and $Ce1⊥$⁠. First, the 3 + 1 fluid model gives us⁷ 

where $ωce=eB0/me, πe∥=2(pe∥−pe⊥)/3$ and $Te||=pe||/ne$⁠. After linearization, the x component of $qe1∥$ is

where $rte=2Te0/3(Te0+Ti0)$⁠. For $qe⊥$⁠, we derive a closure in  Appendix A, which can be written as

After linearization, the x component of $qe1⊥$ is

For $qe1z∥$ and $qe1z⊥$⁠, we employ a closure for plasmas with arbitrary collisionality, which can be written as²⁵ 

where

Here, $Te1*=Te1+2πe1||/5n0, vte=2Te0/me$ is the electron thermal speed, and $k¯||=k||λc$ is the normalized parallel wave number. The electron collision length is defined as $λc≡vteτee$⁠, and the electron–electron collision time $τee$ is given by

where $ln Λee$ is the Coulomb logarithm for electron–electron collisions and ε₀ is the permittivity of free space. In Eqs. (28) and (29), $K¯AB$ represents a kernel function that is obtained from a 6400 moment solution.²⁵ The kernel function $K¯AB$ has the following form:

where the values of coefficients, such as a, α, and δ in Eq. (31), are given in Table I in Ji and Joseph.²⁵ For a negative $k¯||, K¯AB(k¯||)=K¯AB(−k¯||)$ if α = 0 or α = 2. When α = 1, $K¯AB(k¯||)=−K¯AB(−k¯||)$⁠. These closures are consistent with those of Hammett and Perkins²⁹ in the collisionless limit, and they become consistent with those of Braginskii³⁰ in the collisional limit.

The heat generated by the collision terms $Ce1||$ and $Ce1⊥$ also needs a closure and can be written as

where $Qe$ is the heat generated by collisions and $Se||$ is related to the temperature anisotropy.²⁵ The closure for $Se1||$ is given by²⁵ 

The heat generated by collisions can be written as²⁶ 

where $τei$ is the electron–ion collision time and $uei=ue−ui$ is the relative flow velocity between electrons and ions. Assuming the ion charge status $Zi$ is unity, $τei$ is

where $ln Λei$ is the Coulomb logarithm for electron–ion collisions. Linearizing $Qe$ yields

We also need an expression for the resistivity. Since there is no temperature gradient in the equilibrium quantities, the zeroth-order resistivity $Re0$ can be written as²⁶ 

For $Zi=1$⁠, the two coefficients are²⁶ 

where $r=ωceτee$⁠. There are additional terms in $Re1$ since temperature gradients exist in the first order. The parallel (z) component of $Re1$ is²⁵ 

Equation (41) can be written as

where

The x component of $Re1$ is²⁶ 

where $β⊥$ for $Zi=1$ is given by²⁶ 

Finally, the y component of $Re1$ is given by $Re1×=α×men0uei1y/τei$⁠. Here, the coefficient $α×$ for $Zi=1$ is²⁶ 

With these closures, the first-order momentum equation [Eq. (8)] can be used to obtain the perturbed electron current density $Je1$⁠. Then, the Maxwell equation [Eq. (5)] can be written as

The detailed derivation of each component of tensor D can be found in  Appendix B.

Dispersion relations for the lower hybrid drift waves are obtained from $|D|=0$⁠, where $|D|$ is the determinant of the tensor D; from this equation, the normalized angular frequency Ω is computed numerically for the given k and θ. Required input parameters are B₀, n₀, $Te0, Ti0, ue0z$⁠, and $ue0x$⁠. In addition, the ion mass has to be specified.

Compared to the previous collisionless model in Yoo et al.,⁷ there are two significant changes in the current model: the inclusion of the first-order perturbation of the perpendicular electron temperature (⁠$Te1⊥$⁠) and the use of collisional closures. To understand the effects of each change, we obtain dispersion relations from four different models—(i) the collisionless model in Ref. Yoo et al.,⁷ (ii) a model with collisional closures but without $Te1⊥$⁠, (iii) the current model in the collisionless limit $τee→∞$⁠, and (iv) the current model.

First, we obtain dispersion relations with typical plasma and field parameters near the electron diffusion region of the MRX during reconnection with a guide field; $B0=180$ Gauss, $n0=2×1013$ cm⁻³, $Te0=Ti0=10$ eV, $ue0z=−130$ km/s, and $ue0x=50$ km/s. Here, the ion species is singly ionized helium. Justified by previous measurements in MRX,^19,31 we assume that $Zi=1$⁠. With these parameters, $τeeωce=157, βe$ is 0.25 and $VA$ is 44 km/s. Note that $ue0x$ exceeds $VA$⁠, which is a necessary condition for LHDWs to have large growth rates.

Figure 2 shows dispersion relations from the four models. Left (right) panels are contour plots of the real (imaginary) part of the angular frequency as a function of $kρe$ and θ. Here $ρe=vte/ωce$ is the electron gyroradius. From now on, ω represents the real part of the angular frequency and γ represents the imaginary part. Both ω and γ are normalized to the (angular) lower hybrid frequency, $ωLH$⁠. All four models are qualitatively similar, showing strong growth rates (⁠$γ≲0.6ωLH$⁠) for the ES-LHDW. The ES-LHDW propagates almost perpendicular to $B0$ (⁠$θ∼90°$⁠) with $ω≲ωLH$⁠. The peak growth rate occurs at $kρe∼0.7$ and $θ∼91°$⁠. Here $kρe∼0.7$ corresponds to $λ∼0.6$ cm. These similarities among the four models indicate that the effects of Coulomb collisions on the ES-LHDW are limited for typical MRX parameters. Moreover, inclusion of $Te1⊥$ also has a limited impact on the dispersion.

For a better comparison between the four models, the dispersion relation and growth rate of the ES-LHDW are presented in Fig. 3 for $θ=91°$⁠. It is worth noting that including Coulomb collisions decreases the growth rate γ. This is understandable since collisions decrease the reaction of electrons to the external perturbation, such that they reduce the positive feedback from the plasma. The change in ω is not straightforward but is related to frequency shift due to additional terms of $ue1x$ and $ue1z$⁠. For example, the parallel force balance equation Eq. (B48) has the resistivity $Re1||$⁠, which adds additional terms in $αez$ in Eq. (B50). These additional terms can cause a shift in ω (note that $αez$ has a dependency on ω via $αe$⁠).

It is interesting to see that including $Te1⊥$ in the electron dynamics decreases both ω and γ of the ES-LHDW. Interpreting this trend is complicated, because $Te1⊥$ impacts both the x and z components of the electron momentum equation. For the x component, the first term (⁠$ik⊥n0Te1⊥$⁠) on the right side of Eq. (B55), which is the perturbed perpendicular electron pressure gradient term, directly contains $Te1⊥$⁠. For the parallel momentum balance of Eq. (B48), $Te1⊥$ affects $Te1||$ via $qe1x||$ in Eq. (23). The parallel resistivity [Eq. (42)] also has a term with $Te1⊥$ (⁠$−ik∥n0γez⊥Te1⊥$⁠).

The dispersion relation is calculated after setting $γez⊥=0$ to remove contributions from $Te1⊥$ in the z component of the electron force balance equation. As shown in Fig. 4, this change (green line) decreases ω and increases γ, compared to the reference case with $Te1⊥$ (red line). Changes in ω and γ are not significant.

The change in ω with $Te1⊥$ is caused by the $ik⊥n0Te1⊥$ term in the x component of the electron momentum equation. As shown in Fig. 4(a), without the term (magenta line), ω increases significantly compared to the reference case with $Te1⊥$ (red line). Removing the $ik⊥n0Te1⊥$ term also increases γ for most values of k. Again, these changes are caused by the frequency shift due to the additional term with $ue1x$⁠; from Eqs. (B35) and (B55), the inertial term effectively changes from $imen0(ω−k·ue0)ue1x$ to $imen0(ω−k·ue0−0.5c¯ux⊥k⊥vte)ue1x$⁠.

We have repeated the dispersion calculation for the EM-LHDW that propagates obliquely to $B0$⁠. The plasma and field parameters used for calculations are $B0=30$ Gauss, $n0=2×1013$ cm⁻³, $Te0=Ti0=10$ eV, $ue0z=−50$ km/s, and $ue0x=130$ km/s. Again, the ion species is singly ionized helium and $Zi=1$⁠. With these parameters, $τeeωce=26.2, βe$ is 8.9 and $VA$ is 7.3 km/s. These parameters represent typical MRX values near the electron diffusion region during reconnection with a negligible guide field.

As shown in Fig. 5, dispersion relations from the four models again qualitatively agree with each other; these models expect positive growth rates for the EM-LHDW. Models without $Te1⊥$ have the maximum growth rate around $kρe∼0.6$ and $θ∼55°$⁠, while those with $Te1⊥$ have the maximum growth rate around $kρe∼0.5$ and $θ∼50°$⁠. The wavelength with the largest growth rate is about 4 cm. In is interesting to see that all models expect that the mode has frequency significantly less than $ωLH$ in the ion rest frame. This agrees with measurements in MRX and numerical simulations that show that most of the power of the EM-LHDW exists below $ωLH$⁠.^9,16

For comparison between the four models, ω and γ as a function of k for $θ=55°$ are presented in Fig. 6. Similar to the ES-LHDW case, collisional effects decrease γ regardless of the existence of $Te1⊥$ in the model. This is consistent with the aforementioned explanation; collisions decrease the reaction of electrons to the external perturbation, thereby decreasing the positive feedback. For the EM-LHDW, collisions generally decrease ω especially when $Te1⊥$ is not included in the model (blue lines). Including $Te1⊥$ further decreases both ω and γ for this mode (red lines).

In summary, we have developed a local, linear model of LHDWs that includes effects of Coulomb collisions and $Te1⊥$⁠. This model works best for plasmas with weak collisionality. Without collisions, some assumptions for the 3 + 1 model may not be valid, as the zeroth-order distribution function is not close to a Maxwellian. In addition, in the collisionless plasma, agyrotropy can be developed, while a gyrotropic electron pressure tensor is assumed in this model. For collisional plasmas, we need to consider the zeroth-order electric field along the x and z directions; for the zeroth-order electron force balance, additional components of $E0$ are needed to balance the zeroth-order resistivity $Re0$⁠. If there are too many collisions, we need additional first-order terms (⁠$eE0xne1$ and $eE0zne1$⁠) in the x and z components of the electron momentum equation [Eq. (8)]. From Eq. (38), required equilibrium electric field components are given by $E0z=−α||B0ue0z/ωceτei$ and $E0x=−α⊥B0ue0x/ωceτei$⁠. From Eq. (3), $E0x/E0$ is given by

because $α⊥∼Te0/(Te0+Ti0)∼1$ and $τei∼τee$ for $Zi=1$⁠. This means that $E0x$ is negligible compared to E₀, as long as electrons are fully magnetized (⁠$ωceτee≫1$⁠), which is one of the basic assumptions of this model. From a similar argument, $E0z$ is also negligible unless $|ue0z|≫|ue0x|$⁠. For the two cases presented here, the effects of both $E0x$ and $E0z$ are expected to be minimal since $|ue0z|∼|ue0x|$ and $ωceτee≫1$⁠.

To verify this argument, we have calculated dispersion relations of LHDWs after including two additional terms (⁠$eE0xne1$ and $eE0zne1$⁠) and have found that impacts from these terms are actually negligible. The basic reason for not including additional components of $E0$ in the current model is that including $E0x$ may require an additional electron flow component along the y direction, since there will be a corresponding $E×B$ drift of electrons, while ions are unmagnetized. This means that collisions may impact the dynamics of LHDWs by changing the equilibrium itself. A future work will address this effect in a self-consistent manner. As the main purpose of the current study is to study collisional effects on LHDWs, we minimize other changes for simplicity. The parallel component of the equilibrium electric field $E0z$⁠, on the other hand, can be easily added in the model without creating complexity. Moreover, $E0z$ in the electron diffusion region during reconnection with a strong guide field may significantly exceed the value required to balance the classical resistivity.³² In the future, we will study the possible impacts of $E0z$ on LHDWs with values measured in MRX during guide field reconnection.

With this model, we have calculated two sets of LHDW dispersion relations for typical MRX parameters. The first case uses parameters from the electron diffusion region during reconnection with a significant guide field, while the second one uses those with a negligible guide field. Due to the presence of the guide field, the first case has a low electron beta (⁠$βe=0.25$⁠), such that the ES-LHDW is unstable in that region. For the second case (⁠$βe=8.9$⁠), on the other hand, the ES-LHDW is stabilized by the high beta effect¹⁷ and the EM-LHDW is unstable instead.

It will be interesting to study the critical value of β_e that determines whether the ES- or EM-LHDW is unstable. Initial studies show that the critical value is determined by the value of $ue0x/VA$⁠; for a relatively low (⁠$∼1$⁠) value of $ue0x/VA$ like the first case, $βe$ also has to be low (⁠$≲0.5$⁠) to have the ES-LHDW unstable. For a high value (>10) of $ue0x/VA$⁠, on the other hand, the ES-LHDW exists at the higher $βe∼1$⁠. We plan to conduct a statistical study with data from MMS and/or MRX, which will be compared to the results from the current theoretical model.

Based on the two cases we have studied, collisional effects on LHDWs in typical MRX current sheets are limited. In both cases, including Coulomb collisions in the model decreases the growth rate. However, the difference in γ is relatively small (⁠$≲20$%). This is because the wavelengths of LHDWs (0.5–5 cm) are smaller than the mean free path of electrons (⁠$∼10$ cm) and electrons are fully magnetized (⁠$ωceτee≫1$⁠) for these parameters.

To further investigate how collisions may impact on the dispersion relation, we have artificially varied $τee$ and $τei$⁠. For the ES-LHDW, artificially high collisions significantly affect the dispersion relation and the growth rate, as shown in Figs. 7(a) and 7(b). When the collisions are enhanced by a factor of 5 (red dashed line), the real frequency becomes larger for $kρe>0.2$ than the reference value (blue solid line). There is also a significant decrease in the growth rate for $kρe>0.7$⁠. Changes in less collisional cases, on the other hand (green solid and dashed lines), are minimal. With the reduced collision time (⁠$τee→0.2τee$⁠), the mean free path (⁠$τeevte$⁠) becomes about 2 cm, which corresponds to $kρe∼0.2$⁠. This supports the insertion that collisions have large impacts on modes with a wavelength comparable to the mean free path (⁠$λ∼2πτeevte$⁠).