We have developed a local, linear theoretical model for lower hybrid drift waves that can be used for plasmas in the weakly collisional regime. Two cases with typical plasma and field parameters for the current sheet of the magnetic reconnection experiment have been studied. For a case with a low electron beta (, high guide field case), the quasi-electrostatic lower hybrid drift wave is unstable, while the electromagnetic lower hybrid drift wave has a positive growth rate for a high- case (, low guide field case). For both cases, including the effects of Coulomb collisions reduces the growth rate but collisional impacts on the dispersion and growth rate are limited (%).
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 (; k is the magnitude of the wave vector k, 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 (; 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) simulations15,16 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 models7,24 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 (), 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, . For the low- case, which represents conditions near the electron diffusion region during reconnection with a strong guide field, the ES-LHDW is unstable. For the high- 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 ( %). Finally, in Sec. IV, we discuss the results and propose future research.
II. DERIVATION OF THE DISPERSION RELATION
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 () 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 ky. 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, is the equilibrium ion temperature, and is the inverse of the density gradient scale. From the y component of the electron momentum equation, we have
where is the x component of the equilibrium electron flow velocity and is the equilibrium electron temperature. Then, the equilibrium electric field is
The inverse of the gradient scale is given by
All perturbed quantities have a normal mode decomposition proportional to with the wave vector . 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 () is given by24
where is the ion mass, is the ion thermal speed, , and 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 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, , and ; and 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 is the perturbed electron pressure tensor and is the perturbed resistivity. The perturbed electron density is given by the electron continuity equation, which is
To close the momentum equation, we need closures for and . For , we only need closures for and , since we assume a gyrotropic pressure tensor as mentioned earlier. To obtain and , we start from the following kinetic equation:
where is the electron distribution function and is the collision operator. First, multiplying the kinetic equation with and integrating over the velocity space yield
Similarly, multiplying the kinetic equation with and integrating over the velocity space yield
Linearizing Eq. (11) yields
Similarly, linearizing Eq. (15) yields
We now need fluid closures for , and . First, the 3 + 1 fluid model gives us7
where and . After linearization, the x component of is
where . For , we derive a closure in Appendix A, which can be written as
After linearization, the x component of is
For and , we employ a closure for plasmas with arbitrary collisionality, which can be written as25
Here, is the electron thermal speed, and is the normalized parallel wave number. The electron collision length is defined as , and the electron–electron collision time is given by
where is the Coulomb logarithm for electron–electron collisions and ε0 is the permittivity of free space. In Eqs. (28) and (29), represents a kernel function that is obtained from a 6400 moment solution.25 The kernel function 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 if α = 0 or α = 2. When α = 1, . 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 and also needs a closure and can be written as
The heat generated by collisions can be written as26
where is the electron–ion collision time and is the relative flow velocity between electrons and ions. Assuming the ion charge status is unity, is
where is the Coulomb logarithm for electron–ion collisions. Linearizing yields
We also need an expression for the resistivity. Since there is no temperature gradient in the equilibrium quantities, the zeroth-order resistivity can be written as26
For , the two coefficients are26
where . There are additional terms in since temperature gradients exist in the first order. The parallel (z) component of is25
Equation (41) can be written as
The x component of is26
where for is given by26
Finally, the y component of is given by . Here, the coefficient for is26
The detailed derivation of each component of tensor D can be found in Appendix B.
III. COLLISIONAL EFFECTS ON THE DISPERSION
Dispersion relations for the lower hybrid drift waves are obtained from , where 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, , and . 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 () 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 , (iii) the current model in the collisionless limit , 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; Gauss, cm−3, eV, km/s, and km/s. Here, the ion species is singly ionized helium. Justified by previous measurements in MRX,19,31 we assume that . With these parameters, is 0.25 and is 44 km/s. Note that exceeds , 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 and θ. Here 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, . All four models are qualitatively similar, showing strong growth rates () for the ES-LHDW. The ES-LHDW propagates almost perpendicular to () with . The peak growth rate occurs at and . Here corresponds to 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 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 . 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 and . For example, the parallel force balance equation Eq. (B48) has the resistivity , which adds additional terms in in Eq. (B50). These additional terms can cause a shift in ω (note that has a dependency on ω via ).
It is interesting to see that including in the electron dynamics decreases both ω and γ of the ES-LHDW. Interpreting this trend is complicated, because impacts both the x and z components of the electron momentum equation. For the x component, the first term () on the right side of Eq. (B55), which is the perturbed perpendicular electron pressure gradient term, directly contains . For the parallel momentum balance of Eq. (B48), affects via in Eq. (23). The parallel resistivity [Eq. (42)] also has a term with ().
The dispersion relation is calculated after setting to remove contributions from 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 (red line). Changes in ω and γ are not significant.
The change in ω with is caused by the 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 (red line). Removing the term also increases γ for most values of k. Again, these changes are caused by the frequency shift due to the additional term with ; from Eqs. (B35) and (B55), the inertial term effectively changes from to .
We have repeated the dispersion calculation for the EM-LHDW that propagates obliquely to . The plasma and field parameters used for calculations are Gauss, cm−3, eV, km/s, and km/s. Again, the ion species is singly ionized helium and . With these parameters, is 8.9 and 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 have the maximum growth rate around and , while those with have the maximum growth rate around and . 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 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 .9,16
For comparison between the four models, ω and γ as a function of k for are presented in Fig. 6. Similar to the ES-LHDW case, collisional effects decrease γ regardless of the existence of 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 is not included in the model (blue lines). Including further decreases both ω and γ for this mode (red lines).
IV. SUMMARY AND DISCUSSION
In summary, we have developed a local, linear model of LHDWs that includes effects of Coulomb collisions and . 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 are needed to balance the zeroth-order resistivity . If there are too many collisions, we need additional first-order terms ( and ) in the x and z components of the electron momentum equation [Eq. (8)]. From Eq. (38), required equilibrium electric field components are given by and . From Eq. (3), is given by
because and for . This means that is negligible compared to E0, as long as electrons are fully magnetized (), which is one of the basic assumptions of this model. From a similar argument, is also negligible unless . For the two cases presented here, the effects of both and are expected to be minimal since and .
To verify this argument, we have calculated dispersion relations of LHDWs after including two additional terms ( and ) and have found that impacts from these terms are actually negligible. The basic reason for not including additional components of in the current model is that including may require an additional electron flow component along the y direction, since there will be a corresponding 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 , on the other hand, can be easily added in the model without creating complexity. Moreover, 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 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 (), such that the ES-LHDW is unstable in that region. For the second case (), 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 βe that determines whether the ES- or EM-LHDW is unstable. Initial studies show that the critical value is determined by the value of ; for a relatively low () value of like the first case, also has to be low () to have the ES-LHDW unstable. For a high value (>10) of , on the other hand, the ES-LHDW exists at the higher . 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 (%). This is because the wavelengths of LHDWs (0.5–5 cm) are smaller than the mean free path of electrons ( cm) and electrons are fully magnetized () for these parameters.
To further investigate how collisions may impact on the dispersion relation, we have artificially varied and . 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 than the reference value (blue solid line). There is also a significant decrease in the growth rate for . Changes in less collisional cases, on the other hand (green solid and dashed lines), are minimal. With the reduced collision time (), the mean free path () becomes about 2 cm, which corresponds to . This supports the insertion that collisions have large impacts on modes with a wavelength comparable to the mean free path ().
For the case of the EM-LHDW, the effects from collisions become significant when collisions are enhanced by a factor of 5 or more ( and ). As denoted by the red line in Fig. 7(c), the overall shape of the dispersion relation changes noticeably, when is reduced to . The mean free path with is about 2 cm (the same electron temperature and density as the first case), and the change starts around . When reduces even further to (red dashed line), the deviation from the reference line starts around . For both cases, there are also significant reductions in γ, as shown in Fig. 7(d) especially for .
This means that parameters for the two cases studied here are actually in the weakly collisional regime and that the dynamics of LHDWs are susceptible to collisional effects only when collisions are strong. For example, if the base electron temperature for both cases is 3 eV, the dispersion relation from this collisional model will be vastly different from that of the collisionless model.
Including in the model has limited impacts on the dispersion; it generally decreases the frequency and growth rate of LHDWs, but changes in ω and γ are less than 20% for both cases. These changes mostly come from the additional pressure gradient term () in the electron momentum equation along the x direction. This limited impact is related to the existence of Lorentz force terms along the perpendicular direction;7 because of these terms, the electron force balance is less sensitive to the pressure gradient term along the perpendicular direction.
It should be noted that the current theoretical model ignores the global structure of the current sheet by assuming that there is no wave propagation along the density gradient direction (y direction in Fig. 1). To address the effects from the global current sheet structure, an eigenmode analysis21,33 or numerical simulations22,23 will have to be carried out, which will be one of our future works. In MRX, where the current sheet is actually broader (; is electron skin depth), this local approximation is generally valid, as the length scale along the y direction is larger than the wavelength of LHDWs.
This model assumes that there is no equilibrium temperature gradient across the current sheet. In MRX, electrons are locally heated in the current sheet.20,34 However, inside the current sheet the temperature gradient is rather small, compared to that of density. Therefore, the effects of the temperature gradient are expected to be negligible.24
This study will provide a theoretical framework for quantifying anomalous terms and heating associated with LHDWs in MRX. With the solved dispersion relation, we can express every fluctuating quantity in terms of a measurable quantity. For example, the first-order density perturbation [Eq. (B81)] can be expressed in terms of the fluctuation in the reconnection electric field () that can be measured with a probe.8,35 Then, the wave-associated anomalous drag term 36 can be estimated by measuring . Here, the assumption is that the linear relation holds, such that we can use . Furthermore, this model can provide direct estimates of wave-associated heating in Eq. (35) via the same quasi-linear argument. This estimate cannot be done with other collisionless models. In the future, we will establish quasi-linear calculations and conduct measurements of LHDWs in MRX to find out how LHDWs affect the electron and reconnection dynamics.
This work was supported by DOE Contract No. DE-AC0209CH11466, NASA Grant Nos. NNH20ZDA001N and 80HQTR21T0060, NNSFC Contract No. 11975163, the Priority Academic Program Development of Jiangsu Higher Education Institutions (PAPD), and a DOE Grant No. DE-FG02–04ER54746. Digital data used are available in the DataSpace of Princeton University (http://arks.princeton.edu/ark:/88435/dsp01x920g025r).
Conflict of Interest
The authors have no conflicts to disclose.
The data that support the findings of this study are openly available in the DataSpace of Princeton University http://arks.princeton.edu/ark:/88435/dsp01b2773z812, Ref. 38.
APPENDIX A: DERIVATION OF THE HEAT FLUX CLOSURE
From the kinetic equation in the coordinates (V is the fluid velocity),
For the fluid equation, we need to obtain the closure,
Note that can be obtained from
We adopt the closure (transport) ordering and the linear response theory, linear in thermodynamic drives, i.e., , and .
We take the moments of the kinetic equation:
We should decompose into orthogonal polynomials (see Ji and Held37) for the consistent truncation in the expansion of a distribution function.
In terms of orthogonal basis
Hereafter → will be used to drop b terms, which will be nullified by the operation,
For the term
The final equation becomes up to
Since we are interested in up to , we consider only the CGL viscosity, which is
One can rewrite equations in terms of and using
APPENDIX B: DERIVATION OF TENSOR D
where six dimensionless parameters are defined as
Here, is the first-order relative flow velocity along the z direction.
where six dimensionless parameters are given by
The additional ion terms and can be expressed as
The z component of Eq. (8) is
The x component of Eq. (8) is
where , and are
Here, two dimensionless parameters are given by
Similarly, the y component of Eq. (8) is
where , and are
Similarly, is given by
Then, can be written as
Now, we are ready for computing the dispersion relation. Equation (5) is
By multiplying by ( is the ion skin depth; is ion plasma frequency), the above equation can be written as
where and .
From Eq. (6), each component of is
where , and . Since there is no y component in is simply
In terms of dimensionless parameters, , and can be written as
and is the Alfvén speed.
where three dimensionless parameters are given by
Here, two additional parameters and are defined as
where three dimensionless parameters are given by
Two additional parameters and are
The last ion term is .
Equations (B88)–(B90) can be written as
Each component of the tensor D is
APPENDIX C: COMPARISON WITH CLASSICAL MODEL
Since the current model has been established independently, benchmarking with the classical model is desirable. Here, we used the well-known model by Davidson et al.17 For this benchmarking, we set both and to be zero as in the classical model.
As shown in Fig. 8, the results from both collisional (blue line) and collisionless (red line) models do not agree with results from the classical model (black line). In particular, our models expect an almost linear dispersion relation, but ω increases slowly for small in the classical model. Another interesting difference is that the peak growth rate occurs around in our models, while it is around in the classical model. This discrepancy is not due to the choice of our heat flux closures; there is not much difference between our two models, which shows the insensitivity of the dispersion to . Moreover, the dispersion relation is independent of when . We also have confirmed that this discrepancy is not due to the inclusion of the perturbed ion current density, which is ignored in the classical model.
We note that the basic set of equations used in the classical model by Davidson et al.17 is different. The biggest difference is that Poisson's equation is used in the classical model, while we used Faraday's induction law. To understand the cause of this discrepancy, we have developed another model to calculate the dispersion relation. In this model, we follow the basic equations of the classical model, while using our results for the perturbed density and current density.
In our geometry, the first-order equations in Davidson et al.17 can be written as
where , which is from the displacement current. This contribution is ignored, since the phase velocity of LHDWs is much smaller than the speed of light (). We have confirmed that the dispersion relation is insensitive to the inclusion of .
For , and , we use the results from our models. The perturbed ion density is given by24
For the perturbed electron density, we will use one from the collisionless model for simplicity, as there is not much difference between two models. We also assume that . With and can be expressed as7
The y component of the perturbed ion current is24
The y component of the perturbed electron current is7
where is the ion Debye Length. The dispersion relation can be obtained by setting .
The dispersion relation from this simplified model (green line) agrees with the classical model, as shown in Fig. 8(a). This means that the discrepancy is due to the use of Poisson's equation, where the Faraday induction term is ignored. With the parameters for the ES-LHDW, is about 0.25, which means that perturbed magnetic field due to the perturbed plasma current may not be negligible. This argument is supported by observations in laboratory and space,7,10 where magnetic field fluctuations exist when there are strong electric field fluctuations associated with ES-LHDW.
It is interesting to see that the growth rate from the simplified model is considerably lower than that from the classical model, as shown in Fig. 8(b). This difference is likely related to the lack of a rigorous modeling of the heat flux in this simplified model. Although the magnitude is different, both models show that the peak growth rate is around .
This comparison shows that the use of electron fluid equations is acceptable for dynamics of LHDWs. It should be also noted that only our models include full electromagnetic effects, since the induction term is included. These effects are important when β is not negligible.