Magnetized High Velocity Clouds in the Galactic Halo: A New Distance Constraint

In this section, we will restrict our discussion to simulations L200x4, H200x4, and L200x12, all of which correspond to a cloud initially moving at 200 km s⁻¹, but with different initial density and initial position. This is because we found no significant differences with distance between the clouds initially moving at 200 and 300 km s⁻¹.⁵

Figure 4 shows the position of the cloud's CoM along its orbit (${z}_{{\rm{CM}}}$) as a function of time. The dashed line corresponds to motion at a constant velocity of 200 km s⁻¹ toward the plane, and is included for reference. Clearly, the clouds move at essentially constant velocity along a large fraction of their orbit in all cases. Low density clouds are eventually hampered by hydrodynamic and magnetic drag, which become significant at z ∼ 15 kpc, efficiently decelerating the cloud to the point that it becomes comoving with the wind, i.e., stationary in z, at z ≈ 10 kpc. Clouds falling further away from the center at x₀ = 12 kpc are able to travel slightly larger distances due to the weaker drag resulting from a lower halo density at larger radii. High-density clouds move nearly unimpeded with a velocity close to its initial value all the way to the smallest allowed distance of z = 1.5 kpc, potentially reaching the disk. Note that, in all cases, the motion of the cloud is (highly) supersonic, with Mach numbers in the range of ∼2–12.⁶

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The fundamental difference between HD and ideal MHD is the presence in the latter formalism of the induction equation,

$$\begin{eqnarray}&&\displaystyle \frac{\partial {\boldsymbol{B}}}{\partial t}={\boldsymbol{\nabla }}\times ({\boldsymbol{v}}\times {\boldsymbol{B}}),\end{eqnarray} \tag{ 5 }$$

which describes the evolution of a magnetic field embedded in a fluid in motion. In essence, the induction equation is a statement of the fact that a magnetic field moving with a fluid, no matter how small and regardless of the details of the motion, will eventually be amplified, potentially by factors of several orders of magnitude. In numerical experiments like ours, the clouds move through a weakly magnetized medium and "sweep up" the ambient field along their orbit, compressing and amplifying it along the direction of motion at their leading edge and in the transverse direction in their wake (see also e.g., Dursi & Pfrommer 2008). Motivated by the fact that enhanced fields have been observationally associated with some of the Galactic HVCs (see Section 1), in the following, we perform a detailed analysis of the evolution of the magnetic field in our simulations, especially paying attention to its behavior around the cloud.

The importance of the magnetodynamic effects relative to hydrodynamical effects can be measured by the ratio of gas pressure to magnetic pressure β = 8πP/B². In our simulations, β ≈ 2 × 10³ everywhere initially, implying that the magnetic field is dynamically irrelevant. Note that our assumption of an isobaric halo (P fixed) implies $\beta \propto {B}^{-2}\propto {(1+{[(z-{z}_{a})/{z}_{b}]}^{2})}^{2}$ in the halo. Thus, for χ = 2500 and at x₀ = 4 kpc, where the constant of proportionality is $8\pi {{PB}}_{0}^{-2}\approx 0.1$, β < 1 for z < 8 kpc, i.e., close to the plane. In general, the value of β anywhere in the volume remains well above unity throughout the majority of the evolution of the clouds, regardless of their initial density or velocity, and thus hydrodynamic effects dominate overall. Nevertheless, magnetodynamic effects rapidly gain importance with time, as indicated by the evolution of β, which reaches β ≈ 0.4 in simulation H200x4 and β ≈ 0.07 in simulation L200x4 for all clouds near the Galactic plane, at the tenth percentile level over the whole volume. The relevance of magnetodynamic effects is attested as well by the ratio of ram pressure to magnetic pressure, $8\pi {\rho }_{h}{v}^{2}/{B}^{2}$ (ρ_h being the mass density of the halo at the leading edge of the cloud). This decreases by several orders of magnitude from an initial value of order 10⁴ to ∼10 for $| B{| }_{90}$ by the end of the simulation H200x4 (and vanishes by the end of the low density cloud simulations as they become comoving with the wind).

The evolution of the magnetic field strength in the volume, $| B|$, as a function of the cloud's position along its orbit is shown in Figure 5. For ease of discussion, in what follows, we characterize the magnetic field strength $| B|$ using essentially two quantities: (1) the maximum value of $| B|$ in the simulation at a given ${z}_{{\rm{CM}}}$ (i.e., time) denoted by $| B{| }_{\max }$ and (2) the ninetieth percentile of the distribution of magnetic field strength across the volume denoted by $| B{| }_{90}$. The latter is calculated from the distribution, which results after removing all cells within one standard deviation of the local halo field. This approach removes outliers and makes $| B{| }_{90}$ a robust statistic. In contrast, $| B{| }_{\max }$ may be subject to strong local and temporal fluctuations, but is still useful as an absolute upper limit to the magnetic field strength at any given time.

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In Figure 5, solid (dashed) lines correspond to $| B{| }_{90}$ ($| B{| }_{\max }$). For low density clouds, $| B{| }_{90}$ keeps increasing throughout their journey, while the growth of $| B{| }_{\max }$ levels off as the cloud slows down in the wind frame near the Galactic plane and then decays slightly. In contrast, high-density clouds experience an ongoing field growth in both $| B{| }_{90}$ and $| B{| }_{\max }$ all the way to the disk–halo interface.

In order to get a sense of the spatial distribution of the amplified magnetic field and to make a connection with observations, we show in Figure 6 the evolution of $| B|$ on a slice through the yz plane at x = 4 kpc in simulation H200x4. Interestingly, there are two regimes where the field becomes amplified: (1) at the leading edge of the cloud, where the field is amplified rather quickly and continues to grow at all times and (2) behind the cloud, where an enhanced field of comparable magnitude develops along a planar, coherent double tail. The magnetic field there moves in opposite directions along the tail components thus creating a current sheet, where the magnetic field annihilates, a feature typical of MHD flow around a sphere (see Romanelli et al. 2014; Banda-Barragán et al. 2016). At t ≈ 100 Myr, the magnetic tail loses coherence and becomes turbulent, but it becomes strongly collimated again closer to the Galactic plane as a result of the increased magnetic field there (see Section 3.3). The amplified field at the leading edge of the cloud completely dominates over the tail field for t ≳ 150 Myr.

The relative strengths can be appreciated more quantitatively in Figure 7. Here we show the evolution of the ratio between the maximum value of $| B|$ on the leading edge and the $| B{| }_{\max }$ in the tail, defined as ${z}_{\mathrm{front}}\lt {z}_{{\rm{CM}}}$ and ${z}_{\mathrm{tail}}\gt {z}_{{\rm{CM}}}$, respectively. Generally, the tail field dominates at early times, while the field at the leading edge dominates at late times. Thus, the front-to-tail field ratio provides some history on the cloud's past interaction. This could be useful in future high-resolution observations that would be able to separately measure the front and tail fields associated with an HVC.

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Higher density clouds moving further away from the Galactic center develop stronger tail fields that dominate over the leading amplified field across larger distances. In all cases, the tail field is rapidly amplified at early times only to drop off as the tail becomes elongated and loses its coherence, eventually being overtaken in strength by the field at the leading edge. The leading edge field grows steadily along most of the cloud's orbit, with a slight decrease at late times for low density clouds. This is a consequence of the stronger drag operating on these clouds close the plane, which efficiently decelerates the clouds and leads to a decrease in the amount of field lines being swept up.

Guided by the few available measurements of magnetic fields associated with HVCs so far (McClure-Griffiths et al. 2010; Hill et al. 2013), we now consider to be relevant only simulations in which the field (1) has been amplified beyond the local halo field and (2) is comparable to the observed fields of order 1 μG. The former is motivated by the fact that the field around the cloud will always be at least equal to the local value. In order to quantify the importance of the field amplification, we consider the ratio of the amplified field to the local field quantified by the parameter $\alpha =| {\text{}}B| /| {\text{}}B{| }_{\mathrm{halo}}$. In general, α is always highest in the cloud's tail, as material stripped from the cloud, which carries the magnetic field may be present out to large distances from the Galactic plane where the halo field is much weaker. Therefore, we focus on the more interesting case of amplification in front of the cloud. We define α_max as the amplification factor of $| B{| }_{\max ,\mathrm{front}}$ at $z\lt {z}_{{\rm{CM}}}$ and α₉₀ as the amplification of $| B{| }_{90,\mathrm{front}}$ at $z\lt {z}_{{\rm{CM}}}$. Here the subscript "front" refers to $z\lt {z}_{{\rm{CM}}}$, i.e., ahead of the cloud's motion. The evolution of these quantities is presented in Figure 8. Note that this is not generally the same as the maximum and 90th percentile amplification in front of the cloud. We choose this definition because we are only interested in amplification that actually leads to strong fields. Low density HVCs experience a more significant field amplification compared to higher density HVCs, moving at roughly the same speed. However, high-density clouds carry an amplified field at their leading edge for larger distances, i.e., closer to the Galactic plane, because drag does not affect their motion as significantly as it affects lower density clouds. In either case, the amplified field at a cloud's leading edge reaches values that are a factor of at least a few higher than the local field at the 90th percentile level (and at least an order of magnitude higher if the maximum amplification is considered) at intermediate distances.

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In an attempt to condense all of our results so far regarding the field amplification, we show in Figure 9 the relation between the ratio of the field at the cloud's leading edge relative to the tail (see Figure 7) and α (see Figure 8). In addition, we split the data points into three categories depending on the distance of the cloud's CoM relative to the Galactic plane. As can be seen, the low density clouds trace a path through this plane where they move from a tail dominated phase with maximum amplification of α_max ∼ 10 to a front dominated phase with α_max reaching ∼40. They then move back down this path and end in a front dominated but relatively low amplification phase. For x₀ = 12, slightly higher amplifications are reached during the intermediate phase. The high-density clouds trace out a very different path, skipping the intermediate phase of high α. Instead, they move mostly vertically in this plane and end up being highly front dominated. In any case, although the greatest amplification occurs at intermediate distances, only relatively close to the plane does the amplified field reach values on the order of μG.

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Thus, taking our results at face value, we conclude that the observed field strengths of ${B}_{| | }\gtrsim 5\,\mu {\rm{G}}$ associated with HVCs likely indicate that the clouds must be close to the disk, i.e., z ≲ 10 kpc. It is worth noting that there is little difference in the results between clouds moving at different velocities and having different initial densities, with the exception of the fact that low density clouds hardly reach field values close to 1 μG (ignoring the peak in $| B{| }_{90}$ at the end of simulation L200x4, which is caused by a numerical artifact). This comes about because—as previously discussed—they are unable to reach z ≲ 10 kpc, while the high-density clouds travel all the way to our minimum allowed distance of z = 1.5 kpc, where the halo field is much stronger.

The question of whether a gas cloud subjected to (magneto)hydrodynamic interactions has "survived" from a given initial state has no well defined answer, as it is not trivial to arrive at a robust definition of "survival." A common approach is to quantify the evolution of the cloud's mixing with the ambient medium during its evolution. The intuition behind this is that the more a cloud mixes with the ambient medium, the more severe the ablation it has experienced. A cloud's mixing can be measured in different ways, for example, in terms of its density dispersion relative to its average density over a given volume volume (e.g., McCourt et al. 2015). Here, we introduce a different measure of a cloud's ablation, namely, its half-mass radius, i.e., the radius that encloses half of the mass of the initial cloud material at any given time. The idea is that the larger this radius, the more dispersed, and therefore the more "destroyed," a cloud is. We define the half-mass radius as the radius of an infinitely long cylinder with a symmetry axis along one of the coordinates centered on the cloud CoM, which contains half of the cloud's initial mass; we denote it by ${r}_{50,a}$ ($a\in \{x,y,z\}$). Note that measuring ${r}_{50,a}$ along different orthogonal axes is useful to separately assess the tail elongation and the transverse expansion. We define a second quantity, ${r}_{50,\mathrm{sph}}$, as the radius of a sphere centered on the cloud CoM encompassing half of the cloud's initial mass. Note that ${r}_{50,a}$ provides a more direct link to what can actually be measured. Indeed, radio observations can only provide spatial information on the plane of the sky, and are limited to kinematic information along the line of sight.

We use ${r}_{50,\mathrm{sph}}$ to estimate the evolution of the cloud–halo density contrast χ relative to the initial density contrast (see Figure 10). Low density clouds experience an initial shock due to the supersonic wind. This compression in turn leads to a slight increase in the cloud's density relative to the background. Such an effect is also present in high-density clouds, but it is weaker due to the lower Mach number of the shock. In all cases, the density contrast declines monotonically throughout the majority of the cloud's journey toward the disk as the cloud is ablated. At t ≈ τ_cc, where the "cloud crushing time," ${{\tau }}_{\mathrm{cc}}=2{{r}}_{{\rm{c}}}\sqrt{{\boldsymbol{\chi }}}/{{v}}_{\mathrm{wind}}$, is based on the initial density contrast and is roughly the time it takes for the shock to cross the cloud (Jones et al. 1996), clouds experience significant disruption. This happens earlier for low density clouds, at ∼100 Myr. For the high-density clouds τ_cc is higher by a factor of $\sqrt{5}$, so roughly 220 Myr. Cloud crushing times for the v_wind = 300 km s⁻¹ simulations can be found by dividing by 3/2. Note that the cloud crushing time is often defined with respect to the cloud radius rather than diameter in which case they will be a factor of two smaller.

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The above holds true for clouds of similar density moving through media with different densities. Indeed, making use of our newly defined metric, ${r}_{50,a}$, we find that clouds traveling through a less dense medium (L200x12) experience less ablation with respect to clouds in a high-density environment (L200x4; see Figure 11). This is consistent with intuition, thus validating the use of ${r}_{50,a}$ as a quantitative measure of the survivability of HVCs moving through a magnetized halo. When the density of the medium is high enough (i.e., n_h ∼ 10⁻³ cm⁻³ at ${z}_{0}\mbox{--}{z}_{{\rm{CM}}}\approx 30\,\mathrm{kpc}$), the cloud experiences significant ablation, as indicated by the dramatic increase in its half-mass radius. High-density clouds show a different evolution altogether. Indeed, regardless of the increasing density along their orbit, high-density HVCs remain remarkably compact all along their journey. This is consistent with the fact that the cloud crushing timescale is longer for high-density clouds compared to lower density clouds because of their greater density contrasts. There is no significant difference in the half-mass radii along transverse axes, as the half-mass radii are dominated by the elongation along the tail in these cases. But there is a difference in the actual elongation along x and y at large ${z}_{{\rm{CM}}}-{z}_{0}$ with the dispersion of cloud material along x being about 20% higher than along y. This is a well known effect that occurs because the magnetic field inhibits RT instabilities along y and z and so the cloud expands more in the x direction (Gregori et al. 1999). Our results suggest that for high-density clouds, despite their elongated head–tail morphology (Putman et al. 2011), much of the cloud's mass remains in the core. This, in turn, suggests that it is in fact not at all a poor approximation to estimate a cloud's (distance dependent) mass from its angular size alone.

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As the reader may recall, we mentioned in the Introduction that previous studies are somewhat inconclusive with respect to the question of whether magnetic fields prevent or enhance the destruction of HVCs. The most straightforward way we can address this is to compare simulations that include magnetic fields to simulations without magnetic fields, but which are otherwise identical in their initial conditions. Such a comparison can be realized in a quantitative, objective way, e.g., by calculating the evolution of ${r}_{50,a}$ in the purely HD case relative to the corresponding MHD case. We perform such a comparison for two runs with different initial cloud densities, n_c = 0.1 cm⁻³ and n_c = 0.5 cm⁻³, but fixed x₀ = 4 kpc and v = 200 km s⁻¹. The result of this exercise is shown in Figure 12.

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The first thing that becomes apparent is that clouds moving through a magnetized medium do tend to remain compact for a longer time, and more so if their density is high. In addition to breaking up earlier, low density clouds moving in a non-magnetic halo do so in a quite asymmetric fashion, being significantly more dispersed than their magnetized counterparts in the direction perpendicular to their orbit (and the ambient magnetic field). The behavior is clearly shown in the density projections shown in Figure 13.

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Low density clouds are essentially destroyed well before they get close to the Galactic plane, while high-density clouds remain compact all the way to z ≈ 1.5 kpc (at which point, we stop the simulation). Low density clouds get destroyed despite the presence of an ambient magnetic field, though less severely than in its absence. High-density clouds display a more compact morphology, and a stronger collimated tail, when moving through a magnetized medium. As noted before, their tail becomes coherent close to the Galactic plane as a result of the increased magnetic field in the halo there. In contrast, non-magnetized clouds display a diffuse, turbulent tail at all times. We did not show a comparison between HD and MHD simulations for the density contrast, but they differ in a way that is expected from the current comparison. Throughout the majority of the simulations the density contrasts are similar but they become lower for the HD simulations at later stages with a roughly 50% difference relative to their MHD counterparts by the end.

A caveat to this analysis is that, in our simulations, the magnetic field is completely in the transverse direction, which maximizes its amplification. However, while a magnetic field parallel to the cloud's direction of motion will lead to significantly less amplification, oblique fields angled at 45° have been shown to lead to almost as much amplification as in the transverse case (Banda-Barragán et al. 2016). We therefore conclude that, in fact, magnetic fields delay the break-up of a cloud as it travels through the halo, but perhaps not enough to guarantee that it reaches the disk.

Magnetic field constraints are currently only available for two HVCs. However, with future surveys such as POSSUM (Gaensler et al. 2010), we expect that magnetic field strengths will become available for a much larger amount. There are two primary methods for deriving constraints on magnetic fields from observations: rotation measures and Zeeman splitting.

The rotation measure is the measure of the change of polarization angle of emission observed at wavelength λ. It is proportional to the line-of-sight integral of the product of the magnetic field parallel to the line of sight and the free electron density. Lower limits on the magnetic field strength along the line of sight have been derived from rotation measures in two HVCs: the Smith Cloud (Hill et al. 2013) and an HVC in the Leading Arm (HVC 287.5+22.5+240, McClure-Griffiths et al. 2010). These studies assume a distance to the HVC; however, this is only used to estimate the size of the cloud and rough limits can still be derived without assuming a distance. For the Smith Cloud, ${B}_{| | }\gtrsim 8\,\mu {\rm{G}}$ is obtained. The location of the Smith Cloud is known to quite good accuracy, z = −2.9 ± 0.3 kpc and R = 7.6 ± 0.9 kpc (Lockman et al. 2008), which is in agreement with the conclusion from our model that it must be within ∼10 kpc of the Galactic disk. For HVC 287.5+22.5+240, ${B}_{| | }\gtrsim 6\,\mu {\rm{G}}$ is obtained. Neither the distance to this HVC nor the distance to the Leading Arm II complex that it is a part of are known. Nonetheless, it is assumed to be closer than the Large Magellanic Cloud, which has a well constrained Galactocentric distance of d ≈ 50 kpc, with a vertical component z = 28 kpc (Pietrzyński et al. 2013). Thus, the upper limit on its associated field is in agreement with our results.

Zeeman splitting causes a spectral line to split into multiple lines in the presence of a magnetic field. This effect can be used to estimate the magnetic field strength along the line of sight, but it requires higher signal-to-noise ratios than the rotation measure method. As a result, no robust constraints on HVC magnetic fields have yet been derived using this method. An early promising candidate in the literature was HVC 132+23-212 (Kazes et al. 1991), but this was later shown to be the result of an instrumental artifact (Verschuur 1995). In short, of the two HVCs with available magnetic field constraints, one is within a few kiloparsecs of the Galactic plane in agreement with our model for its relatively high field strength of at least several μG. The other one has no known distance but must be at z < 28 kpc and is thus consistent with our model, which predicts that this HVC should be close to the disk.

The underlying assumption for this work is that the measured magnetic field corresponds to the ambient field that has been swept up and amplified at the cloud's leading edge. But this interpretation may not apply in general. Indeed, in some cases, such as the magnetic field recently measured in the Magellanic Bridge (${B}_{| | }=0.3\pm 0.3\,\mu {\rm{G}}$; Kaczmarek et al. 2017), it is more likely to represent the magnetic field that has been pulled from the Magellanic clouds. In cases where distances are already available through a more reliable method, our method can be used to examine the origin of the cloud's observationally derived magnetic field by assessing whether the field is consistent with being swept up halo field or if it must have another source.

Figure 7 is especially relevant for observations because it provides a prediction of a relative measure of the variation of the magnetic field across the cloud and so is independent of instrumental artifacts and of a detailed knowledge of the ambient field.

In order to assess whether our standard resolution of 16 cells per cloud radius, ${ \mathcal R }\equiv {r}_{c}/{\rm{\Delta }}x=16$, is sufficient to resolve the amplification of the magnetic field around the cloud, we perform a crude convergence test. More specifically, we compare the evolution of $| B{| }_{90}$ and $| B{| }_{\max }$ in runs with different spatial resolutions. To this end, we rerun simulation L200x4 with 1.5 and 2 times the default resolution, i.e., ${ \mathcal R }=24$ and ${ \mathcal R }=32$, respectively. A quantitative comparison is achieved by computing the ratio of a given quantity at our highest resolution ${ \mathcal R }=32$, say $| B{| }_{90}$ (denoted by $| B{| }_{90,{ \mathcal R }32}$), with the corresponding quantity at a different resolution, $| B{| }_{90,{ \mathcal R }}$. We show such a comparison in Figure 14. Clearly, $| B{| }_{90,{ \mathcal R }32}$ is only slightly higher than $| B{| }_{90,{ \mathcal R }24}$, and the latter is only slightly higher than $| B{| }_{90,{ \mathcal R }16}$ at all times, except perhaps around t ≈ 150 Myr. This indicates that $| B{| }_{90}$ is converging, though it has not yet fully converged at ${ \mathcal R }=32$. $| B{| }_{\max }$ does not show a convergence as good as $| B{| }_{90}$, though this is not surprising because it is a one-pixel statistic, and is thus subject to large fluctuations, in contrast to $| B{| }_{90}$, which is robust. Nonetheless, $| B{| }_{\max ,{ \mathcal R }16}$ and $| B{| }_{\max ,{ \mathcal R }24}$ remain within about a factor of two at all times. We conclude that a standard resolution of ${ \mathcal R }=16$ is sufficient to carry out experiments like ours.

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It is worth mentioning that Dursi & Pfrommer (2008) found that, in order to resolve the magnetic field at the cloud's leading edge in their simulations, at least ${ \mathcal R }=32$ was necessary. Likewise, Goldsmith & Pittard (2016) found that the cloud mass and velocity was reasonably converged at ${ \mathcal R }=32$ as well. But care must be taken when comparing this type of simulations at different resolutions. Resolutions are usually stated in terms of the cloud radius; however, definitions of this differ because cloud density profiles vary between studies. The density profile is typically smooth. It might consist of an inner region of essentially constant density surrounded by a transition layer, as in our simulations, or it might decline immediately outside r = 0. For instance, Dursi & Pfrommer (2008) and Banda-Barragán et al. (2016) both use profiles with wide transition layers and include these in the cloud radius (in fact our profile is similar to the one used in Banda-Barragán et al. (2016) except that they set the steepness to s = 10). If the radius is taken to follow our definition n(r_c) ≈ n(0)/2, then the radius and thus resolution in these studies should be halved. Goldsmith & Pittard (2016) also included the entire transition region in their definition of the cloud radius; however, this transition region was narrower than in the previously mentioned studies. When the differences in the density profiles and definitions of cloud radius are taken into account, our result that the magnetic field amplification is reasonably converged at a resolution of ${ \mathcal R }=16$ is consistent with previous results.

There is a degeneracy in the magnetic field strength between the radius in the Galactic plane, R (denoted by x in our simulations), and the distance from the Galactic plane, z. However, comparing the amplification of the halo field in the cloud for our x₀ = 4 kpc and x₀ = 12 kpc simulations (see Figure 8) shows that it is roughly equal. If we assume that this holds for all R, then we can generate a full three-dimensional model of the cloud magnetic field strengths in the galaxy and upper limits on R and z can be estimated from the point where the line of sight crosses the contour of the upper limit of the magnetic field strength. At this stage, however, we are only concerned with constraining distances to within a factor of several mainly to establish whether a given HVC is relatively "near" or "far."

The amplification of the magnetic field in front of the cloud depends on the angle of the field lines with respect to the cloud's trajectory. The reference orbit we have adopted in all of our simulations—i.e., perpendicular to the Galactic plane—corresponds to an extreme case where the halo magnetic field is everywhere (roughly) perpendicular to the cloud's motion, which in turn yields the maximum possible amplification. However, an oblique field at an angle of 45° in fact leads to an amplification similar to our maximal case, albeit for clouds moving through a uniform medium (Banda-Barragán et al. 2016).

As mentioned in Section 2, we ignore the radial variation of the magnetic field and density of the halo. The density, strength, and direction of the magnetic field change significantly across the 8 kpc² xy plane of our simulations. However, only the variations across the cloud is important for our purposes. These are relatively small over the 0.5 kpc initial radius of the cloud. However, the clouds do increase in size during the simulations (see Section 3.3), so, to check that our approximation is valid throughout the simulations, we ran L200x4-rv, H200x4-rv, and L200x12-rv in which the radial variation is included. We found no significant differences between these three simulations and their radially invariant counterparts.

Our simulations include many simplifications. A significant omission is not including the gravity of the Galaxy. The simplest way to do this would be to assume some dark matter density profile and then calculate the gravitational acceleration of the cloud's CoM from the static potential of the sum of the dark matter and halo gas density profiles (more components could be included for increased accuracy, e.g., McMillan 2017). However, this would complicate the analysis considerably. The acceleration would pull the cloud horizontally in the x direction as well as vertically, which would lead to the clouds having different paths depending on x₀. It would also mean that our approximation that the magnetic field strength and density only depends on z would no longer be appropriate. With a gravitational potential included and only taking the velocity into account, the low density clouds would presumably be able to reach the disk rather than being stopped by drag at z ≈ 10 kpc. However, based on our results, they would probably be destroyed before reaching the disk.

We use the ideal MHD approximation to evolve the magnetic field in the simulations. In doing so, we assume that the gas is sufficiently ionized to be described as a single fluid with negligible ambipolar diffusion and diamagnetic current terms (see Pandey & Wardle 2008). The halo gas in the simulations has temperatures of T ≳ 10⁴ K at all times and can therefore be assumed to be highly ionized. The halo temperature would drop below 10⁴ K for simulation L200x4 at the smallest allowed distance from the plane of z = 1.5 kpc; however, the cloud is stopped by drag before reaching parts of the halo at these temperatures. The mass of ionized gas in HVCs is comparable to the H i mass (Putman et al. 2012) and surrounding ionized gas is observed in, e.g., the Smith Cloud (Hill et al. 2009) and Complex C (Fox et al. 2004; Collins et al. 2007). Thus we would expect the ionization fraction of an HVC to be high in the outskirts and low in the core. In our simulations, we are concerned with the magnetic field amplification, which occurs in front of the cloud and in the tail behind the cloud and the magnetic field remains weak compared to the halo field in the inner parts of the cloud (see Figure 6). Thus our assumption of the high ionization fraction is reasonable in the regions of interest in our simulations.

Hall drift is an effect that, for a fully ionized gas, is caused by the difference in inertia between ions and electrons. The length scale at which Hall drift becomes significant is ${L}_{H}=\tfrac{{v}_{A}}{{\omega }_{H}}$, where ${v}_{A}=B/\sqrt{4\pi \rho }$ is the Alfvén velocity and ω_H is the Hall frequency (Pandey & Wardle 2008). For fully ionized low-metallicity gas L_H ≈ 1.8 × 10⁻⁵ cm^−1/2 g^1/2 × ρ^−1/2 (B cancels out as ω_H ∝ B). For the densities present in our simulations, this is much less than the smallest length scale Δx ∼ 10 pc that is resolved in our highest resolution simulation. The further assumption of ideal MHD is that the resistivity is negligible. The validity of this assumption can be assessed through the magnetic Reynolds number R_m = vL/η, where v and L are typical velocity and length scales and η is the resistivity that approximately depends on the temperature through η ∼ T^−3/2 (Spitzer & Härm 1953). Even at the smallest length scale we can resolve in the highest resolution simulation and the lowest temperatures present (the initial temperature in the cloud) R_m ≫ 1 for the velocities considered in our simulations (including when the velocity is chosen to be the Alfvén velocity and R_m becomes the Lundquist number) indicating that resistivity is relatively unimportant. Our simulations do, however, have numerical resistivity as do all grid-based MHD simulations analogous to numerical viscosity (see, e.g., Fromang & Papaloizou 2007). Non-ideal MHD effects can still be important for the field topology through magnetic reconnection; however, in this paper, we only consider the amplitude of the magnetic amplification around the cloud and the overall effect of the magnetic field on cloud survival.

Finally, although we are aware that radiative cooling may be important for simulations like ours (see, e.g., Mellema et al. 2002; Cooper et al. 2009; Scannapieco & Brüggen 2015), we have chosen to ignore this process for now. Including an advanced treatment of cooling in three-dimensional simulations with reasonable resolution—using, e.g., MAPPINGS (Sutherland & Dopita 1993) or CLOUDY (Ferland et al. 2013)—is computationally costly, but possible. Doing so renders simulations like ours more realistic, but it also introduces a high level of complexity, which may obscure other effects. Thus, for the sake of clarity, we have run our simulations adiabatically, deferring the more challenging task to include cooling for future work. In addition to cooling, other physical effects that may be relevant when modeling cloud–wind interactions include: thermal conduction (Armillotta et al. 2017; Brüggen & Scannapieco 2016), turbulence (e.g Pittard & Parkin 2016), and fractal density structure (e.g., Bland-Hawthorn et al. 2007; Schneider & Robertson 2017). Taking all of these effects into account in MHD simulations is computationally expensive, often limiting the geometry to two-dimensional simulations, which may be even less realistic than full 3D simulations ignoring these processes.