Gas Sloshing Regulates and Records the Evolution of the Fornax Cluster

We include all the existing XMM-Newton observations within 1° of NGC 1399 as listed in Table 1. Basic data reductions including screening and background modeling were performed using the Science Analysis System (SAS) version xmmsas-20160201. All the ODF files were processed using emchain and epchain to ensure the latest calibrations. Soft flares were filtered from MOS data and pn data using the XMM-ESAS tools mos-filter and pn-filter respectively (Snowden & Kuntz 2013). The effective exposure time of each detector is listed in Table 1. The combined exposure time is ≈100 ks at the center and ≳200 ks at the outskirts. We only include events files with FLAG = 0 and PATTERN <= 12 for MOS data and with FLAG = 0 and PATTERN <= 4 for pn data. Point sources detected by edetect_chain and confirmed by eye were excluded from further analysis. Out-of-time pn events were removed from both spectral and imaging analyses. Modeling of the astrophysical background (AXB) and the Non-X-ray background (NXB) is presented in the Appendix.

Spectral fit of regions of interest was restricted to the 0.5–7.0 keV energy band. These spectra were fit to two sets of models simultaneously. The first model set takes the form of phabs × (pow_CXB+apec_MW+vapec_ICM)+apec_LB. Parameters of the AXB models were fixed to the values listed in Table 3, which were determined with offset pointings. The thermal vapec_ICM model is for the cluster emission. The abundances of O, Ne, Mg, Si, S, Fe, and Ni were allowed to vary freely; all other elements were tied to Fe. The second model set is to characterize the NXB components (Table 3) and their spectra were not folded through the Auxiliary Response Files (ARF). Parameters of the NXB models were fixed to the best fits determined for each observation.

We created images in the 0.5–2.0 keV energy band. Individual detector images were created using the tasks mos-spectra and pn-spectra. Point sources detected by the cheese routine were removed. We used the XMM-ESAS tasks comb and adapt_900 to create a background-subtracted, vignetting-corrected EPIC mosaic image, binned by a factor of 2 and adaptively smoothed with a minimum of 50 counts per bin. The resulting image is shown in Figure 1 (left).

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We analyzed a combination of 250 ks Chandra observations centered on NGC 1399 (Obs-ID: 319, 4172, 9530, 14527, 14529, and 16639). We refer interested readers to Paper 1 for details of the observations, data preparation, and the deprojected spectral analysis. All the data were reduced using CIAO 4.9 and CALDB 4.6.9 following standard procedures. We produced the blank-sky background-subtracted, exposure-corrected, and point source removed image, which covers the entire weak cool core (r < 25 kpc; t_cool < 7.7 Gyr), as presented in Figure 2 (top left). In order to enhance detailed structures in the X-ray surface brightness, we divide this 0.5–2.0 keV image by its best-fit double-β model. The resulting residual image is shown in Figure 2 (top right). Readout artifacts were subtracted in both imaging and spectral analyses. Spectral fits were performed with XSPEC 12.7 and C-statistics; the solar abundance standard of Asplund et al. (2006) was adopted. We use the thermal emission model vapec to model the hot gas component and a power-law model with an index of 1.6, pow_1.6, to describe the unresolved low-mass X-ray binaries (LMXB; Irwin et al. 2003). The deprojected temperature is below 1 keV at the cluster center and rises to 1.5 keV beyond 10 kpc.

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Within its strong cool core (r < 4.5 kpc; t_cool < 1 Gyr), the contribution to the X-ray emissions from the diffuse stellar emission and unresolved LMXB may become progressively more important. We calibrate their contribution to the X-ray surface brightness based on the K-band surface brightness profile using the Two Micron All Sky Survey (2MASS; Skrutskie et al. 2006) archived image (see Su et al. 2017a), which amounts to about 1% of the total X-ray luminosity. We then subtracted the diffuse stellar emission and unresolved LMXB emission from the 0.5–2.0 keV image as shown in Figure 3 (top left).

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With Chandra observations, we performed a two-dimensional spectroscopic analysis using the Weighted Voronoi Tesselation (WVT) binning (Diehl & Statler 2006) based on the Voroni binning algorithm presented in Cappellari & Copin (2003). We generated a WVT binning image containing 137 regions for the Chandra image in the 0.5–2.0 keV band (Figure 2 top left). Each bin has an S/N of 80. We applied a model containing two-temperature components to probe its gas metallicity distribution, otherwise "Fe-bias" would be caused by fitting a single thermal model to multi-phase gas (Buote 2000). The two-temperature model takes the form of phabs×(vapec+vapec+pow_1.6). The metallicities of the two vapec components were linked to each other. The two-temperature fit cannot be well constrained for all regions and we find it necessary to fix one temperature at 1.5 keV. The other temperature and the normalizations are allowed to vary independently. The resulting Fe abundance map is shown in Figure 2 (bottom left). The southwest side of NGC 1399 is more metal-enriched than the northeast side and the metal distribution traces the spiral morphology of the sloshing front.

To probe the thermal structure of the hot gas within its strong cool core (r < 4.5 kpc), we produced a binned image containing 126 regions for the image in the 0.5–2.0 keV band (Figure 3, top left). Each bin has an S/N of 36. The spectra were fit with the model phabs ×(apec+pow_1.6). The abundance was fixed to the solar abundance, which is approximately the average metal abundance at the cluster center. The resulting temperature map is presented in Figure 3 (bottom left).

Sloshing cold fronts are identified by eye in images and confirmed by abrupt changes in surface brightness and temperature. The XMM-Newton mosaic image as shown in Figure 1 (left) reveals several edges in the X-ray surface brightness in Fornax: the outermost one is at 200 kpc (30') to the east (F1, red), the second outermost one at 30 kpc (5') to the southwest (F2, magenta), and an inner one at about 10 kpc (2') to the northeast (F3, blue). These fronts stand out in the matching residual image as shown in Figure 1 (right), obtained by dividing the XMM-Newton image with its azimuthal average. Together, they form a spiral pattern wrapping around the bright cluster core. The deep Chandra observation covers F3 and the extended emission to the southwest (Figure 2). Chandra reveals another possible front at ≲3 kpc (05) to the west (F4, black) as shown in Figure 3 (top left). We present the surface brightness profiles over a radial range of 1–250 kpc in the northeast and southwest directions (Figure 4). All four edges can be identified spanning two orders of magnitude in radius. These features are suggestive of gas sloshing triggered by perturbations of minor mergers or off-axis mergers on a large scale.

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We present the temperature profiles of the east and southwest directions crossing the outer fronts in Figure 5. The two edges, F1 and F2, are associated with cooler gas relative to the gas at the same radii but on the opposite side of the cluster. The northeast front at r = 10.6 kpc (116''), F3, is the most evident front (Figure 2, top left). We apply a Gaussian Gradient Magnitude (GGM) filter to highlight sharp edges in the Chandra X-ray image, which determines the magnitude of surface brightness gradients using Gaussian derivatives (Sanders et al. 2016a; Walker et al. 2016). Brighter regions correspond to sharp features in surface brightness. The resulting GGM image is shown in Figure 2 (bottom right), obtained by combining images on scales of 2σ, 4σ, 8σ, 16σ, and 32σ. F3 stands out as a sharp edge at r = 10 kpc to the northeast on the GGM image. We compare the Chandra X-ray surface brightnesses derived in annular sectors with a radial bin width of 2'' from the cluster center to the northeast (133°–159°, marked in the red sector in Figure 2, top right) and to the southwest (290°–390°, marked in the black sector in Figure 2, top right). As shown in Figure 6, the surface brightness profile of the southwest sector declines more smoothly than that of the northeast sector. We fit a broken power-law density model to the northeast profile and we obtain a break at 116 ± 1'' (10.6 kpc) relative to the cluster center, which corresponds to a gas density jump of 2.1 ± 0.2. We convolve this power-law density model with a Gaussian component and obtain a best-fit width of σ = 5'' (450 pc). The smoothed model does not provide a better fit with a F-test probability of 0.11. We extracted spectra from seven concentric annular sectors across this northeast edge (marked in the red annular sectors in Figure 2, top right). The spectra were fit to a single-temperature model. For comparison, we performed the same analysis for the southwest direction (marked in the black annular sectors in Figure 2, top right). The resulting temperature profiles are presented in Figure 6. The gas on the brighter side of the northeast edge is cooler than that on the fainter side, suggesting that this edge is a cold front. The Fe abundance is higher within the edge than that outside as shown in Figure 2 (bottom left). Hot, diffuse, and relatively pristine cluster gas from the larger radii may have been brought into contact with the cool, dense, and enriched gas by sloshing. The distribution of the gas to the southwest over the same radial range displays a relatively uniform thermal and chemical distribution.

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The X-ray emitting structure within the strong cool core of Fornax, tracing the pure hot gas distribution, is shown in Figure 3 (top left). We note a surface brightness discontinuity to the west with a best-fit edge at r = 2.9 kpc (Figure 7-top), while the surface brightness profile to the east is relatively smooth. We derive the temperature profile across this west edge as shown in Figure 7 (bottom). It rises abruptly from ∼1 keV to ∼1.5 keV over a radial range of 1–2 kpc, manifesting the presence of a cold front, consistent with the temperature map (Figure 3-bottom-left). In contrast, the temperature profile over the same radial range to the east varies from ∼1.1 keV to ∼1.2 keV.

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We produce a GGM image to probe substructures in the innermost 5 kpc by combining GGM images on scales of 1σ, 2σ, 4σ, 8σ, and 16σ. As shown in Figure 3 (top right), a subkiloparsec region of enhanced surface brightness is visible just outside the cold front (F4) to the west (blue box). Two 3–5 kpc long bright filaments can be identified to the east at a similar radius (Figure 3 green and magenta boxes). We perform spectral analysis for these structures to determine their natures. We apply local background spectra extracted from neighboring regions as shown in Figure 3 (top left; cyan dashed box for S1; cyan dashed circle for S2 and S3). The spectra were fit to the model phabs×(apec+pow_1.6). Their metal abundance was fixed at the solar abundance. The best-fit temperatures of S2 and S3 are ${1.26}_{-0.18}^{+0.19}\,\mathrm{keV}$ and ${0.82}_{-0.18}^{+0.19}\,\mathrm{keV}$, respectively (Figure 7-bottom). Their ambient ICM has a temperature of ∼1.3 keV. To calculate their densities, we assume these structures are cylinders in 3D with a volume V = l · π(w/2)², which is a common approximation for elongated substructures at cluster centers (e.g., David et al. 2017). We derive the pressure of the ambient ICM using the deprojected density profile of NGC 1399 determined in Paper 1. S3 is in pressure equilibrium with the ambient ICM. S2 would be over-pressurized unless its temperature is near its lower limit of ∼1 keV. The temperature of S1 cannot be constrained. Assuming S1 is also in pressure equilibrium, its temperature would be ∼1 keV. The low temperature (cooler than the ambient ICM) of these subkiloparsec features meets our expectation for KHI eddies growing at cold fronts (e.g., NGC 1404, Su et al. 2017a). Their properties and locations are also consistent with being low-entropy filaments induced by thermal instability (e.g., NGC 5044, David et al. 2014, 2017).

We identify four edges in the X-ray surface brightness at radii of 200 kpc, 30 kpc (Figures 1 and 5), 10 kpc (Figures 2 and 6), and 3 kpc (Figures 3 and 7), all distributed along the SW–NE direction (Figures 1 and 4). The brighter sides of these edges comprise cooler gas. We infer that they are sloshing cold fronts induced by minor or offset mergers. Gas motion around the cluster center approximates the oscillating flow in a statically stable environment. We calculate the Brunt–Väsälä frequency (buoyancy frequency; Cox 1980) at each radius, r (Balbus & Soker 1990):

$$\begin{eqnarray}&&{\omega }_{\mathrm{BV}}={{\rm{\Omega }}}_{{\rm{K}}}\sqrt{\displaystyle \frac{1}{\gamma }\displaystyle \frac{d\mathrm{ln}K}{d\mathrm{ln}r}},\end{eqnarray} \tag{ 1 }$$

where K = kT/n^2/3 is the gas entropy, ${{\rm{\Omega }}}_{{\rm{K}}}=\sqrt{\tfrac{{GM}}{{r}^{3}}}$ is the Keplerian frequency, and γ = 5/3. The density and temperature profiles of the cluster center, measured with Chandra observations, are used to calculate K and to derive the hydrostatic mass profile, M(r); for comparison, we also derive the total mass using the stellar velocity dispersion (Paper 1). Gas properties outside the weak cool core are measured with XMM-Newton; the best-fit Navarro–Frenk–White (NFW) dark matter profile of Navarro et al. (1997) is used to calculate Ω_K. The resulting sloshing period of P = 2π/ω_BV is shown in Figure 8, which increases with radius. The sloshing timescale (P or P/2) provides an order-of-magnitude estimate of the age of the sloshing front (Churazov et al. 2003). To verify this approximation, we compare these timescales to the evolution of the sloshing simulated for a Virgo-like cluster (Figure 7 in Roediger et al. 2011). The ages of the simulated sloshing fronts at radii of 100 kpc (NW), 40 kpc (SE), and 20 kpc (NW) are 1.7 Gyr, 0.8 Gyr, and 0.6 Gyr, respectively (fronts within 10 kpc are not resolved). These scales are comparable to the sloshing fronts observed in Fornax in this work at 200 kpc (E), 30 kpc (SW), and 10 kpc (NE) with their ages (P/2 ∼ P) corresponding to 2.5–5.0 Gyr, 0.4–0.7 Gyr, 0.1–0.2 Gyr, respectively, as shown in Figure 8. Our approach thus leads to reasonable approximations. In future work, we plan to refine our estimates with simulations specifically tailored to Fornax (A. Sheardown et al. 2017, in preparation).

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Simulations reveal that the infall of one subcluster is capable of inducing multiple sloshing cold fronts as the displaced gas peak oscillates back and forth around the dark matter center of the main cluster (ZuHone et al. 2010, 2011). That F1, F2, and F3 fall on the same spiral structure is typical for sloshing cold fronts produced in a single merger event, as seen in simulations (e.g., Roediger et al. 2011). NGC 1404 and NGC 1387 are the second and third brightest member galaxies in Fornax (Figure 1). NGC 1387 is more than 5× fainter in X-rays than NGC 1404 and its radial velocity relative to NGC 1399 is ≳100 km s⁻¹, 5× smaller than that of NGC 1404. Therefore, the infall of NGC 1404 may have initiated the gas sloshing. Adopting a timescale of P/2 ∼ P = 2.5 ∼ 5 Gyr for the outermost front at r = 200 kpc (Figure 8), we estimate that the infall of NGC 1404, took place more than 2 Gyr ago, longer than the typical crossing time of galaxy clusters. This is consistent with our previous study of NGC 1404 approaching the inner region of Fornax for the second time (Su et al. 2017c). Fornax is one of the few clusters where the perturber that has initiated the sloshing can be identified, providing an ideal clean case for tailored simulations.

The two-dimensional Fe abundance distribution of NGC 1399 (Figure 2, bottom left) shows that the hot gas with enhanced metallicity (>1.0 Z_⊙) reaches radii of approximately 5–10 kpc to the north and the east and it is more extended to the south and the west, reaching beyond 12 kpc. The extended distribution of enriched ICM may reflect an AGN outburst or gas sloshing. The enriched gas to the north is most likely due to the AGN outburst along the north–south direction. Kirkpatrick & McNamara (2015) calibrated an empirical relation between the maximum projected radius of the uplifted gas and the cavity enthalpy

$$\begin{eqnarray}&&{R}_{\mathrm{Fe}}=(57\pm 30)\times {H}^{0.33\pm 0.08}\,\mathrm{kpc},\end{eqnarray} \tag{ 2 }$$

where H is in units of 10⁵⁹ erg. Substituting the cavity enthalpy of NGC 1399 (Paper 1), we obtain a R_Fe of 3–10 kpc, consistent with the observed enriched gas extent to the north. The enriched gas extended from the south to the west follows the spiral pattern of the gas sloshing, strongly suggesting that gas sloshing is driving the metal redistribution at the center cluster.

While AGN bubbles are very vulnerable to gas motions at the cluster center (Morsony et al. 2010), the two bubbles in NGC 1399 are remarkably symmetric and intact in both X-ray and radio. We thus infer that the AGN outburst has taken place later than the gas sloshing. This is consistent with the cycle of AGN outbursts (a few tens of megayears) being much shorter than the timescale of gas sloshing (∼1 Gyr; ZuHone et al. 2010). The enriched gas along the southwest front is more metal abundant, more extended, and distributed to a larger radius than the enriched gas to the north. We therefore conclude that gas sloshing, rather than AGN uplift, has played a primary role in redistributing the enriched gas, at least in this particular case.

While the cooling time drops below 1 Gyr at the center of Fornax, as in many more massive cool-core clusters, the star formation rate is negligible in NGC 1399 (Vaddi et al. 2016). AGN feedback is by far the most viable solution to the cooling problem. The mechanical energy provided by AGN is correlated with the cool-core luminosity (Bîrzan et al. 2004). AGN activity can respond to cooling through precipitation of cooled gas, as proposed by McCourt et al. (2012). Still, it is unclear how the jet power is transformed into thermal energy. One route is by the dissipation in the ICM of turbulence generated in the wakes of rising bubbles (Churazov et al. 2002). Then again, Zhuravleva et al. (2017) found that in the cool cores of some clusters (e.g., the Perseus Cluster and Abell 1795), gas perturbations are associated with gas motion (isobaric) rather than AGN bubbles (isothermal), suggesting that the bulk of the turbulence may not be generated by radio outbursts. Apart from AGN outbursts, gas sloshing has been considered as a means of quenching cooling flows (Fujita et al. 2004; ZuHone et al. 2010). Frequent gas sloshings do not necessarily require a high merging rate in that multiple sloshing fronts can be induced by just one subcluster infall (ZuHone et al. 2010, 2011). NGC 1399 displays a series of sloshing cold fronts at various radii. As demonstrated in Figure 8, the sloshing timescale is ≈10× shorter than the cooling time over the entire cluster center, making it a promising mechanism to heat the cool core. However, it would be far fetched for gas sloshing to be the primary mechanism of preventing the gas from cooling, unless the frequency of gas sloshing responds to the state of the gas. Nevertheless, with AGN feedback being the primary solution, gas sloshing may still provide supplemental heating to the cool core, as long as heat can be transported effectively between gases of different entropy. Below we discuss two transport processes that may occur at the interface.

4.3.1. Conduction

Thermal conduction is expected to erase temperature gradients outside the strong cool core (1 Gyr < t_cool < 14 Gyr; Voit et al. 2015), where most sloshing cold fronts reside. If conduction is reduced, low-entropy gas brought out by sloshing would eventually fall back to the cluster center (Ghizzardi et al. 2014).

We calculate the characteristic mean-free path (mfp) of electrons, λ_e, in the hot plasma at the leading edge. Sarazin (1988) gives

$$\begin{eqnarray}&&{\lambda }_{e}=\displaystyle \frac{{3}^{3/2}{({{kT}}_{e})}^{2}}{4{\pi }^{1/2}{n}_{e}{{q}_{e}}^{4}\mathrm{ln}{\rm{\Lambda }}}.\end{eqnarray} \tag{ 3 }$$

where n_e is the electron density and the Coulomb logarithm is

$$\begin{eqnarray}&&\mathrm{ln}{\rm{\Lambda }}=35.7+\mathrm{ln}\left[\left(\displaystyle \frac{{{kT}}_{e}}{1\,\mathrm{keV}}\right){\left(\displaystyle \frac{{n}_{e}}{{10}^{-3}{\mathrm{cm}}^{-3}}\right)}^{-1/2}\right].\end{eqnarray} \tag{ 4 }$$

Equation (3) can be approximated by

$$\begin{eqnarray}&&{\lambda }_{e}\approx 340\,\mathrm{pc}{\left(\displaystyle \frac{{{kT}}_{e}}{1\mathrm{keV}}\right)}^{2}{\left(\displaystyle \frac{{n}_{e}}{{10}^{-3}{\mathrm{cm}}^{-3}}\right)}^{-1},\end{eqnarray} \tag{ 5 }$$

assuming the temperatures of electrons, ions, and gas are equal. Substituting a temperature of 1.4 keV and a gas density of 0.004 cm⁻³ for the northeast cold front at r = 10 kpc (F3, the most prominent cold front), we obtain a λ_e of 150 pc. This value is smaller than we can resolve in either our imaging or spectral analysis. Therefore, we are unable to strictly determine the effective conductivity in the ICM in this current work. Following Sarazin (1988) and Markevitch et al. (2003), we estimate the timescale for the thermal conduction to operate in the Spitzer regime

$$\begin{eqnarray}{t}_{\mathrm{cond}} & \sim & {{kl}}^{2}{n}_{{\rm{e}}}/{\kappa }_{{\rm{s}}}\sim 12\left(\displaystyle \frac{{n}_{e}}{0.002\,{\mathrm{cm}}^{-3}}\right)\\ & & \times {\left(\displaystyle \frac{l}{100\mathrm{kpc}}\right)}^{2}{\left(\displaystyle \frac{T}{10\mathrm{keV}}\right)}^{-5/2}\,\mathrm{Myr},\end{eqnarray} \tag{ 6 }$$

where κ_s is the Spitzer value (1956), l is the size of the cold front, and T is the ICM temperature. Applying the conditions of F3, we obtain t_cond ≲ 10 Myr, shorter than the age of the bubbles (∼15 Myr, Paper 1). As we discussed in Section 4.2, AGN outburst is more recent than the event that caused the sloshing. The northeast sloshing front, residing at 10 kpc, has a sloshing timescale of 100 ∼ 200 Myr (Figure 8). The conductivity, possibly being reduced by a factor of (κ/κ_s)⁻¹ ∼ t_age/t_cond ≳ 10, is not sufficient to wipe out temperature gradient. This is consistent with the edge-fitting of the surface brightness profile that convolving the power-law density model with a Gaussian component is not required (Section 3.2). Komarov et al. (2014) argue that fluid elements along the presumably incompressible cold front are stretched, which tend to enhance the temperature gradients and align the originally random magnetic field lines along the front. Consequentially, and counterintuitively, heat flux can be reduced at the cold front where the temperature gradient is the largest. Then again, using MHD simulations, ZuHone et al. (2013, 2015) found that the temperature gradient can nevertheless be reduced if the conduction is anisotropic even in the presence of this magnetic layer.

4.3.2. KHI and Turbulent Mixing

The innermost sloshing cold front (F4) resides at r ≲ 3 kpc to the west while the two AGN bubbles are more than 5 kpc away from the cluster center. Sloshing brings high-entropy gas to the innermost regions where cooling is most severe (t_cool < 1 Gyr). This innermost sloshing cold front is not sharp with a subkiloparsec bright structure, a KHI eddy candidate, visible at the interface (blue box in Figure 3). Its temperature could be ≈1 keV, similar to the gas temperature inside the interface.

Either viscosity or ordered magnetic field can damp out the growth of KHI. The presence of a KHI roll allows us to put upper limits on the viscosity and the magnetic field strength using the criteria presented in Roediger et al. (2013) and Vikhlinin et al. (2001), respectively. For a shear flow velocity with a Mach number ${ \mathcal M }$, the fraction of the viscosity relative to the Spitzer value, f_ν, needs to be smaller than

$$\begin{eqnarray}&&{f}_{\nu }\lt \displaystyle \frac{10}{16\sqrt{{\rm{\Delta }}}}\cdot \displaystyle \frac{{ \mathcal M }{c}_{s}}{400\,\mathrm{km}\,{{\rm{s}}}^{-1}}\cdot \displaystyle \frac{{\ell }}{10\,\mathrm{kpc}}\\ &&\qquad \cdot \,\displaystyle \frac{{n}_{e,h}}{{10}^{-3}\,{\mathrm{cm}}^{-3}}{\left(\displaystyle \frac{{{kT}}_{h}}{2.4\mathrm{keV}}\right)}^{-5/2},\end{eqnarray} \tag{ 7 }$$

where ${c}_{s}(=\sqrt{\gamma {{kT}}_{h}/\mu {m}_{p}})$ is the sound speed, T_h and n_e,h are the temperature and density of the ICM on the high entropy side, ℓ is a characteristic length scale, and ${\rm{\Delta }}=\tfrac{{({\rho }_{h}+{\rho }_{c})}^{2}}{{\rho }_{h}{\rho }_{c}}$ characterizes the contrast of the gas densities on each side of the interface. This KHI eddy candidate has a height of h ∼ 0.3 kpc; we estimate ℓ ≈ 2h ∼ 0.6 kpc (Roediger et al. 2013). Likewise, the strength of an ordered magnetic field needs to be smaller than

$$\begin{eqnarray}&&B\lt {\left[8\pi \displaystyle \frac{\gamma {T}_{h}{n}_{e,h}}{1+{T}_{c}/{T}_{h}}\right]}^{\tfrac{1}{2}}{ \mathcal M }.\end{eqnarray} \tag{ 8 }$$

Sloshing cold fronts are found to be subsonic in simulations and the motion slows down toward smaller radii (Ascasibar & Markevitch 2006; ZuHone et al. 2010, 2011). Roediger et al. (2011) found that the sloshing cold fronts propagate at ${ \mathcal M }\sim 0.1$ in a simulation tailored to the Virgo Cluster. We note that there is no offset between the X-ray and the infrared centroids (Figure 3, bottom right), disfavoring a recent sloshing event in Fornax. We assume that this innermost front is 30–60 Myr old, that is the sloshing timescale at r = 3 kpc (Figure 8). This corresponds to an average sloshing velocity of 50–100 km s⁻¹ (${ \mathcal M }\lesssim 0.2$). Here, we conservatively adopt ${ \mathcal M }\lt 0.5$ (ZuHone et al. 2011). The upper limits on f_ν and B as a function of ${ \mathcal M }$ are presented in Figure 9. A full Spitzer viscosity can be ruled out and the strength of an ordered magnetic field should be smaller than 10 μG. However, as demonstrated in ZuHone et al. (2013), the suppression due to magnetic draping is only partial since the field lines may not be perfectly aligned.

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More evident ≲1 kpc structures are present to the east at a similar radius (green and magenta boxes in Figure 3). In particular, S3 has a best-fit temperature similar to that of the gas within the interface and is in pressure equilibrium with the external gas. We note that the temperature gradient to the east is much smoother than that in the west (Figure 7). This difference may be due to more evident KHI to the east, which may have disrupted a previously existing cold front. KHI accelerates the turbulent mixing of gas of different phases, as we expect from theory and simulations (ZuHone et al. 2011; Roediger et al. 2015a, 2015b). Large KHI rolls (above certain wavelengths, ℓ_max) are expected to be suppressed by gravity (Chandrasekhar 1961). We do not expect to observe KHI rolls larger than

$$\begin{eqnarray}{{\ell }}_{\max } & \approx & 18\,\mathrm{kpc}{\left(\displaystyle \frac{{\rho }_{c}/{\rho }_{h}}{1.5}\right)}^{-1}{\left(\displaystyle \frac{U}{200\mathrm{km}{{\rm{s}}}^{-1}}\right)}^{2}\\ & & \times {\left(\displaystyle \frac{g}{3\times {10}^{-8}\mathrm{cm}{{\rm{s}}}^{-2}}\right)}^{-1},\end{eqnarray} \tag{ 9 }$$

where U is the shear velocity at the front and g is the gravitational acceleration (Roediger et al. 2012b). We assume ρ_c = 1.5ρ_h and U(r) = 0.3c_s(r); c_s and g(r) were calculated from the best-fit deprojected temperature profile and the X-ray hydrostatic mass profile (Paper 1). We present the maximum KHI length scale as a function of radius at the cluster center in Figure 10. The size of these features stays below this upper limit. A potential ghost bubble of ≳5 kpc diameter resides at r = 13 kpc (r = 10 kpc in projection) to the northwest (Figure 2, see Paper 1 for details). Interestingly, Walker et al. (2017) propose that such giant X-ray surface brightness decrement may be due to KHI instead of AGN bubbles. The position and size of this ghost bubble candidate are also in the allowed parameter space for a giant KHI roll.

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Alternatively, S1, S2, and S3 may be thermally unstable X-ray filaments resulting from gas cooling (David et al. 2014, 2017). The growth time of KHI can be estimated from (Roediger et al. 2012a)

$$\begin{eqnarray}&&{\tau }_{\mathrm{KH}}=\displaystyle \frac{\sqrt{{\rm{\Delta }}}}{2\pi }\displaystyle \frac{{\ell }}{U}=3.9\,\mathrm{Myr}\sqrt{{\rm{\Delta }}}\displaystyle \frac{{\ell }}{10\,\mathrm{kpc}}\left(\displaystyle \frac{U}{400\,\mathrm{km}\,{{\rm{s}}}^{-1}}\right).\end{eqnarray} \tag{ 10 }$$

We calculate τ_KH for the maximum KHI length scale, the maximum growth time, at each radius as shown in Figure 8. The maximum growth time of KHI is shorter than the cooling time by two orders of magnitude. KHI is thus somewhat favored as their origin. The timescale of KHI is also significantly shorter than the dynamical timescale (free-fall time or buoyant rising time), implying that KHI, coupled with gas sloshing, may have played an important role in regulating the thermal state of the hot plasma at cluster centers.