Stability and Ly 𝛼 emission of Cold Stream in the Circumgalactic Medium: impact of magnetic fields and thermal conduction

Cold streams of gas with temperatures around 10 4 K play a crucial role in the gas accretion on to high-redshift galaxies. The current resolution of cosmological simulations is insufficient to fully capture the stability and Ly 𝛼 emission characteristics of cold stream accretion, underscoring the imperative need for conducting idealized high-resolution simulations. We investigate the impact of magnetic fields at various angles and anisotropic thermal conduction (TC) on the dynamics of radiatively cooling streams through a comprehensive suite of two-dimensional high-resolution simulations. An initially small magnetic field ( ∼ 10 − 3 µ G), oriented non-parallel to the stream, can grow significantly, providing stability against Kelvin-Helmholtz instabilities and reducing the Ly 𝛼 emission by a factor of < 20 compared to the hydrodynamics case. With TC, the stream evolution can be categorised into three regimes: (1) the Diffusing Stream regime, where the stream diffuses into the surrounding hot circumgalactic medium; (2) the Intermediate regime, where TC diffuses the mixing layer, resulting in enhanced stabilization and reduced emissions; (3) the Condensing Stream regime, where the impact of magnetic field and TC on the stream’s emission and evolution becomes negligible. Extrapolating our findings to the cosmological context suggests that cold streams with a radius of ≤ 1 kpc may fuel galaxies with cold, metal-enriched, magnetized gas ( 𝐵 ∼ 0 . 1–1 µ G) for a longer time, leading to a broad range of Ly 𝛼 luminosity signatures of ∼ 10 37 –10 40 erg s − 1 .


INTRODUCTION
In the framework of the Λ cold dark matter (CDM) paradigm, the process of gas accretion on to high-redshift galaxies at  > 2 is primarily driven by cold gas flowing along dark matter filaments of the cosmic web.This cold gas is considered to be one of the main components contributing to the overall gas accretion phenomenon in the high-redshift universe (e.g.Fardal et al. 2001;Kereš et al. 2005;Dekel & Birnboim 2006;Dekel et al. 2009).Such accretion, often referred to as cold streams or cold flows, plays a crucial role in fueling galaxies with gas that is readily available for collapse and subsequent star formation.Dekel & Birnboim (2006) provided key insights into the conditions required for the survival of cold streams within massive haloes.On top of being ubiquitous in cosmological simulations, the cold stream accretion scenario provides a key physical mechanism for explaining the observed cosmic star-formation history (e.g.Reddy & Steidel 2009;Cucciati et al. 2012;Gruppioni et al. 2013;Madau & Dickinson 2014), the low-redshift galaxy color bimodality (e.g.Strateva et al. 2001;Kauffmann et al. 2003;Blanton et al. 2003;Baldry et al. 2004;Bell et al. 2004;Nelson et al. 2018), * E-mail: ledos@astro-osaka.jp and the acquisition of galaxy angular momentum (e.g.Danovich et al. 2015).
Recent observational data provide growing support for the cold stream accretion scenario (Behroozi et al. 2019;Daddi et al. 2022b,a).However, direct observations of cold streams remain relatively scarce.Observational support for cold accretion using absorption line spectroscopy of background quasars or galaxies primarily involves Mg ii and Fe ii lines (Giavalisco et al. 2011;Rubin et al. 2012;Martin et al. 2012;Bouché et al. 2013Bouché et al. , 2016;;Zabl et al. 2019), as well as H i gas (Turner et al. 2017;Chen et al. 2020;Fu et al. 2021).The limited number of detections of cold accreting gas through absorption line systems can be attributed to their small covering factor compared to the surface area of the halo (Faucher-Giguère & Kereš 2011).On the other hand, emissions-line observations of Ly emitters have revealed filamentary structures in or around the halos of high-redshift galaxies (e.g., Cantalupo et al. 2014;Fumagalli et al. 2016;Borisova et al. 2016;Umehata et al. 2019), as well as emission from cold gas consistent with cold stream emission(e.g., Arrigoni-Battaia et al. 2018;Martin et al. 2019).Daddi et al. (2021) utilized observations of Ly emission to identify the presence of clear cold filamentary gas structures surrounding massive galaxies at a redshift of  = 2.9.Emonts et al. (2023), on the other hand, detected cold filamentary gas structures using observations of neutral carbon (C i) emission at  = 3.8.In contrast, Zhang et al. (2023), through Ly and metal lines, detected emissions consistent with inspiraling streams around a galaxy at redshift  = 2.3.These observations provide direct evidence for the existence of cold streams near these galaxies, although they also highlight the challenge of relating the emissions to cold streams.Hence, comprehending the emission signature of cold streams is a crucial step in establishing their widespread occurrence beyond the realm of cosmological simulations.
Concurrently, there have been recent endeavors to investigate the influence of simulation resolution on the properties of gas within the halo, specifically the circumgalactic medium (CGM) (Peeples et al. 2019;van de Voort et al. 2019;Hummels et al. 2019;Suresh et al. 2019;Nelson et al. 2020;Bennett & Sĳacki 2020).Bennett & Sĳacki (2020) find that increasing their mass resolution by a factor of 512 near shocks increases the cold gas content in the CGM by ∼ 50% and the inflow rate of cold gas by ∼ 25% compared to typical case, which gives a much more multiphase and turbulent picture of the CGM than the usual one from the state-of-the-art cosmological simulations.However, they do not fully achieve convergence at their finest resolution.Nelson et al. (2020) also shows that a resolution of Δ < 100 pc is needed to fully resolve the small-scale cold gas structure in the CGM of massive galactic haloes ( h > 10 12  ⊙ ) at redshift  = 0.5.In the case of cold stream accretion, the lack of resolution does not allow the development of Kelvin-Helmholtz Instabilities (KHI) at the interface between the cold dense gas of the stream and the hot diffuse CGM.These instabilities can shorten the lifetime of the stream as first studied by Mandelker et al. (2016).Given that the typical resolution of CGM in most cosmological simulations near the virial radius is around ∼ 1 kpc, it also suggests the need to study the cold stream evolution further using high-resolution simulations.
To understand both the evolution and the emission signatures from cold streams, numerous works performed high-resolution simulations considering idealized stream geometry: Mandelker et al. (2016) (linear analysis), Padnos et al. (2018) (2D hydrodynamic (HD) simulations), and Mandelker et al. (2019) (2D and 3D HD simulations) initiated such work with HD simulations and revealed that cold streams could be disrupted by both KHI surface modes and body modes (also called reflective modes).They concluded that surface modes have the highest growth rate and were the dominant mode that could alter the cold stream evolution.Aung et al. (2019) targeted the impact of self-gravity, which can cause the stream to fragment from gravitational instabilities.With 2D and 3D magnetohydrodynamic (MHD) simulations, Berlok & Pfrommer (2019) studied the impact of the magnetic fields parallel to the stream, and found that it can help the stream to survive KHI if the field strength is strong enough ( ≳ 0.3-0.8µG).Vossberg et al. (2019) investigated via 2D HD simulations the appearance of over-dense stream regions from the growth of KHI.While these works all mainly investigated the cold stream evolution, Mandelker et al. (2020a) provided valuable insights into the emission signature of cold streams by incorporating radiative cooling into HD simulations, showing that the cooling emission scales with the cooling time as ∝  −1/4 cool .Further analytical considerations (Mandelker et al. 2020b) demonstrate that the cold streams, with radius  s > 3kpc, exhibit Ly luminosities exceeding  Ly > 10 42 erg s −1 for halo masses  h > 10 12 M ⊙ at  ∼ 2.
The key emission mechanism comes from the mixing of the hot CGM gas ( cgm ∼ 10 6 K) and the cold stream gas ( s ∼ 10 4 K) which creates a gas mixture at an intermediate temperature ( mix < 10 5 K) whose cooling rate becomes orders of magnitude higher (Begelman 1990).Gronke & Oh (2018, 2020) investigated this mechanism from simulations of cold clouds embedded in a hot wind.The latter work found that both the cooling emission and condensation/mixing velocity of the cloud scaled with the cooling time in the mixing layer as ∝  −1/4 cool .From higher resolution shear layer simulations, Fielding et al. (2020) and Tan et al. (2021, for strong cooling) retrieve such scaling while Ji et al. (2019) (and Tan et al. (2021, for weak cooling)) found a scaling of ∝  −1/2 cool .Further simulations also investigated the impact of high Mach numbers (Yang & Ji 2023).Hence, in the case of HD simulations with radiative cooling, one may predict the evolution of the cold stream in terms of mass flux and emission thanks to the estimated cooling time in the mixing layer.
The impact of additional physics, such as magnetic fields and thermal conduction, remains unanswered when combined with radiative cooling for studying cold streams.One can find some insights from simulations of cold clouds in the CGM.Armillotta et al. (2017, 2D HD simulations) found that isotropic conduction can hinder the growth of KHI and increase the survival time of the cloud.Hidalgo-Pineda et al. ( 2023) investigated with 3D simulations the impact of magnetic field and concluded that the magnetic field could help stabilize the cloud against KHI, allowing it to survive for a longer time scale.Brüggen & Scannapieco (2023) investigated both isotropic and anisotropic thermal conduction from 3D MHD simulations with different magnetic field angles.They found that both the magnetic field and thermal conduction can lower the KHI growth, i.e., the mixing of the cold cloud gas and the CGM, allowing the cloud to survive longer.Sander & Hensler (2021) studied high-velocity clouds inside the CGM from 3D MHD simulations, including self-gravity, star formation, thermal conduction, and additional physics.They concluded that thermal conduction also helps stabilize the cloud but that it also diffuses the cold gas substructure, which has been detached from the cloud (in the cloud tail, for example).
To sum up, the mixing of the CGM gas and the streaming gas triggered by KHI is a crucial mechanism that can explain the stream emission signature and its evolution.In particular, the emission from the mixing layer is dominated by Ly which might be linked to observed Ly emitters.However, from simulations of cold clouds in the hot CGM, it appears that magnetic fields with various angles and thermal conduction can each affect the growth of the KHI.Reducing the KHI growth rate can increase the survival time of cold streams but may also decrease its emission.We hence intend to address this issue by performing a large suite of 2D MHD simulations (∼ 120 simulations), including radiative cooling and anisotropic thermal conduction.Focusing on 2D simulations allows us to cover a wide range of parameters with different stream velocities, CGM/stream densities, and magnetic field angles, on HD, MHD, and MHD with thermal conduction (MHD+TC) simulations.
We start by describing the idealized cold stream model and the relevant time-scales in Section 2. The numerical setup and the initial conditions of the numerical experiments are then described in Section 3. Section 4 presents our results and analysis on the impact of magnetic fields and thermal conduction on the evolution of and emission from cold streams.Finally, we discuss the extrapolation of our results in a cosmological context and the caveats of our work in Section 5 before concluding in Section 6.

COLD STREAM MODEL AND RELEVANT TIME-SCALES
This section discusses the cold stream model and the relevant timescales for our simulations.We describe the model of the cold stream and the chosen parameters (Sec.2.1), the radiative cooling-heating model (Sec.2.2), the evolution of the mixing layer (Sec.2.3), and the thermal conduction (Sec.2.4).The section ends with a definition of the stream evolution regimes based on the important time-scales (Sec.2.5).

Typical Parameters of Cold Streams
Our choice of parameters for the stream model is similar to previous numerical studies of idealised cold streams (see, e.g., Berlok & Pfrommer 2019;Mandelker et al. 2020a).Observations of cold inflow in the CGM typically target massive haloes with halo masses of  h ≳ 10 11 M ⊙ , covering redshifts from  ∼ 0.4 (Martin et al. 2012(Martin et al. , 2019) ) to  ∼ 3.8 (Emonts et al. 2023).The inferred H i column densities of cold streams span over a wide range of  HI ∼ 10 17 −10 21 cm −2 with metallicities  ∼ 10 −3.8 Z ⊙ -1 Z ⊙ .
Building upon a cosmological simulation, Dekel et al. (2013) developed a simplified model for star-forming galaxies within massive haloes ( h > 10 11 M ⊙ ) at  > 1 with the following virial radius and velocity: where  v is the virial mass of the halo and is taken as 10 12 M ⊙ .
Consistently with previous work, we target three number densities  s = 10 −3 , 10 −2 , 10 −1 cm −3 , along with two density ratios  ≡  s / cgm = 30, 100, one set of stream/CGM metallicities  s ,  cgm = 10 −1.5 , 10 −1 Z ⊙ , and three different stream Mach number M = M cgm = 0.5, 1, 2. We also choose to fix the stream radius to  s = 1 kpc.While our chosen value  s is below the analytical estimate of Mandelker et al. (2020b), using such a small value allows us to better study the effects of thermal conduction on stream evolution.Furthermore, cosmological simulations that hyper-refine streams in the CGM suggest that these may contain smaller-scale stream-like structures (Bennett & Sĳacki 2020).The impact of a larger radius is discussed in Sec.2.5 and in Sec. 6.
Our understanding of the magnetic field in the large-scale structure of the Universe remains incomplete.The primordial magnetic field has been constrained to a lower limit of ∼ 10 −10 -10 −9 µG (Neronov & Vovk 2010;Dolag et al. 2011).Although the evolution of the primordial magnetic field has been theorized (Saveliev et al. 2012), it may be more reliable to focus on magnetic fields that have been studied in recent simulations and observations of the CGM.
Little is known about the properties of the magnetic field in the CGM of high-redshift galaxies.Most zoom-in simulations are focused on the magnetic field strength growth (Rieder & Teyssier 2017;Martin-Alvarez et al. 2018) and morphology (Pakmor et al. 2017(Pakmor et al. , 2018;;Steinwandel et al. 2019) in the galactic disc due to the small-scale dynamo.Therefore, to better understand the composition and magnetic morphology of the CGM, we may rely on CGMfocused simulations.Simulations from the Auriga project (Pakmor et al. 2020) and FIRE project (Hopkins et al. 2020) have investigated the evolution of the magnetic field in the CGM, providing estimates of the magnetic field strength ranging from 10 −3 µG to 10 −2 µG at the virial radius and for redshift  ∼ 2.
Observations of the near-centre CGM gas have put upper constraints on the magnetic field strength, but probing the magnetic field in high-redshift haloes is challenging.Using fast radio bursts, Prochaska et al. (2019) have constrained the magnetic field to a range of 6 × 10 −2 -2 µG for electron density range of 10 −5 -10 −3 cm −3 inside the hot halo ( cgm ∼ 10 6 K).The study by Lan & Prochaska (2020) of over 1000 Faraday Rotation Measures in low-redshift galaxies ( < 1) provides a lower constraint of 2 µG for the upper limit of the coherent magnetic field.Since the magnetic field in the halo grows with time, these upper limits motivate us to investigate the effects of a low magnetic field.
Berlok & Pfrommer (2019) investigated the impact of a magnetic field aligned with the cold stream on the growth of KHI, using a magnetic field strength of approximately 1 µG.This relatively high magnetic field strength ( < 100) was chosen to explore a range of dynamically relevant magnetic field strengths for cold streams without radiative cooling, when the magnetic field is parallel to the stream.In our work, we investigate a magnetic field with various angles in which cases an additional amplification of the magnetic field strength can be expected due to the stretching of the magnetic field lines from the velocity difference at the interface between the stream and the CGM.Hence, we focus on a lower magnetic field strength of  ∼ 10 −3 µG defined by an initial ratio of thermal pressure over magnetic pressure  = / mag = 10 5 for our study.This gives an Alfvenic time-scale, with the Alfven speed  a =  0 / √ .Defining the sound crossing time of the stream as  sc = 2 s / s , we have  a ∼ 0.5   s /(2) 0.5 ∼ 100  sc , meaning that the Alfvenic time can be initially ignored.

Radiative cooling-heating
Tabulated cooling and heating rates are derived from the photoionization code CLOUDY (Ferland et al. 2017), which accounts for both atomic and metal cooling processes.The heating rates from Haardt & Madau (2012) are employed for the UV background radiation from galaxies and quasars at redshift  = 2.
Fig. 1 presents the resulting cooling/heating map for the assumed CGM metallicity of  = 0.1 Z ⊙ .The left panel presents logarithmic scales of the heating and cooling rates in red and blue color maps, respectively, where the white region denotes near-equilibrium states.The red region is dominated by heating, while the blue region is dominated by cooling.The right panel shows the net cooling/heating curves of a typical gas in the mixing layer with number density  mix = 10 −3 cm −3 .The colored lines represent the total cooling from the main species in our temperature region.
To ensure the thermal equilibrium of the cold stream in our initial conditions of the simulation, we determine its initial temperature from where H is the heating rate and Λ is the cooling rates, respectively.This gives stream temperatures of  s ∼ (1.3, 1.9, 3.1) × 10 4 K for  s = (10 −1 , 10 −2 , 10 −3 ) cm −3 , respectively.Assuming an isobaric cooling, the resulting cooling time for gas in the mixing layer at temperature  mix is where the net cooling-heating rate Λ net,mix = H  mix ( mix ) − Λ  mix ( mix ) is defined by the number density and temperature in the mixing layer between the stream and the CGM.The value of the mixing layer number density and temperature are defined in the subsequent section.

Mixing layer
We hereby summarise previous studies on the mixing layer relevant to our work.Begelman (1990) and Slavin et al. (1993) first described the physical properties of the mixing of two gases from a shear interface in the case of the interstellar medium.In particular, considering the mass accretion rate  c and  h of the cold and the hot phase, they defined the temperature of the mixed phase as, assuming ideal mass accretion rate as  h−c ∼  h−c  t,h−c for both phases with  t,h−c being the turbulent velocity for the hot and the cold phases, linked by equating their kinetic energies,  c  2 t,c =  h  2 t,h 1 .This was later followed by detailed studies from simulations on interface geometry (e.g.Kwak & Shelton 2010;Ji et al. 2019;Fielding et al. 2020;Tan et al. 2021;Yang & Ji 2023), on the cold gas entrained in hot wind (Gronke & Oh 2018, 2020), and on cold streams (Mandelker et al. 2020a).
In the case where the radiative cooling is strong enough to condense gas from the hot phase, the mixing velocity or the inflow velocity of CGM gas into the mixing layer scales as (Gronke & Oh 2020;Mandelker et al. 2020a;Fielding et al. 2020;Tan et al. 2021) The inflow of the hot gas in the mixing layer occurs at a steady rate.Once at the temperature  mix , the gas mixture cools efficiently, leading to a steady cooling emission which also scales as From  in , one can also recover the mass evolution of the cold stream for sufficiently strong cooling due to the condensation of CGM gas, where  is the surface between the stream and the CGM.We note that a different scaling is found by Ji et al. (2019) with  in ∝  −1/2 cool , similarly to the weak cooling case in Tan et al. (2021).While we discuss the scaling in our simulations, higher resolution simulations targeting specifically the mixing layer (Fielding et al. 2020;Tan et al. 2021) might be needed to investigate the origin of the scaling properly in the presence of magnetic field and thermal conduction.Such work is beyond the scope of this paper.
The evolution of the radiative mixing layer is also important as it can stabilize the stream against the KHI.The initial growth of the mixing layer before its steady evolution due to cooling can be described by the shearing time (Mandelker et al. 2019(Mandelker et al. , 2020a)), with the dimensionless growth rate defined by the empirical fitting value  ∼ 0.042 + 0.168exp −3M 2 tot (Dimotakis 1991), where the total Mach number is defined with the sound speed of both phases M tot =  s /  s +  cgm .

Thermal conduction
The thermal conduction time-scale for the stream is defined as where we assume the Spitzer thermal conduction coefficient  =  0  2.5 (Spitzer 1962).Note that the coefficient is defined based on the mixing phase, as the temperature of the diffusion front is approximately the same.This diffusion time, along with the cooling time, determines whether the stream will grow in mass or diffuse.However, we are also interested in the impact of thermal conduction on the mixing between the stream and the CGM due to the growth of the KHI.Therefore, we also define the diffusion time for a small perturbation of size 0.1 s , as with the subscript  standing for perturbation and where the diffusivity is also defined from the mixing phase.

Time-scale comparison
From the physical processes presented above, we can assume the evolution of the stream by considering the different time-scales  sh ,  cool ,  s , and  p .An important parameter to describe the stream evolution is the ratio of the cooling time over the shearing time, Fig. 2 plots contours of the ratio  on the plane of stream density  s and density ratio  =  s / cgm .In the hydrodynamic case, the stream evolution can be defined by  as follows: • Disrupting stream regime:  > 1, • Condensing stream regime:  < 1.
In the presence of thermal conduction, the above categorization is modified by the ratios  s / cool and  p / cool , both shown in the plot.
Those two ratios are an equivalent formulation to the Field length, which defines the limit at which a structure either diffuses or condenses.Hence,  s / cool = 1 and  p / cool = 1 mark the limit for which the stream or a cold clump, respectively, either diffuse or survive.
Including thermal conduction with our cold stream parameters, we end up with three different regimes of interest: • Diffusing stream regime:  p <  s <  sh <  cool In this case, the diffusion from thermal conduction (TC) is too rapid and overcomes all other processes.The stream diffuses faster than it can condense gas from itself or the CGM.This regime is analogous  to a Field length larger than the stream radius.In practice for our model, this regime is obtained with the condition  ≳ 8.
• Intermediate regime:  p <  sh <  cool <  s In this regime, the cooling time and the shearing time are about the same magnitude and are bigger than the diffusion time  p .Hence, the diffusion of perturbations or small clumps smaller than 0.1 s , happens faster than their growth, potentially shutting off the mixing and the subsequent condensation of CGM gas.In this regime, the mixing layer should diffuse and the stream remains at a constant mass while possibly fragmenting due to long-wavelength KHI modes.From our parameters, this regime can be roughly defined by 0.3 <  < 8.
• Condensing stream regime:  cool <  sh <  p <  s In this regime, radiative cooling occurs faster than both diffusion and KHI growth.The gas in the mixing layer cools efficiently, hence condensing on to the stream.We found that this regime is satisfied for  ≲ 0.3.
The regime map of Fig. 2 is also dependent on the assumed stream radius  s and the metallicities of the CGM and stream.For a bigger or smaller radius, the condensing stream regime region in the  s −  plane would widen or shrink, respectively.Similarly, increasing or decreasing the metalicities would widen or shrink the condensing stream regime region, respectively.

Governing Equations and Computational Methods
We solve the following normalized MHD equations in conservative form using the finite volume mesh code Athena++ (Stone et al. 2020), in which we implemented anisotropic thermal conduction and radiative cooling: where I is the identity matrix.The total pressure  T is defined as Notations , u, B, , , stand for the mass density, velocity vector, magnetic vector, total energy density, and pressure, respectively.The total energy density is defined as The anisotropic thermal conduction term Q is defined as with b = B/ the magnetic field unit vector, and  =  0 2.5 the Spitzer thermal conduction coefficient (Spitzer 1962).The radiative cooling terms H − Λ are interpolated from tables of our model described in Sec.2.2.We summarise the normalisation units in Table 1.
We also trace the mass fraction of gas in the initial stream and CGM gas using a passive scalar transport equation, defined as where  s =  s / is, in other words, the mass fraction of the gas at the initial stream metallicity.In practice, we use this scalar to compute the metallicity of the gas when computing the cooling rate.

Normalised unit Values
Length The conservative MHD formulation is solved using the HLLD Riemann solver from Miyoshi & Kusano (2005) using the spatial PLM reconstruction which is second-order accurate.The divergence-free constraint of the magnetic field is ensured with the Constrained-Transport-scheme introduced in Gardiner & Stone (2005, 2008).For time integration, the second-order Runge-Kutta scheme is used for simulations without thermal conduction.In the case of thermal conduction, the conduction time-step is proportional to Δ 2 , leading to high computational costs.To reduce the computational cost, the super-time-stepping (STS) Runge-Kutta-Legendre second-order solver from Meyer et al. (2014) is used to solve the thermal conduction equation.We apply the limiting scheme developed by Sharma & Hammett (2007) which avoids overestimation of the conduction flux when this one is anisotropic.

Initial and boundary conditions
Our initial conditions are similar to those used in the simulations from Mandelker et al. (2020a) and are summarised here.Fig. 3 shows the initial setup.
The stream is initialized at the centre of a two-dimensional rectangular domain of size 64 s × 32 s , the stream axis being the -axis and with  s = 1 kpc.Boundary conditions are set as periodic in the stream parallel direction and as fixed CGM fluid values along the perpendicular directions.Compared to previous works (Mandelker et al. 2019;Berlok & Pfrommer 2019;Mandelker et al. 2020a), we extended the domain transversal to the stream by a factor of two to avoid boundary effects on the stream due to thermal conduction and the fixed boundary conditions.About the resolution, we use static mesh refinement with the highest resolution of Δ = Δ =  s /64 defined in the near-stream region (< 2 s ).The grid size is then doubled up every 2 s along the -axis up to a maximum cell size of  s .
The stream is defined in the density field by where ℎ determines the smoothness of the transition 2 between the stream and the CGM gas, with ℎ =  s /32.In a similar way, the stream velocity is defined as where  ,0 = M cgm , and M is the Mach number of the stream based on the sound speed in the CGM.We considered values of M ∈ [0.5, 1.0, 2.0].To initialize the KHI, the transverse velocity field is initially seeded by perturbations, where  k is the total number of perturbations defined by   = 2/  with   ∈ [0.5 s , 16 s ], and  0 = 0.01 s is the amplitude3 of the perturbed modes.
Table 2. List of simulation parameters.From left to right, the physics ID stands for hydrodynamic (HD), magnetohydrodynamic (MHD), and MHD with anisotropic thermal conduction (MHD+TC); the density ratio ; the stream number density  s ; the magnetic field angle , such that  = 0 • is a field parallel to the stream; the magnetic field initial strength, the stream Mach number with respect to the CGM sound speed, M = M cgm ; the CGM sound speed; the ratio of the virial crossing time and  for M = 1; the stream temperature; the ratio of the cooling time over the shearing time  =  cool / sh ; and the temperature at which the cooling is maximum.For each row, simulations are performed for the three Mach numbers M = (0.5, 1, 2) or less if specified.The total number of simulations is about 120.The magnetic field is defined by an initial angle , such as  = 0 • corresponds to the case where the magnetic field is parallel to the stream axis ŷ (anti-parallel to the flow) and perpendicular to the temperature gradient, and  = 90 • corresponds to the case where the magnetic field is transverse to the stream (and aligned with the temperature gradient).As the magnetic field is constant over the entire domain, hydrostatic equilibrium gives us a constant pressure  0 both in the CGM and in the stream.
simulation with  0 = 0.5 s exhibits stronger mixing between the stream and the CGM leading to higher cold stream mass growth.
Table 2 lists all simulations and their parameters.For each row of parameters, unless specified, three simulations with M = 0.5, 1 and 2 are run, leading to a total of about 120 simulations.The resulting stream velocities span over 48-534 kms −1 in good agreement with observations.Simulations are run for a total time of  end = 22 0 ∼ 0.8-1.2Gyr.

RESULTS
We first describe the general evolution for HD, MHD and MHD+TC simulations, as well as the cooling emission signature and mass of the stream (Sec.4.1).Then, we focus on the impact of the magnetic field and thermal conduction (Sec.4.2.1, the magnetic field evolution (Sec.4.2.2), and the turbulence in the mixing layer (Sec.4.2.3).

General evolution of HD and MHD cases without TC
We present a brief showcase of the general evolution of the stream based on our HD and MHD simulations.In Fig. 4, we show temperature maps for HD simulations with different values of the ratio  =  cool / sh , at the early, median, and final time, namely  = 0.8, 11, 22 0 , respectively.The displayed maps are rotated by 90 • compared to the illustration in Fig. 3.The fate of the stream is determined by the value of .As described in Sec.2.5, when  < 1, (condensing stream regime), the CGM gas condenses onto the stream, resulting in the growth of the cold-stream mass.For  > 1 (the disrupting stream regime), an increasing amount of initially cold stream gas mixes into the CGM, leading to stream disruption.Such stream evolution is in good agreement with the findings presented by Mandelker et al. (2020a).The simulation with  ∼ 1 represents the limit where the cooling is strong enough to sustain the cold mass but not sufficient to cool the gas in the mixing layer efficiently, leading to a relatively thick mixing layer.At such a limit, the stream also fragments into cold and rather large clouds (>  s ) at a later time.
Fig. 5 illustrates the temperature maps of the MHD simulations at half-time  = 11 0 for M = 1 and  =  s / cgm = 100, considering different values of  =  cool / sh and various initial magnetic field angles.Comparing these with the middle column of Fig. 4, we see that the magnetic field significantly impacts the stream evolution only for an initial magnetic field not parallel to the stream ( ≠ 0 • ) and when the stream is not in the condensing stream regime ( ≳ 1).The main visual difference is the decrease in the amount of gas in the mixing layer for  ≠ 0 • , particularly in the disrupting stream regime ( > 1).The magnetic field strength is initially insignificant ( = 105 ), highlighting the need for a drastic increase of the field strength to impact the stream evolution, as one shall see in Sec.4.2.2.

General evolution of MHD cases with TC
Fig. 6 presents the temperature maps for MHD+TC simulations with varying ratios of  =  cool / sh , considering an initial magnetic field angle of  = 45 • , at early, median and final time, of  = 0.8, 11, 22 0 , respectively.In the diffusing stream regime ( > 8), as predicted, TC effectively hinders the growth of instabilities and diffuses the stream into the CGM.The rate of diffusion depends on the magnetic field angle and can be explicitly expressed as an efficiency parameter for the thermal conduction flux Q.As the simulation progresses, the magnetic field lines bend at the interface between the stream and the CGM, diminishing the efficiency of the conduction.Consequently, at  = 22 0 , the thermal conduction efficiency drops to a point where it no longer overcomes the cooling, even leading to a small cold gas mass increase at later time 4 .As discussed later, the diffusion efficiency of the stream depends on  and on the stream velocity.In the intermediate regime (0.3 <  < 8), TC diffuses the mixing layer and erases any small-scale perturbations.This is consistent with the fastest time-scale  p in the intermediate regime which defines the diffusion of a small cold clump with a size of 0.1 s .Hence, the stream can be stabilized against small-scale surface modes, also meaning that 4 Such increase is also visible in the time profile of the cold stream mass in Appendix A the mixing layer and any small structures or perturbations diffuse in the CGM.In the condensing stream regime ( < 0.3), there are no substantial differences between HD, MHD and MHD+TC simulations.Notably, at the later stages of the MHD+TC simulations, the stream starts to fragment into large cold clumps.This evolution is also seen in the MHD simulations and originates from the magnetic field tension force which inhibits the smaller-scale perturbations, i.e., the short wavelength KHI modes (Berlok & Pfrommer 2019).As a result, as time progresses, the longer wavelength KHI modes grow sufficiently to induce stream fragmentation.

Cooling signature and stream mass
To assess the cooling occurring in the mixing layer, we compute the net cooling emission of the gas in each cell below a temperature threshold  u =  mix +  cgm /2.This threshold targets only the emission in the mixing layer 5 .The gas in the stream, being in equilibrium between heating from the UV background and cooling, produces negligible net cooling, and therefore we do not define any lower bound.We found that a small variation of the threshold does not affect our results.Note that we exclusively consider net cooling.Including cooling induced by photonionization from the UV background might result in an overestimation of the total cooling, as our simulations do not account for self-shielding.The cooling rate is integrated over all gas under  u and subsequently averaged over time, starting at the point where the mixing layer reaches the quasi-steady state:6 where the factor  s represents integrating around the stream axis to mimic a cylindrical stream.The stream radius is  s = 1 kpc in all our simulations.In practice, the integral over the mixing layer of an arbitrary variable  on the domain ML is done such that ∬ ML dd = ΔΔ × bool  , where the sum is done over all cells and where bool  = 1 for a cell temperature The brackets define the time log-average 7 , ⟨⟩ log 10 = 10 log 10 (  )/ (23) with  the total number of averaged variables   .The resulting net cooling emission is presented in Fig. 7 as a function of  for all simulations.The HD simulations generally exhibit the highest values of  net compared to the MHD and MHD+TC simulations for almost all  values.The net cooling emissions in the HD simulations are in relative agreement but lower compared to those reported by Mandelker et al. (2020a, Fig. 13) from their threedimensional simulations.The majority of their simulations correspond to our  s = 10 −2 cm −3 cases, but with a larger radius of  s = 3 kpc.Our  net values are relatively close to theirs after considering a factor of approximately 10 multiplication, which accounts in the time profile plot in Appendix A and Appendix D, the mixing layer in all simulations reaches a quasi-steady state (Equation 24), i.e., a roughly constant stream mass growth/loss, net cooling emissions, and magnetic field growth. 7In a few simulations with strong cooling (  < 10 −2 ), the stream grows in size, and the mixing layer is shifted to regions with lower resolutions, leading to excessive cooling due to the lack of resolution (see Fielding et al. 2020).The log-average allows us to smooth out the effect of those peaks, without changing results for all other simulations.Physically, this quasi-steady state is the balance between the flux of kinetic energy and enthalpy and the net radiative cooling rate.In the hydrodynamic case, the quasi-steady state can be described as, which, upon integration over the mixing layer, yields, with  referring to the surface between the stream and the background, and  cool the cooling emission in the mixing layer.To obtain  in , we first compute the stream mass growth rate  s from the simulations (later shown in equation 28), and then considering Equation 8, we compute the simulation's  in,sim taking the surface of a cylindrical stream with  = 32 s × 2 s kpc 2 .The mixing velocity  in is then directly fitted from the simulations values  in,sim obtained using Equation 8. From the fit we obtain, Both models can roughly reproduce the expected emissions of the HD simulations, although they exhibit discrepancies.Our model shows a mean factor difference of ∼ 1.5 and a maximum factor difference of ∼ 7.3.In contrast, the models presented by Mandelker et al. (2020a, equations 31 and 37) appear to exhibit a flat trend.This difference can be attributed to their utilisation of different definitions for the relevant timescales8 .Furthermore, their model assumes that the cooling emission at late times is primarily a result of thermal energy loss by the hot CGM gas.In contrast, in our cases, we consider the energetic quasi-steady state within the mixing layer, linking the cooling emission to the mixing layer's gas enthalpy.
To investigate the stream mass evolution, we calculate the total cold mass in each simulation.The cold stream mass is defined by the gas below a temperature threshold  d = ( s  mix ) 1/2 9 .The final mass  s,final is obtained through integration over all domains.Fig. 8 displays the final stream mass normalized by the initial value for all simulations.Once again, the HD values represent the maximum case of mass growth and loss among simulations with  < 1 and  > 1, respectively.Differences between HD and MHD/MHD+TC simulacooling time over the stream sound-crossing time.Such cooling time scale removes the dependency on , while the sound-crossing time is independent of M, leading to the flat trend. 9The impact of varying this threshold temperature on the cold mass is found only to affect our results by a few percent.tions are less pronounced than for the cooling emissions, showing a maximum difference of ∼ 0.4  s,0 .Note that, for simulations with  > 1, i.e., the disrupting stream regime for HD and MHD simulations, some simulations have a final stream mass very close to their initial value, indicating that MHD and MHD+TC in some cases can help the stream to stabilize against KHI.
We compare our numerical results to analytical models in Fig. 8. Thanks to our fitted value of  in from equation 26, we recover the steady mass growth  s from Equation 8.The theoretical final stream mass is then obtained as,

Impacts on the cooling emission and stream mass
The net cooling emissions,  net , for our simulations are plotted in Fig. 9.To reduce the number of points displayed for clarity, each point represents an averaged value over the three Mach number M = 0.5, 1, 2 for each set of simulations, i.e., each row of Table 2.The scatter represents the maximum and minimum values across the simulations with the three Mach numbers.A small random offset is added in the abscissa for clarity.
MHD starts to impact the cooling emission  net at  ∼ 0.1.Above this value, the emission gradually decreases for increasing , which can be directly linked to the reduction of the amount of gas in the mixing layer seen in Fig. 5. Across all  values, the MHD simulations with  = 0 • show identical  net compared to the HD ones, while the decrease reaches a factor 10 difference for MHD simulations with  = 90 • .
Similarly to the MHD simulations, MHD+TC also starts to impact the emission  net at  ∼ 0.1, causing a further decrease.In the condensing stream regime ( < 0.3), thermal conduction does not significantly affect the emissions, except when  = 90 • and  ∼ 0.06.In this case, the cooling emission is reduced by a factor of 20 compared to the HD case, but, as indicated by the scatter, such a decrease only occurs for a specific Mach number (M = 2).The scatter can be attributed to the influence of the stream velocity on the magnetic field growth, as one shall see in the next section.
In the intermediate Regime (0.3 <  < 8), thermal conduction further reduces the cooling emissions of MHD+TC by up to a factor of ≳ 2 compared to MHD case, for simulations with  ≠ 0 • .As can be seen in the temperature maps in Fig. 6, and as expected from the definition of the intermediate regime in Sec.2.5, thermal conduction diffuses the mixing layer leading to a smaller amount of gas in the temperature range  s - mix that can efficiently cool and radiate.Notably, in this regime, thermal conduction also reduces the dependence of  net on M compared to the MHD cases.In the diffusing stream regime ( ≳ 8), the stream exhibits a slight enhancement of its emission for  ≠ 0 • compared to the MHD+TC simulations at  ∼ 3.As the stream is diffusing, a large amount of gas is heated above the radiative temperature equilibrium in the stream,  s ∼ 10 4 K, resulting in a higher amount of gas that can radiate.As shown in Appendix B, in the condensing stream and intermediate regimes, the dominant cooling processes come from hydrogen.However, in the case of the diffusing stream regime, the emission from hydrogen actually decreases and represents ∼ 4-10% of the total cooling emission.The increase in the total cooling emission is due to Helium and numerous metals such as Oxygen and Neon.
We also compute the mean stream mass growth/loss, to provide a better representation of the stream mass evolution, where the ⟨⟩ indicates the linear arithmetic averaging over time from the start of the quasi-steady state of the mixing layer to the end of the simulation, the same time frame over which we took the log-average of  net .
The stream mass growth/loss  s is shown in Fig. 10 for HD, MHD, and MHD+TC cases.In the condensing stream regime ( < 1), MHD simulations exhibit a small decrease of the stream mass growth of the order of ≲ 10% compared to the HD simulations, except at  ∼ 0.06 where MHD simulations gradually decrease the stream mass growth with increasing .There are almost no differences between HD and MHD simulations for  = 0 • .In the disrupting stream regime ( > 1), MHD simulations have a significant impact, as streams with  ≠ 0 • do not experience mass loss compared to HD and MHD with a magnetic field parallel to the stream ( = 0 • ).
Thermal conduction affects the stream mass evolution only for  > 8, i.e., for a stream in the diffusing stream regime.As expected, in such a case, the stream diffuses into the CGM.However, the effective conduction can be greatly reduced by the initial magnetic field angle and the bending of the field line over time, resulting in a stable stream with almost no mass loss as M and  increases.

Magnetic field amplification
We hereby investigate the amplification of the magnetic field.Fig. 11 presents contours of the magnetic field strength and the density for angle  = 0 • and  = 45 • , at different  values.The contours are obtained from MHD+TC simulations with  = 30.They do not exhibit a qualitative difference from the MHD and/or  = 100 ones as long as  < 8, i.e., as long as they are outside the diffusing stream regime.In all contours, the magnetic field increases at the interface of the stream and the CGM and within the stream.
For a field initially parallel to the stream ( = 0 • ), the magnetic field lines are stretched by the eddies that arise from the KHI.As the  value decreases, the size of the mixing layer becomes smaller, leading to a slightly higher magnetic field amplification.
For  = 45 • , there is no significant difference in the magnetic field amplification across different  values.The field is amplified by up to a factor of ∼ 500 0 , which is higher than the  = 0 • simulations.This magnetic field increase can be explained by the presence of a component perpendicular to the stream, which is continuously stretched as the stream moves forward.
To quantify the growth of the magnetic field, the magnetic energy  m is first averaged within the stream and the mixing layer, i.e., below the threshold temperature  u defined in Sec.4.1.3, where the brackets ⟨⟩ represents the linear-arithmetic time averaging as in Equation 28.The results are plotted in Fig. 12 for the component  y .We focus only on this component because it is the one amplified by the velocity shear term which is the dominant amplification mechanism in our case 10 .Considering  x and plotting  does not significantly change our results.The simulations with  = 0 • and the one with  ≠ 0 • exhibit two distinct trends, as qualitatively observed in Fig. 11.Simulations with the magnetic field parallel to the stream reach a maximum of about 20-35 0 at  ≲ 2 × 10 −2 , followed by a decrease down to  ∼ 3 0 for  ∼ 10, following a roughly constant slope defined as  ∝  −0.16 .The scatter remains approximately constant along the slope and is primarily due to the variations in the Mach number M and the density ratio  among the simulations.In the case where the magnetic field has a component perpendicular to the stream ( > 0 • ), the mean magnetic field is amplified to ∼ 30-150 0 across  values, with solely the scatter between the points increasing with .Also, for a given  and  s , the magnetic field increases as  increases (i.e., for increasing M with fixed  and  s ).This trend is consistent with the fact that the growth 10 From the magnetic induction equation in Equation 13, taking our geometry and initial conditions, one can find that initially,  t  y ∼  x,0  x  y,0 is the dominant term when  ≠ 0. This is confirmed by Fig. 12 which shows that for  = 0 • , i.e.  x,0 = 0, the magnetic fields exhibit significantly lower amplification compared to  > 0 • simulations. of the magnetic field is mainly driven by the velocity shear at the interface of the stream and the CGM, for  ≠ 0 • .Assuming that the velocity difference between the stream and the CGM is the main driver of the field line stretching, we model the field amplification by approximating a field line as a stretching flux tube (see Spruit 2013, for example).The details of the models are derived and discussed in Appendix D. As a result, the magnetic field can be expressed as, where  = 10 5 represents the ratio of the thermal over the magnetic pressure, and the term  s /  s +  s,0  indicates the stretching of the field lines over time.For comparison, the average value of our model ⟨ th ⟩ is plotted in Fig. 12 where the  dependency of  th is enforced in equation 30 with  s,0 =  ×  s /( cool ) using equations 9 and 12.
Our model is consistent11 with the simulations where  > 0 • .It is worth noting that a few points for  > 0 • and  > 10 exhibit relatively small magnetic field growth.These points correspond to the MHD+TC simulation in the diffusing stream regime.As the stream diffuses in the CGM, the shear layer expands, which reduces the bending of the field lines due to the velocity difference, resulting in a smaller growth of .
The approximately 100-fold increase in the magnetic field leads to a reduced value of average  down to ∼ 10 from its initial average value of ∼ 10 5 .This is because the magnetic field is not drastically  amplified in the centre of the stream.The average thermal pressure remains constant in the stream centre.However, as observed in Fig. 11, the field can be amplified up to ≳ 200-500 0 near the CGM and stream interface, giving  ∼ 1 and a physical value  > 0.2 µG.This magnified magnetic field can then explain the stream stabilization in the disrupting stream regime, as depicted in the mass rate plot of Fig. 10.These results are consistent with Berlok & Pfrommer (2019), who found that for  ≳ 0.4 µG with an initial field parallel to the stream, the magnetic tension is strong enough to stabilize the stream against KHI.
Therefore, despite an initially low magnitude, the magnetic field can undergo significant amplification, thereby affecting both the  stream emission and its evolution, as long as the magnetic field is not parallel to the stream.

Turbulent velocity in the mixing layer
As mentioned previously, the mixing of CGM and stream gas is the key mechanism of the cold stream emission signature.In this section, we assess the turbulent velocity within the mixing layer using a mean-field approach.We describe below the procedure to compute the turbulent component of a given fluid value.Firstly, the values are averaged over the stream axis length, and for both sides of the stream axis to obtain a radial-dependent averaging, Next, the density-weighted averaged value is computed, as it is better suited for compressible flows (Favre 1969), The fluctuating field is recovered from the mean-field decomposition where  ′ (, ) =  (, ) −  (||).The root-mean-square value of the fluctuating velocity field inside the mixing layer is derived in practice from the turbulent kinetic energy, y , and where the mixing layer is defined between the temperature thresholds  d and  u (see appendix C for the threshold discussion).The mean mixing layer size is defined as Here,  ML represents the size of the mixing layer.It is important to note that our definition of the turbulent kinetic energy considers the mean radial velocity as part of the mean flow, so that the inflow velocity of the CGM gas  in is accounted for in the bulk motion (cf.Sec.2.3), rather than in the turbulent velocity term.The results are presented in Fig. 13 for all simulations except the MHD+TC simulations in the diffusing stream regime 12 .The turbulent velocity exhibits a clear decreasing trend as  increases.For  < 1, the general decrease observed for the HD simulations can be fitted as  ′ ML ∝  s,0  −1/5 , while for  > 1 we found a roughly constant value around 0.11 s,0 with a very weak dependency on .This decreasing trend is consistent with the analytical model of Tan et al. (2021, equation 42, Figure 12), which was derived from previous simulations of cold-hot interface geometry.From their simulations, they also found the relation cool .Furthermore, in line with the findings in previous sections, the HD simulations exhibit higher turbulent velocities than their MHD and MHD+TC counterparts.As observed previously, the presence of magnetic fields and thermal conduction hinders the growth of KHI for  ≠ 0 • , which directly correlates with the decrease of the magnitude of turbulence in the mixing layer.In terms of the magnitude of 12 For the MHD+TC simulations in the diffusing stream regime, the thermal conduction diffuses all instabilities, resulting in a negligible turbulent velocity  ′ ML < 10 −2  s,0 .For clarity, we have omitted these simulations from the plot.
turbulence, we found similar values to previous cold streams simulations from Mandelker et al. (2019, without radiative cooling) with  ′ ML ∼ 0.2 s,0 .However, our turbulent velocities are higher than in their simulations with radiative cooling (Mandelker et al. 2020a) which exhibit  ′ ML ∼ 0.2 s,0  −1/2 ∼ 0.02 − 0.03 s,0 .This difference could be attributed to the different domains of integration.In their case, they consider both the stream and the mixing layer, while in our case, we strictly confine the domain inside the temperature thresholds of the mixing layer.When including the stream in the integral domain of Equation 33, we find  ′ ML ∼ 0.6 s,0  −1/2 which is smaller than our value within the mixing layer.Their simulations are in three dimensions while ours are in two dimensions; therefore, we do not expect a perfect match in the magnitude of the turbulence.
We found that the impact of MHD or MHD+TC on the turbulence magnitude correlates with the stream mass evolution.As the angle  tends to 90 • , the strength of the turbulence decreases.We applied the fitting approach of the HD simulations in Fig. 13 to the combined sample of MHD and MHD+TC simulations, with  ≠ 0 • .We exclude from the fits the simulations with  = 0 • because their magnetic field does not increase enough to have a significant impact compared with the HD simulations.The resulting fit gives us, The decrease in the turbulence magnitude is evident from the fit.Both HD, MHD and MHD+TC fits start at a similar turbulence magnitude at  ∼ 10 −2 with  ′ ML ∼ 0.6 s,0 .However, the MHD and MHD+TC fit exhibits a steeper slope, resulting in a further decrease in the turbulent velocity down to  ′ ML ∼ 0.1 s,0 .The more pronounced decrease in turbulent velocity illustrates the direct impact of MHD and TC as they dampen the KHI growth.In the case of a magnetic field, the amplified field creates a tension force that counteracts the KHI growth, hence, stabilising the stream.For thermal conduction, the diffusion of either the mixing layer or the stream can further reduce the KHI growth, thereby stabilising the stream against KHI and reducing the mixing of the stream and the CGM.

DISCUSSION
In this section, we extend our findings to the cosmological context of cold streams entering the halo of massive galaxies.We first discuss Ly emission within the halo (Sec.5.1), followed by the properties of the cold streams penetrating the halo (Sec.5.2).Lastly, we address various limitations and caveats of our work (Sec.5.3).

Emission inside the halo
In Section 4.1, we discussed the emission properties of a cold stream with a fixed radius of  s = 1 kpc at the virial radius  v = 100 kpc of a massive galaxy residing in a 10 12 M ⊙ halo.As the stream survives and penetrates deeper into the halo, it becomes denser leading to an increase in its emission by a factor 10 2 − 10 3 (Mandelker et al. 2020b).
In the condensing stream regime, characterised by high density, high metallicity, and/or large  s , the impact of magnetic fields and thermal conduction on the stream's emission is negligible.However, in the intermediate or diffusing stream regimes, where the stream exhibits low number density, and/or low metallicity, and/or a small radius, the presence of magnetic fields and thermal conduction leads to a significant reduction in the stream emission by a factor of 10−20.
Our analysis reveals that, on average, the hydrogen contribution to the stream's emission is ∼ 20%, with the highest cases reaching around 45% (see Appendix B).Given that the cooling emission is predominantly collisional, it is expected that the Ly emission contributes to ∼ 50% of the total hydrogen cooling rate (Dĳkstra 2017, see figure 7).Therefore, for MHD+TC simulations (considering only cases with  ≠ 0 • , which are more realistic than a purely parallel magnetic field), the total net emission from a cold stream near the galaxy (0.1 v from the halo centre) is in the range of  net (0.1 v ) = (100-1000) ×  net (1 v ).The subsequent Ly emission in the vicinity of the galaxy should be, 10 38 -5 × 10 41 erg s −1 for  < 0.3, 4 × 10 37 -2.5 × 10 40 erg s −1 for  > 0.3.(35) with  H = 0.2-0.45 the hydrogen contribution to the total net cooling emission, and  Ly = 0.5 representing the Ly contribution to the hydrogen cooling emission.The two cases  < 0.3 and  > 0.3 refer to the condensing stream regime, and the intermediate and diffusing stream regimes, respectively.The stream's emission originates from the cooling layer, and as the stream radius increases, the volume occupied by the mixing layer surrounding the stream also increases.Consequently, the net emission  net scales with  2 s , i.e., the cross-sectional area of the stream.For  s = 10 kpc, this implies that  Ly ∼ 4 × 10 39 -2.5 × 10 42 erg s −1 for a stream in the intermediate or diffusing stream regime, and  Ly ∼ 10 40 -5 × 10 43 erg s −1 for a stream in the condensing stream regime, where in both case it is assumed that the stream remains in the regime it was with  s = 1 kpc.
Compared to the analytical model13 of Mandelker et al. (2020b), our cold stream emissions from MHD+TC are very similar for the dense and/or thick streams but is smaller by a factor of ∼ 100 for diffuse and/or thin streams.This difference is due to both a lower  net and a lower  H14 .It is worth noting that previous studies, such as Goerdt et al. (e.g. 2010); Mandelker et al. (e.g. 2020b), have often considered the feeding of galaxies by two or three prominent cold streams.Such a picture fits well with current cosmological simulations.However, more recent simulations with higher resolution in galactic haloes (Hummels et al. 2019;Peeples et al. 2019;van de Voort et al. 2019;Bennett & Sĳacki 2020;Nelson et al. 2020) have revealed the presence of a substantial number of small-scale (< 1 kpc) cold structures within the CGM.These smaller features become particularly evident when the resolution is increased near CGM shocks resulting from galactic feedback processes (Bennett & Sĳacki 2020, see figure 5).From Bennett & Sĳacki (2020), the CGM features an almost a cold weblike structure which could increase the emission by a factor > 10.Consequently, the cold flow signature is not solely limited to large filamentary structures but encompasses the emission from a multitude of smaller, less dense streams.This implies that the overall cold flow emission can be significantly enhanced, by a factor of more than 10, due to the contribution of the potential numerous thin cold streams within the CGM.Therefore, the emission properties associated with cold flows are more diverse and complex than previously considered, highlighting the importance of accounting for the full range of cold structures within the CGM.

Properties of the cold stream inflow
We focus on the resulting properties of a stream entering a halo in terms of mass flux, metallicities, and magnetic field.The mean15 cold mass inflow rate  s is computed as, where  y = 32 kpc is the stream length, d = dd the cross-section perpendicular to the stream axis, and the integral is performed only for the gas defined as cold, with  <  d (see Sec. 4.1).The integration through  is written for unit clarity but disappears in practice when normalizing  s by its initial value.We then express the mass flow rate in units of the initial rate  s,0 =  2 s  s,0  s,0 = 0.01-1 M ⊙ yr −1 assuming a cylindrical stream of radius  s .
The metallicity in the stream is computed as a function of the stream mass as Here, we consider that if the stream grows, the additional cold mass is added to the stream with CGM metallicity, and if the stream loses mass, the remaining cold mass dwells at its initial stream metallicity.
The mean magnetic field is computed as in Equation 29and is presented in physical units at the final simulation time.
Fig. 14 illustrates the stream properties as a function of the CGM number density  cgm =  s  −1 , which roughly scales as  −0.5 .Higher density leads to stronger cooling emission and smaller  ratios, and vice versa.For the MHD and MHD+TC simulations, results are only shown for  ≠ 0 • , because there are no significant differences with the HD simulations for  = 0 • .In the left panel, the impact of MHD on the cold mass accretion rate is evident, as it helps to retain ∼ 80% of the initial mass flow in the low-density regime.This demonstrates that the presence of a magnetic field enhances the stability of the stream and sustains the cold mass accretion.
Thermal conduction only acts for very low density with  cgm ∼ 10 −5 cm −3 where the stream diffuses, resulting in a reduced cold mass accretion rate of ∼ 0.5  s,0 .However, as the conduction efficiency decreases with increasing velocity, the stream can maintain a cold mass accretion rate with ∼ 90% of its initial value even at such low densities for M = 2. Therefore, compared to HD simulations, MHD+TC simulations show no significant difference in a high-density CGM with ≳ 2×10 −4 cm −3 , but TC can help the stream to survive below this number density threshold.These trends in our results agree with simulations of cold clouds embedded in galactic winds from MHD simulations by Hidalgo-Pineda et al. (2023) and MHD+TC simulations by Brüggen & Scannapieco (2023), as well as resolution tests on cosmological simulations (Hummels et al. 2019;Nelson et al. 2020;Bennett & Sĳacki 2020) From the metallicity plots, higher CGM density (stronger cooling) leads to higher metallicity in the stream, up to  s,final −  s /  cgm −  s ∼ 60%.By lowering the mixing of the gas, both MHD and MHD+TC simulations lower the metal enrichment of the stream.This metal pollution is counter-intuitive to the ideal picture of cold streams being pristine and can support some of the observed (Bouché et al. 2013(Bouché et al. , 2016) ) or assumed (Giavalisco et al. 2011;Rubin et al. 2012;Martin et al. 2012;Zabl et al. 2019;Emonts et al. 2023) high metallicity of cold inflow.From a cosmological simulation point of view, a higher metal enrichment of the cold inflow would also lead to higher star-formation efficiency.Combined with the cold stream's prolonged stability in the hot CGM, one may expect the star formation of massive galaxies to be sustained for a longer time.
In the case of high-density CGM, the magnetic field in the stream can undergo a significant enhancement, reaching ∼ 1 µG.However, as the CGM density decreases, the magnetic field strength diminishes, reaching a value of 2×10 −2 µG for MHD simulations.For MHD+TC simulations with  cgm ∼ 10 −5 cm −3 , the magnetic field within the cold stream remains close to its initial value as the stream gradually diffuses.
From a cosmological simulation perspective, one may expect the magnetic field to increase again by compression once it reaches the ISM.Such magnetized cold inflow may take a longer time to collapse as they could be magnetically supported.
The simplistic extrapolation of our findings raises an intriguing question about the potential oversights in cosmological simulations resulting from their resolution limitations.In addition to the prolonged sustenance of cold stream inflow, there is an important transformation in the properties of the cold material itself.It transitions from pristine cold streams to metal-enriched magnetized cold streams.

Caveats: 2D vs. 3D and additional physics
The first limitation of our work is that our simulations are conducted in a two-dimensional domain.In three dimensions, the KHI is expected to grow faster due to the appearance of additional instability modes.This faster growth results in a more efficient mixing between the stream and the surrounding medium, thereby altering the evolution of the stream and its observable properties (Padnos et al. 2018;Mandelker et al. 2019).Enhanced mixing would potentially lead to stronger emission and mass growth rates for the streams in the condensing stream regime ( =  cool / sh < 1).
From MHD simulations with a magnetic field aligned with the stream, Berlok & Pfrommer (2019) found that three-dimensional simulations exhibit an increased mixing.This is primarily driven by the growth of azimuthal KHI modes which are not inhibited by any magnetic tension force when the field is parallel with the stream.In future work, we will investigate the impact of magnetic fields not parallel to the stream using three-dimensional simulations.
Additionally, a more realistic model should include additional physics.For example, our model ignores self-gravity, which may affect the stability of the stream (Aung et al. 2019).We also adopt a cooling-heating function (Fig. 1) assuming the gas is optically thin.However, the function should vary with the stream density because dense cold streams will self-shield against the UV background radiation.Since the gas temperature and density affect the importance of self-gravity in the stream (see Ostriker 1964;Aung et al. 2019), a detailed treatment of radiation heating will be important in evaluating the stability of the cold stream.We hypothesize that the self-shielding may not significantly affect the emission from the mixing layer because it is dominated by collisional cooling.

CONCLUSIONS
Recent advancements in idealized high-resolution simulations have contributed to our understanding of cold streams and their emission signatures.To further enhance this knowledge, we conducted an extensive suite of two-dimensional simulations incorporating key physical processes, including radiative cooling, magnetic fields with varying angles, and anisotropic thermal conduction.The combination of these physics has not been explored comprehensively before.The simulations were performed using the Athena++ code and did not account for self-gravity or self-shielding.
In our idealized simulations, we focused on a cold stream situated at the virial radius of a 10 12 , M ⊙ halo at a redshift of  = 2.We consider a stream of radius  s = 1 kpc and an initial magnetic field defined by the ratio of thermal pressure over magnetic pressure  = 10 5 .We summarise our findings in a schematic illustration in Fig. 15.In the HD case, thick or dense streams survive (condensing stream regime) while thin or diffuse streams disrupt within the CGM due to KHI (disrupting stream regime).Ly emissions originate from the mixing layer in both cases.In the MHD+TC case, thick dense streams persist, providing magnetized cold material to the galaxy (condensing stream regime).For thin diffuse streams, the combined effects of magnetic field amplification and thermal conduction stabilize the stream against Kelvin-Helmholtz instability (KHI), resulting in significantly reduced Ly emissions (intermediate regime).If the stream is too thin or diffuse, it may evaporate within the CGM, potentially impeding its reach towards the central galaxy (diffusing stream regime).
Cold streams regimes: By including thermal conduction, the behaviour of the stream can be categorized into three regimes (see Sec. 2.5), depending roughly on the ratio  =  cool / sh , with  cool the cooling time defined from the mixing layer (Equation 5), and  sh the shearing time (Equation 9).
(1) The diffusing stream regime ( > 8) corresponds to the thinnest and/or diffuse cold streams in Fig. 15.In this regime, thermal conduction dominates over other processes, impeding the growth of KHI and causing the stream to diffuse within the CGM.A faster stream can however significantly reduce the efficiency of the thermal conduction while still stabilizing it against KHI, allowing it to potentially reach the central galaxy.
(2) In the intermediate regime (8 >  > 0.3), radiative cooling can overcome thermal conduction within the stream, but not in the mixing layer, which continuously diffuses in the CGM.
(3) The condensing stream regime ( < 0.3) involves dense and/or thick cold streams as shown in Fig. 15.In this scenario, cooling is highly efficient, leading to the condensation of CGM gas on to the stream.The key distinction from the hydrodynamic (HD) case in the condensing stream regime is that the stream becomes magnetized upon reaching the central galaxy.
Emission signature: In the intermediate and diffusing stream regimes, the emission signature experiences a significant decrease in the MHD+TC case compared to the HD case, by a factor of up to 20 (see Sec. 4.1).This reduction in emission is attributed to two factors: the amplification of the magnetic field at the stream interface and the diffusion of the mixing layer, which is the source of the cooling emission.In the diffusing stream regime, the emitting gas becomes hotter, increasing the cooling emissions from metals but a decrease in those from hydrogen.In the condensing stream regime, the impact of thermal conduction and magnetic fields on the stream's emission is insignificant.We observed that, outside the diffusing stream regime, approximately 20% of the stream's cooling emission originates from hydrogen, which is lower than previous estimations found in the literature.This further diminishes the expected Ly luminosity of cold streams.
Cold stream evolution: In the intermediate regime, the presence of magnetic fields and thermal conduction effectively suppresses the growth of the KHI (Sec.4.2.2 and 4.2.3).As a result, the stream remains stable and does not experience mass loss, allowing it to survive for longer periods compared to the hydrodynamic case (Sec.4.1).In the diffusing stream regime, although the stream undergoes mass loss, the efficiency of thermal conduction decreases significantly as the stream velocity increases (see Sec. 4.1.2).As a result, streams with a Mach number M = 2 experience only minimal mass loss.It is worth noting that, similar to the emission signature, the presence of magnetic fields and thermal conduction has negligible effects on the evolution of the stream in the condensing stream regime or when the magnetic field is parallel to the stream ( = 0 • ).
Cosmological implications: By extrapolating our findings from idealized simulations to a cosmological context (Sec.5), we determined that the Ly luminosity of cold streams within haloes falls within the range of 4 × 10 37 erg s −1 to 2.5 × 10 40 erg s −1 , specifi-cally for relatively small cold streams with a radius of  s = 1 kpc.Furthermore, we observed that the inflowing gas in these streams becomes enriched with metals and is magnetized, with the mean magnetic field strength in the stream reaching approximately 1 µG.These results provide insights into the properties and characteristics of cold streams in the cosmological context.Table A1.Simulation runs used for convergence test, along with the relative error of the stream mass expressed as a percentage of the fiducial value (finest grid case).In the last row, the simulation with TC in the diffusing stream regime is shown, where the scaling of ℎ is removed.Note that a good convergence is achieved when the scaling of ℎ is removed, as opposed to when it is kept.The resulting relative error, expressed as a percentage of the fiducial value (finest grid case), is presented in Table A1 for the stream mass.In all cases and when considering the no-scaling case for the MHD+TC simulation in the diffusing stream regime ( = 10 1.3 ), the mean error decreases with increasing resolution reaching a maximum of ∼ 7% by Δ min =  s /32.The purely MHD simulation in the disrupting stream regime ( ∼ 10 1.3 ) exhibits a relatively high error percentage at Δ min =  s /32.However, for all the other simulations, the errors are below a few percent, indicating a relatively good convergence.

Runs
The results obtained with and without scaling of the parameter ℎ (corresponding to the last two rows in the table) exhibit significant differences.In this specific case, removing the scaling of ℎ results in the best convergence.This finding confirms that the broadening of the smooth interface between the stream and the CGM is responsible for the discrepancy observed in Fig. A1.
In conclusion, our analysis indicates that all simulations, except for the MHD simulations in the disrupting stream regime ( ≫ 1), demonstrate good convergence with relative errors below a few percent.For the MHD simulations in the disrupting stream regime, the fact that the simulations do not yet fully converge may change within 7% percent the quantitative conclusion of our work for this particular case.

APPENDIX B: NATURE OF THE COOLING EMISSION
Complementary to the discussion in Sec.5.1, we hereby describe the contribution of Hydrogen to the total net cooling emission.From the cooling model described in Sec.2.2, we compute the ratio of the cooling emission from hydrogen over the total cooling emission for the mixing layer.This ratio is important for estimating the portion of the emission associated with Ly.Both cooling emissions are computed using equation 22.
Fig. B1 compares the ratio  H,net / net between HD and MHD, and between HD and MHD+TC simulations as a function of  =  cool / sh .For HD simulations, the ratio remains roughly constant across  with a value ∼ 0.2, showing little dependency on the stream velocity.The MHD simulations impact this ratio differently depending on the stream regime and for angles  ≠ 0 • .In the condensing stream regime ( < 1), the hydrogen contribution is slightly lower, while in the disrupting stream regime ( > 1), the hydrogen contribution increases to around 0.3.It is worth noting that for increasing , the stream velocities also impact the hydrogen contribution, resulting in  H,net / net up to ∼ 0.45.Thermal conduction does not significantly alter the results in the condensing stream regime.In the intermediate regime, MHD+TC appears to raise  H,net / net ≲ 0.4 as  increases.Finally, in the diffusing stream regime, the hydrogen contribution in the cooling emissions drastically decreases to the range of 0.03-0.1,depending on the stream velocity and the angle , as both parameters directly act on the conduction efficiency.

APPENDIX C: TEMPERATURE THRESHOLDS
We hereby checked that the temperature thresholds introduced in Sec.4.1 are acceptable to compute the stream mass variables and the mixing layer variables.
In Fig. C1, we show the total mass of gas under a given temperature  for HD, MHD, and MHD+TC simulations at M = 1 for density ratios  =  s / cgm = 100, 30.In each panels are plotted the thresholds  l = ( s  mix ) 1/2 ,  mix , and  u = ( cgm +  mix )/2.The is also plotted beneath each panel.For all curves, the mass encompassed inside the stream, the mixing layer and the CGM are clearly identifiable.The first rise of the mass slightly after   results from the stream gas.The relatively flattened phase represents the gas in the mixing layer.The final rise around  cgm =  s comes from the gas in the CGM.The threshold mass  s ( l ) seems acceptable as in all cases, variations of the threshold to temperatures 2 s or 3 s would change the mass by a few percent.Such an argument can also be made for the threshold  u .The only drawback is that such a threshold might introduce a systemic error discriminating mixing layer gas as stream gas.However, such error seems relatively small, and as it is systemic, it would not change significantly our results qualitatively.difference discussed in Sec.4.1.The simulations with  = 30 are in the condensing stream regimes, thus the similar results between HD, MHD and MHD+TC cases.However, for  = 100, the simulations are in the intermediate regime (MHD+TC) or the condensing stream regime (HD, and MHD).The impact of thermal conduction is directly visible as for the MHD+TC simulations, the mixing layer phase is completely flattened, meaning the absence of gas in this phase due to its diffusion which then causes the decrease of cooling emission in Fig. 9.

APPENDIX D: MODELLING OF THE MAGNETIC FIELD GROWTH
We hereby describe the modelling of the magnetic field amplification due to the shear velocity, which is the main mechanism responsible for the magnetic field growth shown in Sec.4.2.2.As our magnetic field is initially uniform, a simple way to consider the growth of the magnetic field is to focus solely on the direction of a magnetic field line and model it as a flux tube (Spruit 2013) with an initial density  i () and length  i .The total length of the field line can be approximated as  () ∼  i +  s,0  with the elongation occurring only near the interface of the stream and the CGM such that  i / s ∼ 1.We focus here on the portion of the field line, which is then elongated.As the stream moves forward, the line will bend at the interface between the stream and the CGM.We assume that this bending only happens for an initial portion of the line of length  s located between each side of the interface [0.5 s , 1.5 s ] 16 .As the field is frozen into the flow, the total mass along the field line and the magnetic flux should remain constant,  () =  (, )  ()  () =  0 , and, Φ =  ()  () = Φ 0 , (D1) where () is the cross-section of the tube.
The total pressure should remain constant as  T =  +  2 /2 =  0 +  2 0 /2.Considering a cold stream without perturbations, we can also assume the tube to have a steady temperature  (, ) =  0 ().Using the ideal gas law and introducing equation D1 into the total pressure term, one may obtain the following solution for the magnetic field growth, where  =  0 /  2 0 /2 = 10 5 represents the ratio of the thermal over the magnetic pressure.It is important to note that in Equation D2, the time dependency is contained inside .
We compare the model with the results of the simulations in Fig. D1.The magnetic field from the simulations is averaged inside the stream and the mixing layer as explained in Sec.4.2.2.For clarity, the figure shows only the simulations where the magnetic field growth is expected to be the highest and lowest, i.e., the highest and lowest stream velocities, which correspond to simulations with M = 2,  = 100 and M = 0.5,  = 30, respectively.The flux tube model is in relatively good agreement with the simulations.It captures the initial rapid growth up until the asymptotic behaviour after  ∼ 5 0 .Notice the impact of the velocity which also shortens the time needed to reach the asymptotic trend.
The discrepancies between our simplified flux tube model and the actual magnetic field evolution in the simulations can be attributed to several factors.
Firstly, our model assumes that the magnetic field grows infinitely and uniformly in the stream.In reality, the field growth is constrained by the energy equipartition, where  2 /2 ∼ Δ 2 /2 and is not uniform, with stronger growth as we approach the interface between the stream and the CGM.In the mixing layer, equipartition gives a constraint of  ≲ (300-500) ×M, which is either higher or similar to the maximum magnetic field observed in our simulations.Because our model reflects the mean magnetic field in the stream, it does not reach such maximum value and thus misses the point at which it should remain constant.Also, in realistic scenarios, the magnetic field can be amplified by turbulent eddies and compression from condensation.These additional mechanisms of magnetic field amplification are not accounted for in our simple model, leading to a faster asymptotic 16 This is relatively consistent with our simulations, as in Fig. 11, for a given stream cross-section, the amplification happens around the interface, and it does not occur near the stream centre.trend in the simulations compared to the model.Furthermore, the simulations may exhibit non-monotonic behavior of the magnetic field, whereas our model assumes a constant magnetic field growth.Factors such as the deceleration of the stream, numerical magnetic diffusion, and the broadening of the mixing length can all influence the growth and behavior of the magnetic field, causing deviations from the idealized model predictions.

Figure 1 .
Figure 1.Net cooling-heating map from our CLOUDY model at a metallicity of  = 0.1 ⊙ .Left panel: the blue and red colormap refer to cooling and heating scaling, respectively.The white dashed line represents the heating-cooling curve at the fixed density of the right panel.Right panel: Net heating-cooling curve at a fixed number density  H = 10 −3 cm −3 .Plain lines and dashed line represents cooling and heating contributions, respectively.Colored lines represent the main species contributing to the cooling under 10 6 K.

Figure 2 .
Figure 2. Contours of the ratio  =  cool / sh (dashed black lines) in log-scale as functions of the density ratio of the stream over CGM, and stream number density.The value log 10 (  ) is displayed on each contour line.The cold stream is assumed to be at the virial radius of a  h = 10 12 M ⊙ halo at  = 2, with  s = 1 kpc, metallicity  s / ⊙ = 10 −1.5 , Mach number M cgm = 1, and CGM metallicity  cgm / ⊙ = 10 −1 .The blue and red curves show the contours for which the diffusion times  s and  p equal  cool .Those limits define three different regimes for the cold stream evolution in the presence of TC: diffusing stream, intermediate, and condensing stream regimes.

Figure 3 .
Figure3.Illustration of the initial conditions of our simulations.The stream region is dense and cold, while the CGM is diffuse and hot.The angle  is the initial angle of the magnetic field with respect to the stream.The stream is moving downward inside the fixed CGM background.The direction of the initial temperature gradient is also shown.

Figure 4 .
Figure 4. Maps of the gas temperature in the HD simulations at different times for the sonic case (M = 1), with a density contrast  = 100, and for three different values of the  ratio (cooling time over shearing time).Each panel shows the 32 kpc full stream axis length in the horizontal direction and a zoomed region of 16 kpc in the vertical direction.The contours displayed are rotated by 90 • compare to the illustration in Fig. 3.

Figure 5 .
Figure 5. Maps of gas temperature in the MHD simulations with  = 0 • , 45 • , 90 • for the sonic case (M = 1), with a density contrast  = 100 and at a fixed time  = 11  0 .The top, middle, and bottom rows correspond to the three different regimes of  .Each panel shows the 32 kpc full stream axis length in the horizontal direction and a zoomed region of 16 kpc in the vertical direction.
with  = 22 0 as a final simulation time.The predicted final mass from our model and the one fromMandelker et al. (2019), for  > 1, andMandelker et al. (2020a), for  < 1, align relatively well with the results for the HD cases.For simulations  > 1, the theoretical prediction does not hold as it assumes condensation of CGM gas on to the stream.Instead, we showcase the expected mass loss rate based on the deceleration of the stream  s () fromMandelker et al. (2019,  equation 10 and 38)  for two dimensions.Quantitatively, the predicted mass loss rate also roughly agrees with the simulated values.

Figure 6 .
Figure 6.Maps of gas temperature in MHD+TC simulations at different times for the sonic case (M = 1) with an initial magnetic field angle  = 45 • , a density contrast  = 100, and three different values of  ratio (cooling time over shearing time) corresponding to the three different stream regimes in the presence of TC, i.e. diffusing stream, intermediate, condensing stream regimes (see Sec. 2.5).Each panel shows the 32 kpc full stream axis length in the horizontal direction and a zoomed region of 16 kpc in the vertical direction.

Figure 7 .
Figure 7. Average cooling emission in the mixing layer for all simulations.Red points are the MHD and MHD+TC simulations, and black points are the HD simulations.Analytical models using our fitted value of the condensation speed,  in , and models from Mandelker et al. (2020a) are shown by the purple and orange lines, respectively.Detailed distribution of the points: Filled and open circles represent simulations with  = 30 and 100, respectively.From left to right, the vertical grey dotted lines delimit simulations with  s = 10 −1 , 10 −2 , 10 −3 cm −3 .Finally, for a subgroup with a given  and  s , points are differentiated from left to right by their Mach number M = 0.5, 1, 2.

Figure 8 .
Figure 8. Final stream mass for all simulations.Red points are the MHD and MHD+TC simulations, and black points are the HD simulations.Analytical models using our fitting value of the condensation speed,  in , and models from Mandelker et al. (2019, 2020a) are shown by the purple and orange lines, respectively.The orange lines shows models from Mandelker et al. (2019) for  > 1 and the one for Mandelker et al. (2020a) for  < 1.The detailed distribution (grey dashed lines, filled and open circles) of the points is the same as in Fig. 7.

Figure 9 .
Figure 9. Net cooling emission in the mixing layer.Left panel: Comparison between HD and MHD simulations.The vertical black dashed line represents the boundary between the condensing stream regime (  < 1) and the disrupting stream regime (  > 1).Right panel: Comparison between HD and MHD+TC simulations.The dashed black lines indicate the limit between the condensing stream regime (  < 0.3), the intermediate regime (0.3 <  < 8), and the diffusing stream regime (  > 8).Each point shows an averaged value over the three Mach numbers M = 0.5, 1, 2, with the error bars representing the range of minimum and maximum values.A small random offset in the x coordinate is included for clarity.

Figure 10 .
Figure 10.Stream mass rate in units of stream initial mass per  0 =  s / s .Left panel: Comparison between HD and MHD simulations.The black dashed line represents the limit between the condensing stream regime (  < 1) and the disrupting stream regime (  > 1).Right panel: Comparison between HD and MHD+TC simulations.The dashed black lines indicate the limit between the condensing stream regime (  < 0.3), the intermediate regime (0.3 <  < 8), and the diffusing stream regime (  > 8).Each point shows an averaged value over the three Mach numbers M = 0.5, 1, 2, with the error bars representing the range of minimum and maximum values.A small random offset in the x coordinate is included for clarity.

Figure 11 .
Figure 11.Comparison of magnetic field (top) and density (bottom) in MHD+TC simulations with  = 0 • (field parallel to the stream) and  = 45 • at the half time  = 11  0 .From left to right, the contours show the simulations with decreasing  .Each panel shows the 32 kpc full stream axis length in the horizontal direction and a zoomed region of 16 kpc in the vertical direction.

Figure 12 .
Figure 12.Mean magnetic field  y normalized by  0 in the stream and mixing layer region for all MHD and MHD+TC simulations as a function of  parameter.The blue-navy points stand for simulations with initial magnetic field angle  = 0 • , and the orange points represent simulations with  ≠ 0 • (i.e., 45 • or 90 • ).The black dashed lines represent the time-average of our model value ⟨ th ⟩.The detailed distribution of the points (grey dashed line, filled and open circles) is the same as in Fig. 7.
dy , with the factor 0.5 accounting for the average of the mixing layers at both sides of the stream and where the integration in performed similarly to the one in equation 22.The integral in equation 33 is done in function of the radius  because the density-weighted averaged values in Equation 32 are a function of .

Figure 13 .
Figure 13.Turbulent velocity in the mixing layer.Red points are the MHD and MHD+TC simulations, and black points are the HD simulations.The thick dashed purple line shows a power-law fit for  < 1, and the thick dotted purple line is a roughly constant fit for  > 1.Both fit use only the HD simulations.The detailed distribution of the points (grey vertical dotted lines, filled and open circles) is the same as in Fig. 7.

Figure 14 .
Figure 14.Properties of the stream entering the halo are depicted in the three panels, each using a common legend.All quantities are plotted as a function of the CGM number density.Left panel: Cold mass accretion rate at the final simulation time in units of initial cold accretion rate,  s,0 =  p  s M cgm   2 s ∼ 0.01-1 M ⊙ yr −1 for  s = 1 kpc and  s = 0.001-0.1 cm −3 .Middle panel: Logarithm of the mean metallicity in the stream at the final time in Solar metallicity units.The initial stream metallicity is log 10  s,0 / ⊙ = −1.5, and the maximum value being the assumed CGM metallicity log 10  cgm,0 / ⊙ = −1.Right panel: Mean magnetic field strength in the stream ( s ).This value  s is reached in the early stage of the simulations at about  ∼ 0.1 end and then remains roughly constant until the final time.

Figure 15 .
Figure15.Schematic illustration comparing the fate of a cold stream in two scenarios: hydrodynamic (HD) and magnetohydrodynamic with thermal conduction (MHD+TC).In the HD case, thick or dense streams survive (condensing stream regime) while thin or diffuse streams disrupt within the CGM due to KHI (disrupting stream regime).Ly emissions originate from the mixing layer in both cases.In the MHD+TC case, thick dense streams persist, providing magnetized cold material to the galaxy (condensing stream regime).For thin diffuse streams, the combined effects of magnetic field amplification and thermal conduction stabilize the stream against Kelvin-Helmholtz instability (KHI), resulting in significantly reduced Ly emissions (intermediate regime).If the stream is too thin or diffuse, it may evaporate within the CGM, potentially impeding its reach towards the central galaxy (diffusing stream regime).

Figure A1 .
Figure A1.Convergence test results for both MHD (top panels) and MHD+TC (bottom rows) cases.Mass evolution of the stream in units of initial stream mass is shown as a function of time for different  values.

Figure B1 .
Figure B1.Ratio of the cooling emission from Hydrogen ( H ) over the total cooling emission for the mixing layer ( net ) as a function of  is shown.Top panel: Comparison between HD and MHD simulations.The vertical black dashed line represents the boundary between the condensing stream regime (  < 1) and the disrupting stream regime.Bottom panel: Comparison between HD and MHD+TC simulations.The black dashed lines represent the boundaries between the condensing stream regime (  < 0.3), the intermediate regime (0.3 <  < 8), and the diffusing stream regime (  > 8).Each point represents an averaged value over the three Mach numbers M = 0.5, 1, 2, and the error bars indicate the range of minimum and maximum values.A small random offset is applied in the horizontal direction to improve visibility and distinguish each point.
Finally, Fig. C can also give some useful insight about the physics

Figure C1 .
Figure C1.Mass threshold check.Mass in the all simulation domain encompassed inside under a given temperature in function of this temperature (upper subplot).The bottom curves show the relative error between our stream mass and the mass computed with the given temperature.(Top) HD, MHD, and MHD+TC simulations for  = 100.(Down) HD, MHD, and MHD+TC simulations for  = 30.In both panels, our chosen stream temperature threshold  l ,  mix and our CGM temperature threshold  u are plotted with plain, dashed and dotted black lines.The relative error with respect to the mass  s ( ) is shown beneath each panel.

Figure D1 .
Figure D1.Comparison of the magnetic field evolution between the simulations and the model from EquationD2.For clarity, we plot only MHD simulations, focusing on the maximum (orange) and minimum (blue-navy) expected growth cases, i.e., the cases with the highest and lowest stream velocities.All the lines for a same color are MHD simulations with the given  s,0 for  = 45 • , 90 • and  s = 10 −3 , 10 −2 , 10 −1 cm −3 (thus 6 lines per color).