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apart from stars , the baryonic matter within the galaxy is in the form of an extremely dilute , turbulent plasma known as the interstellar medium ( ism ) .
the multitude of physical processes within the ism entails a rich heterogeneous structure @xcite .
approximating radiative processes by a simplified cooling prescription , and restricting the computational domain to a local patch , the turbulent ism is now routinely modelled by means of three - dimensional fluid simulations
. one main focus of these simulations has been to obtain filling factors of the different ism phases and compare them to the classical predictions as well as observations ( e.g. , * ? ? ? * ) .
further topics of interest include turbulent mixing @xcite , thermodynamic distribution functions @xcite , and line - of - sight integrated column densities @xcite .
while various simulations discuss the influence of magnetic fields on the ism morphology , little is said about the actual mechanism of field amplification .
@xcite have addressed this question by means of unstratified simulations of sne turbulence .
the authors relate the growth of small - scale magnetic fields to vorticity production in supernova shocks @xcite , and chaotic field line - stretching @xcite .
the fact that vorticity production by colliding shells is almost inevitable in a clumpy and highly structured ism has first been pointed out by . the issue has then been investigated for the simplified case of driven expansion waves by @xcite and , more recently , by del sordo & brandenburg ( this volume ) .
considering turbulence driven by non - helical transverse waves , @xcite have shown that the small - scale dynamo becomes harder to excite in the super - sonic regime , albeit the critical reynolds number for the onset of dynamo action only seems to depend weakly on the mach number .
because the eddy turnover time is short at small scales , a dynamo based on chaotic field line stretching will be fast .
this is in - line with observations , which exhibit dominant turbulent fields .
open issues remain with respect to the mechanism governing the saturation of the small - scale dynamo .
therefore , it is currently unclear whether equipartition field strengths can be obtained by a non - helical dynamo alone .
alternatively , the turbulent field might be explained as a `` shredded '' coherent field , i.e. , as the by - product of a helical mean - field dynamo .
notwithstanding the above , the presence of the observed coherent fields on scales larger than the outer scale of the interstellar turbulence ( @xmath0 , see * ? ? ?
* ) clearly requires the presence of a coherent dynamo .
the favoured candidates for such a dynamo are , in no particular order : ( i ) the kinetic driving by sne , ( ii ) the buoyant cosmic ray - supported parker instability @xcite , ( iii ) the magneto - rotational instability , and ( iv ) gravitational interactions @xcite .
the last of these effects is certainly dominant at the early stages of galaxy formation and will provide a seed field for subsequent processes .
it remains to be shown , however , how important external interactions are in the presence of a realistic feedback from scales currently unresolved in cosmological simulations .
moreover , as have shown , the cosmic ray ( cr ) dynamo critically relies on the anisotropy of the cr diffusion coefficient .
combined simulations , including both sne and crs , will prove whether this anisotropy remains effective for strongly tangled turbulent fields
.    the mri will be important ( at least ) in the outer regions of galaxies , where the star formation activity is low @xcite  even under moderate turbulence , it may operate efficiently .
the stability criterion for a global isothermal disk of thickness @xmath1 leads to the relation @xmath2 for instability . depending on the local rotation frequency , @xmath3 , turbulent diffusivity @xmath4 , and turbulent magnetic prandtl number @xmath5
 which is generally expected to be of order unity  this value may be lower than the critical dynamo number for the supernova - driven dynamo
. it should be worthwhile to address this question within state - of - the - art simulations of the ism , with an alvfn velocity of the external vertical field of the order @xmath6 .     of the top half of the now larger simulation box ( _ upper panels _ ) , and the disc midplane ( _ lower panels _ ) .
the variables shown are : ( a ) number density @xmath7 $ ] , ( b ) column density @xmath8 $ ] , ( c ) temperature @xmath9 $ ] , ( d ) velocity dispersion @xmath10 $ ] , and ( e ) magnetic field strength @xmath11 $ ] .
the logarithmic grey scales cover ranges @xmath12 $ ] , @xmath13 $ ] , @xmath14 $ ] , @xmath15 $ ] , and @xmath16 $ ] , respectively . ]
the energy input through sne into the ism is tremendous .
the corresponding amplitude of the expected dynamo effect has first been estimated by @xcite .
assuming hydrostatic equilibrium and applying quasi - linear theory , these estimates have subsequently been refined by .
a shortcoming of the approach was the neglect of a possible galactic wind .
in general , there was a controversy as to whether the turbulence created by the sne was too vigorous to warrant dynamo action  either because of a too strong wind ( carrying the field away ) , or contrary to this , because of a too strong downward pumping ( enhancing turbulent dissipation near the midplane ) .    in a series of papers ,
analytically derived the electromotive force stemming from isolated remnants ; the line of work was also later supported by simulations of single remnants . to obtain the net @xmath17  effect ,
a convolution with an assumed vertical sn distribution was applied . however , the approach suffered from a too weak dynamo and highly dominant ( upward ) pumping . only when considering the stratified nature of the galactic disc , the issue was somewhat alleviated , although still predicting a strong upward pumping
.    pioneering semi - global simulations based on `` first principles '' where performed by @xcite , and it was only for the fact of a too low dynamo number that no field amplification was observed in their simulations .
the first direct simulations exhibiting dynamo action were reported almost a decade later .
the general morphology of such simulations is illustrated in figure  [ fig : slices ] ; note the apparent correlation between the density  ( a ) and magnetic field amplitude  ( e ) near the midplane , indicating field amplification via compression .
in contrast , away from the midplane ( i.e. , in the diffuse medium ) the field shows more folded structures .    .
the variables are plotted in dark ( @xmath18 ) or light ( @xmath19 ) colours , respectively .
shaded areas indicate @xmath20-fluctuations . in the region for @xmath21 , the results agree well with a previous run applying a much smaller box . ]
the fast growth of the dynamo can be understood rigorously in terms of mean - field theory , i.e. via a reynolds - averaged induction equation @xmath22\ , ,    \label{eq : mf_ind}\ ] ] and assuming a standard parametrisation @xmath23 with tensorial coefficients @xmath24 and @xmath25 . the determination of these closure parameters @xcite hugely benefited from the development of the so - called test field method @xcite . for kinematically forced turbulence , this method allows to determine unambiguously all eight tensor components  and , in fact , none of them can be ignored @xcite .
notably , we find a significant positive rdler effect in the off - diagonal elements of the diffusivity tensor ( see panel d in figure  [ fig : ae ] ) .
the measured @xmath17  effect and turbulent diffusion are of the expected sign and magnitude ( see figure  [ fig : ae ] , panels a and c ) .
the key finding is that of a moderate downward pumping ( panel b ) as predicted by .
this inward pumping has profound implications in compensating the effect of the equally strong upward mean flow ( not shown ) , thus yielding ideal conditions for the dynamo .
moreover , a wind may aid the shedding of small - scale magnetic helicity as discussed in @xcite .
one major concern with the local box approximation was the issue of its limited dimension in the horizontal direction . in the low - pressure ambient medium of the galactic halo ,
supernova remnants can easily expand to several hundred parsec in diameter .
because of the periodic boundaries and the finite domain size , this leads to ( spuriously ) correlated self - interactions .
the simulations of @xcite used @xmath26 , which was justified for the limited vertical extent of their model . for our standard
runs we applied @xmath27 together with a vertical box size of @xmath28 .
even with near @xmath29 horizontal box size , the issue of self - interaction was still seen in the far halo region . to eliminate the possibility of an artificial dependence of the results on the horizontal box size
, we carried out a fiducial simulation run at @xmath30 ( cf .
figures  [ fig : slices ]  &  [ fig : ae ] ) . in the region of overlap
, the extended profiles agree well with their counterparts derived from the smaller boxes ( cf . fig .
4.5 , model f4 in * ? ? ?
beyond @xmath31 , the velocity dispersion increases significantly , enhancing the turbulent diffusion and resulting in noisy profiles for the other coefficients .
note , however , the pronounced systematic @xmath17  effect in the region of strong vertical gradients near the midplane ( panels a and b in figure  [ fig : ae ] ) .
this clearly supports the paradigm of helicity production being the direct result of inhomogeneous turbulence .      beyond the purpose of mere diagnostics , the profiles shown in figure  [ fig :
ae ] serve as a foundation for global large eddy - simulations solving ( [ eq : mf_ind ] ) .
we believe that the semi - global approach of vertically stratified local boxes captures the essential physics behind the turbulent @xmath17  effect in the galactic disc . at the same time
, we are aware that the local geometry implies certain restrictions on the permitted dynamo modes .
identifying global symmetries is however necessary for a direct comparison with observations .    to warrant global mean - field modelling of the @xmath32  dynamo
, the scaling of the measured closure coefficients with the relevant input parameters has to be obtained .
this is important because the angular velocity @xmath3 , the local shear rate @xmath33 , the supernova rate @xmath34 , and the midplane density @xmath35 are functions of the galactocentric distance @xcite .
@xcite already made a first step in this direction by inferring the dependence on the supernova rate .
corresponding mean - field models have been reported in @xcite . further scaling relations with respect to @xmath3 and @xmath35
will be reported elsewhere .
first estimates ( subject to small number statistics ) for a range of 0.11 times the galactic star formation rate lead to the relations @xmath36 this has the consequence that the dynamo number @xmath37 scales inversely with the star formation rate .
therefore we would not expect a stronger amplification of the large - scale field for stronger star formation activity .
lastly , the pitch angle @xmath38 , in the kinematic regime , only seems to depend weakly on the studied parameters .
the strongest dependence here is , however , expected from the local shear rate @xmath39 , suggesting a dedicated parameter survey .
because current numerical simulations are limited to very moderate reynolds numbers , it is important to understand how efficient the observed mechanisms remain under realistic conditions .
to achieve this , it has proven fruitful to study simplified scenarios and run multiple parameter sets ( see * ? ? ? * for a comprehensive review ) .
fortunately , the emerging physical effects are dominated by the outer scale of the turbulence , i.e. , as soon as a rudimentary scale separation is achieved , the turbulent quantities should become independent of the actual micro scale .
the growing complexity of models challenges the distribution of computing time : increasing physical realism leaves little margin for the variation of key parameters , let alone convergence checks or running multiple representations of a single parameter set .
dedicated studies remain mandatory to segregate artificial trends from genuinely physical ones .
the dilemma becomes even more apparent when looking at the recent trend to performing `` resolved '' global simulations .
for these , convergence checks are a rare exception .
while an `` enhanced '' diffusivity may be sufficiently approximated by the numerical truncation error on the grid scale , the diamagnetic pumping term certainly is not .
yet vertical transport has profound implications on the emerging dynamo modes and growth rates .    in conclusion , we advocate a strategy that has been applied with great success in the design of aircraft , namely the concept of large eddy ( or mean - field ) simulations .
global fluid simulations currently can not guarantee scale separation for all relevant physical scales . to obtain quantitatively correct results
, we therefore believe that a sub - grid scale model is inevitable .
present local box simulations are a valuable means to provide a rigorous framework for the calibration of such a model .
the fractal shape and multi - component nature of the interstellar medium together with its vast range of dynamical scales provides one of the great challenges in theoretical and numerical astrophysics . here 
 we will review recent progress in the direct modelling of interstellar hydromagnetic turbulence , focusing on the role of energy injection by supernova explosions . 
 the implications for dynamo theory will be discussed in the context of the mean - field approach .    
 results obtained with the test field - method are confronted with analytical predictions and estimates from quasilinear theory . 
 the simulation results enforce the classical understanding of a turbulent galactic dynamo and , more importantly , yield new quantitative insights . 
 the derived scaling relations enable confident global mean - field modelling .