Large Eddy Simulations in Astrophysics
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Abstract
In this review, the methodology of large eddy simulations (LES) is introduced and applications in astrophysics are discussed. As theoretical framework, the scale decomposition of the dynamical equations for neutral fluids by means of spatial filtering is explained. For cosmological applications, the filtered equations in comoving coordinates are also presented. To obtain a closed set of equations that can be evolved in LES, several subgridscale models for the interactions between numerically resolved and unresolved scales are discussed, in particular the subgridscale turbulence energy equation model. It is then shown how model coefficients can be calculated, either by dynamic procedures or, a priori, from highresolution data. For astrophysical applications, adaptive mesh refinement is often indispensable. It is shown that the subgridscale turbulence energy model allows for a particularly elegant and physically wellmotivated way of preserving momentum and energy conservation in adaptive mesh refinement (AMR) simulations. Moreover, the notion of shearimproved models for inhomogeneous and nonstationary turbulence is introduced. Finally, applications of LES to turbulent combustion in thermonuclear supernovae, star formation and feedback in galaxies, and cosmological structure formation are reviewed.
Keywords
Large eddy simulations (LES) Turbulence1 Introduction
Turbulent flows with high Reynolds numbers are often encountered in computational astrophysics. Examples are the solar wind, stellar convection zones, starforming clouds, and probably the gas in galaxy clusters. This review concentrates on computational methods that treat turbulence in the limit of high Reynolds numbers by explicitly solving the compressible Euler equations for the largescale dynamics of the flow, while incorporating smallscale effects such as viscous dissipation into a subgridscale model. Since the nonlinear turbulent interactions between different scales are at least partially resolved, this type of simulation is called large eddy simulation (LES).
In a numerical simulation of turbulence, the range of length scales is limited by the grid scale Δ, which is simply the linear size of the grid cells. Only if Δ ≲ ℓ_{K}, turbulence can be fully resolved by a socalled direct numerical simulation (DNS).^{1} However, DNS become infeasible for very large Re because the total amount of floating point operations (FLOPs) increases with (L/Δ)^{4} ≳ (L/ℓ_{K})^{4} ∼ Re^{3}. The scaling may differ for highly compressible turbulence, but the basic problem remains the same. For a DNS of solar convection over one dynamical time scale, it would be necessary to perform very roughly 10^{42} FLOPs, which would take far longer than the current age of the Universe on the fastest existing computer.
A mathematical framework for LES is based on the notion of a filter, which separates largescale (ℓ ≳ Δ) from smallscale (ℓ ≲ Δ) fluctuations. Filters can be used to decompose the equations of fluid dynamics into equations for smoothed variables, which have a very similar mathematical structure as the unfiltered equations, and equations for secondorder moments of the fluctuations. The latter are interpreted as subgridscale variables. In Section 2, we will carry out the decomposition of the compressible NavierStokes equation by applying the filter formalism of Germano (1992). This formalism comprises the socalled Reynoldsaveraged NavierStokes (RANS) equations as limiting case if the filter length is comparable to the integral length scale of the flow. This method is equivalent to numerically solving a meanfield theory for turbulent flow. Simulations based on the RANS equations work with low Re_{eff}, while LES have high Re_{eff}. In principle, secondorder moments can be expressed in terms of higherorder moments. Since this would entail an infinite hierarchy of moments, the set of variables is limited by introducing closures. Usually, one attempts to find closures for the secondorder moments by expressing them in terms of the filtered variables. This is what is called a subgridscale (SGS) model.^{3} For example, a complete secondorder closure model for turbulent convection is formulated in Canuto (1994). Much simpler, yet often employed is the oneequation model for the SGS turbulence energy K, i.e., the local kinetic energy of numerically unresolved turbulent eddies. For this reason, it is sometimes called the Kequation model. Closures for the transport and source terms in the SGS turbulence energy equation are presented in some detail in Section 3, followed by a discussion of how the closure coefficients can be determined (Section 4). Of particular importance is the prediction of the local turbulent viscosity, which is is given by Δ√K times a dimensionless coefficient. The turbulent viscosity is required to calculate the turbulent stresses, which enter the equations for the filtered variables analogous to the viscous stresses in the unfiltered NavierStokes equations (see Section 3.1).
Filtering the dynamical equations is usually considered to be equivalent to numerical discretization. The filter length can then be identified with the grid scale Δ. Since the numerical truncation errors of finite difference or finite volume schemes for the computation of compressible astrophysical flows are more or less diffusionlike terms, they produce a numerical viscosity that effectively reduces the Reynolds number to a value comparable to Eq. (3). It is actually a common assumption that numerical viscosity approximates the turbulent viscosity on the grid scale. This leads to the notion of an implicit large eddy simulation (ILES) (Garnier et al., 2009), which is widely used for simulating turbulent flows in astrophysics. Numerous numerical studies demonstrated that ILES is a very robust method, which reliably predicts scaling laws of compressible turbulence at sufficiently high resolution (Sytine et al., 2000; Kritsuk et al., 2007; Benzi et al., 2008; Schmidt et al., 2008; Federrath et al., 2010b; Kritsuk et al., 2013). This is a consequence of the independence of inertialrange scaling from the dissipation mechanism, be it microscopic, turbulent or numerical viscosity, provided that the dynamical range of the simulation is large enough. In simulations of statistically stationary isotropic turbulence, however, the inertial subrange is very narrow for computationally feasible resolutions because the bottleneck effect distorts the spectrum over a large range of high wave numbers below the Nyquist wavenumber (Falkovich, 1994; Dobler et al., 2003; Schmidt et al., 2006a). It appears that LES with an explicit SGS model, such as the Kequation model, can reduce the bottleneck effect to some degree and reproduce scalings from ILES or DNS at lower resolution (Haugen and Brandenburg, 2006; Woodward et al., 2006; Schmidt, 2010). However, more systematic studies covering the parameters space of forced compressible turbulence are necessary to confirm this effect.
There are, of course, alternative methods of scale separation and a large variety of SGS models (for a comprehensive overview, see the monographs Sagaut, 2006; Garnier et al., 2009). An example are the CamassaHolm equations, which follow from the incompressible NavierStokes equations by decomposing the trajectories of fluid elements into mean and fluctuating parts in the Lagrangian framework (Chen et al., 1998). Since the filtered component of the velocity is defined by an inverse Helmholtz operator of the form (1 − α^{2}∇^{2})^{−1}, which is explicitly applied to determine the turbulent stresses in the filtered velocity equation, the resulting model is called Lagrangianaveraged NavierStokes αmodel (LANSα). Depending on the choice of α, the variables computed in LES based on LANSα are typically smoothed over length scales somewhat larger than the grid resolution. In other words, this type of simulation partially resolves the subfilter scales, which improves the controllability of the model. While there is no handle on the competition between the SGS model and numerical truncation errors on the grid scale in convectional LES, LANSα can, in principle, alleviate this problem by adjusting the balance between truncation and model errors (Pietarila Graham et al., 2007). Although the idea is very elegant, the numerical studies discussed in Pietarila Graham et al. (2007, 2008) show that the applicability of LANSα and similar models is limited, particularly for very high Re. Moreover, the generalization to compressible turbulence is not straightforward. Models such as LANSα are not further covered by this review, but they might be an option for magnetohydrodynamical LES (Pietarila Graham et al., 2009).
In astrophysics, LES are mainly applied to complex systems. In simulations of cosmological structure formation, which are discussed in Section 6.3, the length scales on which turbulence is driven by gravity are varying. Although adaptive mesh refinement is applied to track down collapsing structures, it is difficult to to resolve a wide range of length scales between the smallest driving scale and the grid scale at the highest refinement level. In this situation, SGS effects can become fairly large. However, the variable grid scale complicates the scale separation in AMR simulations because energy has to be transferred between the resolved and SGS energy variables if a region is refined or derefined. Section 5.1 describes how to combine LES and AMR. This method, for which the acronym Fearless (Fluid mEchanics for Adaptively Refined Large Eddy SimulationS) was coined in Maier et al. (2009), has been applied to galaxy clusters, the intergalactic medium, and primordial atomic cooling halos. The results from these simulations indicate that the contribution of the numerically unresolved turbulent pressure to the support against gravity is nonnegligible and the turbulent viscosity tends to stabilize disklike structures around collapsed gas clouds. Moreover, the SGS model provides indicators of turbulence production and dissipation and allows for the computation of the turbulent velocity dispersion. A difficulty is that turbulence production by cosmological structure formation is highly inhomogeneous. This entails the problem that the SGS model should dynamically adapt to conditions ranging from laminar flow to developed turbulence. Inhomogeneous and nonstationary turbulence can be treated by dynamical procedures for the calculation of closure coefficients or shearimproved SGS models, which decompose the numerically resolved flow into mean and fluctuating components. These techniques are outlined in Sections 4.2 and 5.2.
Furthermore, SGS models offer unique possibilities for modeling physical processes that are influenced by turbulence. An example is turbulent deflagration, where the turbulent diffusivity predicted by the SGS model dominates the effective flame propagation speed in underresolved numerical simulations. Turbulent deflagration plays a role at least in the initial phase of thermonuclear explosions of white dwarfs (see Section 6.1), which is one of the scenarios that are thought to produce type Ia supernovae. A recent application along similar lines are LES of isolated disk galaxies, where the SGS turbulence energy is a crucial parameter for calculating the star formation rate and the feedback due to supernova blast wave (see Section 6.2). Since the impact of feedback processes on the formation of galaxies and their evolution leaves many questions unanswered, galaxies are a particularly promising field of application.
While great progress has been made for compressible hydrodynamics, magnetohydrodynamical LES are still in their infancy. Several SGS models have been proposed in the context of terrestrial plasma physics (Müller and Carati, 2002a, b; Haugen and Brandenburg, 2006; Chernyshov et al., 2007; Pietarila Graham et al., 2009; Sondak and Oberai, 2012), but their applicability to astrophysical plasmas is unclear. Astrophysical MHD turbulence, particularly in the interstellar medium, extends to the supersonic and superAlfvénic regimes. Moreover, plasmas become collisionless for high temperatures and low densities. A typical example is the solar corona. It is also likely to be the case in the intracluster medium. Since the fluiddynamical description is not applicable in this case, kinetic methods have to be employed. Nevertheless, MHDLES could provide a reasonable approximation on length scales that are sufficiently large compared to the characteristic scales of kinetic processes. In any case, SGS models for MHD turbulence will be a very challenging problem because of the local anisotropy of turbulent fluctuations, the potentially strong backreaction from smaller to larger scales, and complicated dissipative processes such as turbulent reconnection (Brandenburg and Subramanian, 2005; Büchner, 2007; Zweibel and Yamada, 2009). In this area, extensive fundamental studies will be necessary.
2 Scale Separation
Large eddy simulations are based on the notion of scale separation. Although turbulence is a multiscale phenomenon, with interactions among different length scales, a separation into smoothed and fluctuating components can be rigorously defined by means of filter operators. Of course, the filtering of nonlinear terms gives rise to interactions between these components. Filter operators were originally applied in the context of meanfield theories, but can be generalized to LES. For incompressible hydrodynamical turbulence, Germano (1992) introduced a general framework that encompasses mean field theories as limiting case.
 The filter kernel is independent of direction:$$G(x  x^\prime ) = G(r),\quad {\text{where }}\,r = x  x^\prime .$$
 Filtering smoothes out fluctuations on length scales smaller than the filter length Δ_{ G }. Length scales that are large in comparison to Δ_{ G } are not affected. This implies$$G(x  x') \sim \left\{ {\begin{array}{*{20}c} {1/\Delta _G^3 } & { if \left {x  x'} \right \ll \Delta G,} \\ 0 & {if \left {x  x'} \right \gg \Delta G.} \\ \end{array} } \right.$$
 The filter operator is linear, conserves constants, and commutes with spatial derivatives:The simplest lowpass filter is the box or tophat filter. For Cartesian coordinates x_{ i }, the kernel of the box filter is defined by$${\langle \Delta q\rangle _G} = \nabla {\langle q\rangle _G}.$$Usually, Δ_{ i } is assumed to be equal for all spatial dimensions. The mean value of q in a rectangular domain with periodic boundary conditions follows in the limit that Δ_{ i } is the linear size of the domain in each dimension.$$G_{box} (x  x') = \prod\limits_{i = 1}^3 {G_i (x_i  x'_i ), where } G_i (x_i  x'_i ) = \left\{ {\begin{array}{*{20}c} {1/\Delta _i } & {if \left {x_i  x'_i } \right \leqslant \Delta _i /2,} \\ 0 & {otherwise.} \\ \end{array} } \right.$$(5)
2.1 Decomposition of the compressible NavierStokes equations
 Gravitational energy injection on subgrid scales:$$\Gamma =  \langle \rho \tilde u \cdot \nabla \phi \rangle + \tilde u \cdot \langle \rho \nabla \phi \rangle =  \langle \rho \tilde u \cdot \nabla \phi \rangle + \langle \rho \rangle \tilde u \cdot \nabla \langle \phi \rangle  \tilde u \cdot \gamma.$$(28)
 Rate of subgridscale turbulence energy production:^{6}where is defined by Eqs. (19) and \({\tilde S_{ij}}\) is the rateofstrain tensor associated with the Favrefiltered velocity:^{7}$$\Sigma = {\tau _{ij}}{\tilde S_{ij}},$$(29)$${\tilde S_{ij}}: = \frac{1}{2}\left( {\frac{{\partial \tilde u}}{{\partial {x_j}}}\frac{{\partial {{\tilde u}_j}}}{{\partial {x_i}}}} \right).$$(30)
 Rate of viscous energy dissipation in the limit of high Reynolds numbers:^{8}where is defined by Eq. (15), \(\vert{S^{\ast}}{\vert^2} = 2S_{ij}^{\ast}S_{ij}^{\ast}\) is the squared norm of the tracefree rateofstrain tensor \(S_{ij}^{\ast} = {S_{ij}}  {1 \over 3}d{\delta _{ij}}\) and d = S_{ ii }. Although the viscous stresses can be neglected in the filtered momentum equation, viscous dissipation is crucial for the energy balance of turbulent flows.$$\langle \rho \rangle \epsilon = \langle {\sigma _{ij}}{S_{ij}}\rangle  \langle {\sigma _{ij}}\rangle {\tilde S_{ij}} \simeq \langle {\sigma _{ij}}{S_{ij}}\rangle = \langle \eta \vert{S^{\ast}}{\vert^2} + \zeta {d^2}\rangle,$$(31)
 Rate of subgridscale pressure dilatation:where \(\tilde d = {\tilde S_{ii}} = \partial {\tilde u_i}/\partial {x_i}\).$$\langle \rho \rangle \lambda =  \langle dP\rangle + \tilde d\langle P\rangle,$$(32)
 Convective internal energy flux on subgrid scales:^{9}$${\mathfrak{F}^{({\rm{conv}})}} =  \langle \rho ue\rangle + \tilde \rho \tilde u\tilde e.$$(33)
 The flux associated with pressure fluctuations:For ideal gas with adiabatic exponent \(\gamma,\;{\mathfrak{F}^{({\rm{press}})}} = (\gamma  1){\mathfrak{F}^{({\rm{conv}})}}\).$${\mathfrak{F}^{({\rm{press}})}} =  \langle uP\rangle + \tilde u\langle P\rangle.$$(34)
 The diffusive flux of turbulent energy on subgrid scales:For Reynolds operators in the weakly compressible limit, \({\mathfrak{F}^{({\rm{kin}})}}\) can be expressed as a thirdorder moment of the velocity fluctuation: \(2\mathfrak{F}_j^{({\rm{kin}})} \simeq  \rho \langle u_i^{\prime}u_i^{\prime}u_j^{\prime}\rangle \) (Germano, 1992).$${\mathfrak{F}^{({\rm{kin}})}} =  {1 \over 2}\langle \rho {u^2}u\rangle + {1 \over 2}\langle \rho {u^2}\rangle \tilde u  \tilde u \cdot \tau.$$(35)

There is also a viscous flux, which can be neglected relative to other flux terms if the Reynolds number is sufficiently high.
2.2 Cosmological fluid dynamics
3 SubgridScale Models
3.1 Closures for the turbulence stress tensor
3.2 The SarkarSmagorinsky model for weakly compressible turbulence
3.3 The compressible subgridscale turbulence energy model
3.4 Two equation models and gravity
In the framework of the Reynoldsaveraged NavierStokes equations (RANS), which are equivalent to the filtered NavierStokes equation in the limit of a filter scale comparable the integral scale of the flow, the Kε turbulence model can be used to calculate both the turbulence energy K and the dissipation rate ε. Two inhomogeneous PDEs of the advectiondiffusion type determine K and ε (Pope, 2000). In contrast to the simple dimensional closure (58) with a single coefficient of order unity, the diffusion and source terms in the equation for ε come with several additional closure coefficients. This type of model is commonly used for industrial and environmental flows.
The KL model outlined above was adopted as an SGS model for simulations of turbulence driven by active galactic nuclei (AGNs) in galaxy clusters (Scannapieco and Brüggen, 2008). In this case, turbulence is thought to be stirred by hot bubbles rising due to their buoyancy in the intracluster medium (ICM). These bubbles originate from the AGNs in the cluster. For this reason, production through the turbulent cascade is set to zero, i.e., \(\tau _{ij}^{\ast} = 0\).^{17} In contrast to LES based on the consistent decomposition derived in Section 2.1, it follows from the very concept of the KL model that K cannot be interpreted as the kinetic energy associated with velocity fluctuations below the grid scale. Since K is the kinetic energy associated with the dominant eddies driven by the RT instability on a length scale L, where L is a dynamical variable that can become large compared to the grid scale, K generally encompasses some fraction of the numerically resolved turbulence energy on top of the SGS turbulence energy (if L falls below the grid resolution, on the other hand, K will represent only a fraction of the SGS turbulence energy). Consequently, the KL model works as a hybrid model in these simulations, which stands in between RANS and LES. As a result, turbulent fluctuations in the RTunstable bubbles are smeared out over length scales smaller than L and only the coherent structures on larger scales are captured in Scannapieco and Brüggen (2008). LES, on the other hand, always resolve the smallscale structure of turbulent flows down to the grid scale. LES in the sense discussed here, however, would break down if there is no physical turbulence on length scales of the order of the grid scale and below (i.e., if the scale of viscous dissipation is within the range of numerically resolved scales).
For the general case of selfgravitating turbulent gas, no satisfactory closure for Γ has been found yet. A conceptual difficulty is that the acceleration caused by gravity is genuinely anisotropic, while SGS models such as the turbulence energy model are based on local isotropy. The usual solution to this problem is to resolve the flow down to length scales that are not strongly affected by gravity. In AMR simulations, this is achieved by imposing a Truelovelike resolution criterion such that a sufficiently large ratio between the local Jeans length and the grid scale is maintained (Truelove et al., 1997; Federrath et al., 2011). Since the density of gravitationally unstable gas would increase indefinitely, excess mass is usually dumped into sink particles at the highest refinement level (Krumholz et al., 2004; Federrath et al., 2010a; Wang et al., 2010). Thereby, collapsing gas is decoupled from the numerically computed gas dynamics. In a certain sense, a sink particle is nothing but an SGS model for a selfgravitating overdense cloud that collapses down to scales below the minimal grid scale. Despite being a crude model, sink particles are a reasonable approximation to collapsed clouds because they mainly interact through accretion (i.e., mass accumulation) with the numerically resolved gas dynamics. A more complex situation is encountered if the objects represented by the sink particles produce feedback onto the gas. An example is stellar feedback in galaxy simulations, which can be treated with the approach discussed in Section 6.2.
4 Determination of Closure Coefficients
One of the basic assumptions of the Kolmogorov theory is that turbulence is statistically selfsimilar in the inertial subrange (see, for example, Frisch, 1995). With regard to subgridscale closures, the selfsimilarity of turbulence implies that dimensionless coefficients such as C_{ ν } in Eq. (47) should be independent of the chosen filter scale. This is not only a necessary condition for the feasability of LES, but it also allows for the calibration of closure coefficients by explicitly filtering turbulence data. Since closures do not exactly match SGS terms, an improved approximation can be achieved by socalled dynamic procedures, which estimate coefficients from properties of the numerically resolved flow under the assumption of local selfsimilarity.
4.1 Hierarchical filtering
4.2 Dynamic procedures
Subgridscale models in their standard form apply to statistically stationary and isotropic turbulence. But turbulent flows in nature often deviate from this idealization: In terrestrial applications, flow inhomogeneities are inevitably caused by boundary conditions (“walls”). In astrophysics, one of the major energy sources is gravity. It causes matter to clump (galaxies and clusters) or to move under the action of central gravitational fields (stars), which produces inherently inhomogeneous and anisotropic flows. For example, turbulent convection in stars introduces a vertical anisotropy of the flow. Turbulence driven by violent energy release (supernovae) can also be highly inhomogeneous.
One of the solutions to this problem is to localize closures, i.e., to calculate local closure coefficients. This requires local estimators that take properties of the flow in some small region as input. Obviously, this works only if the size of this region is not significantly affected by the flow inhomogeneity on larger scales. In other words, the flow must be asymptotically homogeneous and isotropic at least on length scales of the order of the grid scale. In this case, a socalled test filter 〈 〉_{T} can be applied in LES, with a filter length Δ_{T} that is a small multiple of the grid scale Δ. Test filters are usually implemented as discrete filters over several grid cells (see Section 2.3.2 in Garnier et al., 2009). A multidimensional test filter can be composed as a succession of onedimensional filters.^{18} The test filter length Δ_{T} can be adjusted by varying the weights of the cells. An optimal ratio γ_{T} = Δ_{T}/Δ is given by the closest match between the filter transfer functions of the discrete and analytical box filters with filter length Δ_{T} (Vasilyev et al., 1998). For instance, a test filter with γ_{T} = 2.771 is optimal if a fivepoint stencil is used in each spatial dimension (Schmidt, 2004).
4.3 Global least squares method
5 Adaptive Methods
The most powerful technique for finitevolume codes to resolve localized and anisotropic structures in a flow is adaptive mesh refinement (AMR) (Berger and Oliger, 1984; Berger and Colella, 1989). Even with AMR, however, it is generally not possible to fully resolve turbulence. This entails the problem that the numerically resolved and unresolved turbulence energy fractions vary as regions are refined or derefined. In the following, it shown how to address this problem in adaptively refined LES. In principle, global energy and momentum conservation can be achieved, while reducing the need for artificial changes in the internal energy, which is the standard method to restore energy consistency between different refinement levels in AMR simulations. Apart from that, localized and anisotropic flow structures pose the problem that SGS models with constant coefficients introduce systematic errors because they are usually calibrated for statistically stationary and isotropic turbulence. Shearimproved SGS models can alleviate this problem by adjusting the nonlinear energy transfer across the grid scale to local flow conditions. For example, this is possible by applying an adaptive temporal filter, the socalled Kalman filter.
5.1 Energy and momentum conservation in AMR simulations
5.2 Shearimproved model
 1.Given the error variance \(P_i^{(n)}\) at time t^{(n)}, the prediction for t^{(n+1)} iswhere$$P_i^{(n + 1)\ast} = P_i^{(n)} + \sigma _{\delta [{u_i}]}^{2(n)},$$(122)Here it is assumed that the typical correction of the mean flow, δ[u_{ i }]^{(n)} = [u_{ i }]^{(n)} − [u_{ i }]^{(n−1)}, is of the order 2πΔt^{(n)}u_{c}/(√3T_{c}), where Δt^{(n)} = t_{ n } − t_{ n }_{−1}$$\sigma _{\delta [{u_i}]}^{(n)} = {{2\pi \Delta {t^{(n)}}} \over {\sqrt 3 {T_c}}}{u_c}.$$
 2.The Kalman gain is then given bywhere$$\alpha _i^{(n + 1)} = K_i^{(n + 1)} = {{P_i^{(n + 1)\ast}} \over {P_i^{(n + 1)\ast} + \sigma _{\delta {u_i}}^{2\;(n)}}},$$(123)is the contribution of the fluctuating component \(\delta u_i^{(n)} \equiv u_i^{^{\prime}(n)} = u_i^{(n)}  {[{u_i}]^{(n)}}\) to the error variance. The lower bound on \(\sigma _{\delta {u_i}}^{2\;(n)}\) is necessary to obtain nonvanishing fluctuations from an initially smooth flow with [u_{ i }] = u_{ i }.$$\sigma _{\delta {u_i}}^{2\;(n)} = \max \left({\left\vert {\delta u_i^{(n)}} \right\vert,\;0.1{u_c}} \right){u_c}$$
 3.The corrected error variance for the next step is given byIn a statistically stationary state, the velocity fluctuations should be of the order \(\sigma _{\delta {u_i}}^{(n)} \simeq {u_{\rm{c}}}\). In this case the Kalman filter corresponds to simple exponential smoothing:$$P_i^{(n + 1)} = \left({1  K_i^{(n + 1)}} \right)P_i^{(n + 1)\ast}.$$(124)The two filter parameters, T_{c} and u_{c}, have to be chosen such that u_{c} is roughly the integral velocity of turbulence if the flow enters a steady state and T_{c} is the characteristic time scale over which the flow evolves. In Cahuzac et al. (2011), LES of turbulence produced by the flow past a cylinder were shown to agree well with experimental data if Kalman filtering is applied with T_{c} and u_{c} being set to the inverse of the expected vortexshedding frequency and upstream velocity, respectively.$$\alpha _i^{(n + 1)} \simeq {{\sigma _{\delta [{u_i}]}^{(n)}} \over {\sigma _{\delta {u_i}}^{(n)}}} \simeq {{2\pi \Delta {t^{(n)}}} \over {\sqrt 3 {T_c}}} \ll 1.$$
A similar computational problem in astrophysics is an isothermal spherical cloud, which is bound by a static gravitational potential, in a homogeneous wind. The initial density profile of the cloud is determined by hydrostatic equilibrium. As mass is stripped from the cloud by the wind, a turbulent wake forms in the downstream direction. This problem was originally investigated as a simple model for the infall of small subcluster into the ICM of a much more massive cluster, computed in the frame of reference attached to the center of mass of the subcluster (Iapichino et al., 2008). LES with the shearimproved SGS model are presented in Schmidt et al. (2014), where the Kalman filter parameters are given by the velocity of the wind and the turnover time of the largest eddies. Figure 8 compares the gas density and flow structure for two runs, one with the standard SGS model and the other one with the shearimproved model. In both cases, a 128^{3} root grid and three levels of refinement are used. The AMR control variables are the squared vorticity and the compression rate (Iapichino et al., 2008; Schmidt et al., 2009). The latter is defined by the substantial time derivative of the velocity divergence and tracks down the bow shock in front of the cloud. The top panels in Figure 8 show that the strain associated with the bow shock produces substantial SGS turbulence energy if the standard SGS model is applied. One can also discern the downscaling of the SGS turbulence energy that is transported with the wind, as it enters the refined regions around the cloud. This is based on the algorithm explained in the previous section. KelvinHelmholtz instabilities between the lowdensity wind and the highdensity gas in the cloud cause vortex shedding, which in turn produces turbulence. This becomes manifest in large values of ω and K in the turbulent wake that extends from the cloud towards the right. In contrast to the turbulent wake, however, the shear experienced by the gas when it is passing the bow shock is not associated with turbulence. As a result, the steep increase of K is largely a spurious effect induced by the eddyviscosity closure (49). Indeed, the SGS turbulence energy in the shocked wind is substantially reduced if the shearimproved closure (118) is applied (bottom right panel in Figure 8). In comparison to the standard SGS model, production is also suppressed at the front side of the cloud and, hence, vortex shedding is the dominant source of turbulence production. This has a clearly visible impact on the structure of the turbulent wake.
In general, an appropriate prior choice of the filter parameters might not be obvious. Nevertheless, they can be calibrated a posteriori by performing lowresolution test runs. This is shown for cosmological simulations in Schmidt et al. (2014). In this case, the Kalman filter also has the merit of separating the turbulent flow in clusters from the gravitydriven bulk flows (see Section 6.3.3).
6 Astrophysical Applications
6.1 Thermonuclear combustion in white dwarfs
Among the various possible scenarios for supernovae of type Ia are thermonuclear explosions of gas accreting white dwarfs in close binary systems (Hillebrandt and Niemeyer, 2000; Röpke et al., 2011; Hillebrandt et al., 2013). If the mass of a white dwarf approaches the Chandrasekhar limit, explosive carbon and oxygen burning is ignited (Nonaka et al., 2012). Owing to the degeneracy of white dwarf matter, the thermonuclear reaction zones propagate as thin flame fronts, whose thickness δ_{ f } and propagation speed s_{f} are determined by the very high thermal conductivity of the fuel and the nuclear reaction rates. This mode of burning is called deflagration. Since the burned material has lower density than the fuel, it rises because of its buoyancy. Consequently, the energy released by thermonuclear deflagration drives convection. Since eddies exert strong shear on rising bubbles of burning material, they are deformed into mushroomlike shapes and KelvinHelmholtz instabilities at the surface are rapidly producing turbulence (Malone et al., 2014). Eventually, this results in a very complex flame front with a fractal structure that cannot be resolved in numerical simulations over the full range of length scales.
As an illustration of the level set method with the turbulent flame speed (125), Figure 9 shows snapshots of the flame front for a thermonuclear supernova simulation from Schmidt and Niemeyer (2006). The turbulent velocity fluctuations on subgrid scales, \(\sqrt {2K} \), are shown as color shades at the flame surface. The asymptotic value of turbulent flame speed is \({s_{\rm{t}}} \simeq {C_{\rm{t}}}\sqrt {2K} \) if s_{t} ≫ s_{f}. In this simulation, a Poisson process is used to randomly place small ignition spots in the highdensity core of the white dwarf. The statistics of this process is based on a simple model for temperature fluctuations produced by convection in preignition phase. Although the number of ignition resulting from this model appears to be by far too large in the light of recent numerical studies of the ignition process (Nonaka et al., 2012), the simulation nevertheless demonstrates in an exemplary manner how the turbulent deflagration progresses. At early time (a), one can see a large number of small bubbles generated by the stochastic ignition process at distances of the order 100 km from the center of the white dwarf. As the burning bubbles are rising from the centre, they begin to form the typical RayleighTaylor mushroom shapes (b). At this point, the effective flame propagation speed is already dominated by turbulence. After about half a second (c), the turbulent flames mostly have merged into a single structure of about 1000 km diameter. Then the burning front rapidly expands to much larger radii and causes the white dwarf to explode after roughly one second (d).
6.2 Galaxy simulations
6.3 Cosmological simulations
Despite its shortcomings, the method based on Maier et al. (2009) has its merits as a first approximation. For example, Figure 14 shows the density of the baryonic gas and the smallscale turbulence in an adiabatic cosmological simulation of a cluster from Maier et al. (2009). As indicator of turbulence, the magnitude of the numerically unresolved turbulent velocity fluctuation \(\sqrt K \) is scaled down from refinement level l to the minimal cell size at the maximal level l_{max} via the power law factor \({r^{(l  {l_{\max }})\eta }}\). It is particularly interesting that the infall of a subhalo, which is marked by the small box in slice (a), produces a pronounced turbulent wake, quite similar to the idealized scenario discussed in Section 5.2. However, a shearimproved model was not used in this simulation. Even at z = 0, the turbulent velocity slice (d) in Figure 14 shows a clear trace of the minor merger, which is not discernible in the density slice (c).
6.3.1 Turbulence production and support against gravity
6.3.2 Gravitational collapse of gas in primordial halos
Apart form morphological differences, the SGS model also has a significant influence on the accretion of mass by the protostars that are formed through the collapse of gas clouds in the halos. As mentioned in Section 3.4, it is possible to follow the evolution of gravitationally bound dense objects, such as protostars, by inserting sink particles (Federrath et al., 2010a; Wang et al., 2010). This method is applied in Latif et al. (2013e) to follow the accretion history of protostars in atomic cooling halos. The resulting time evolution of the accretion rate is plotted in Figure 19 for simulations of three different halos, using both ILES and LES. As one can see, the accretion rates reach peak values around 10 M_{⊙} yr^{−1} roughly within 10^{4} yr. This value agrees with the theoretical expectation for BondiHoyle accretion. A comparison of ILES and LES suggests a systematically higher accretion rate for LES. This trend is confirmed by calculating the cumulative masses of the sink particles (see right plot in Figure 19), which reach masses above 10^{5} M_{⊙}. As a result, the SGS model favors the formation of higher black hole masses.
6.3.3 Turbulent velocity dispersion
A resolutiondependent turbulent velocity dispersion is avoided by the — hybrid model for RayleighTaylordriven turbulence (see Section 3.4), however, at the cost of smearing out turbulent fluctuations on numerically resolved length scales. This model was used to simulate the production of turbulence by AGN feedback in galaxy clusters (Scannapieco and Brüggen, 2008; Brüggen and Scannapieco, 2009). Another caveat of both the KL model and the SGS model based on Maier et al. (2009) are the constant closure coefficients. Strictly speaking, constantcoefficient models are applicable only to statistically homogeneous and stationary turbulence. Turbulence produced by cosmological structure formation and AGNs, however, is highly inhomogeneous and nonstationary.
6.4 Concluding remarks
It is a particular difficulty of validating LES in astrophysics that neither DNS nor sufficiently accurate experimental data exist. For virtually all instances of astrophysical turbulence, DNS are infeasible because of the very high Reynolds numbers (maybe, with the exception of turbulence in the ICM) and observations are usually not sensitive enough to clearly discriminate between different numerical models. As a consequence, the standards for model validation in astrophysics are very different compared to engineering or atmospheric sciences, where direct comparisons with measurements or DNS data are commonly made.
Statistical properties of homogeneous turbulence such as twopoint statistics, turbulence energy spectra, or the distribution of mass density fluctuations are not very sensitive on the application of an explicit SGS model, provided that the numerical resolution is high enough. This is necessarily so, because the scale locality of hydrodynamical turbulence implies that properties related to inertialrange dynamics on some scale must be largely independent of effects on much smaller scales. However, the determination of secondorder scaling exponents of compressible turbulence might benefit from the application of an explicit SGS model by reducing the bottleneck effect in the energy spectra (Woodward et al., 2006). Moreover, the SGS turbulence energy has its merits as a buffer variable, which helps to reduce artificial manipulations of the internal energy in AMR simulations (Schmidt et al., 2014). Much more difficult, however, is the assessment of systematic differences in the flow structure compared to ILES. An example are the preferentially disklike structures in LES of collapsing primordial halos (Latif et al., 2013c). Although the correct solution is unknown, we know for sure that there are couplings between numerically resolved scales and subgrid scales in simulations, which are caused by the turbulent stresses on the grid scale. The calculation of these stresses on the basis of scale separation (Section 2) and testable closures (Sections 3 and 4) is at least physically better motivated and provides a more accurate approximation than purely numerical truncation errors.
Most likely, however, the main utility of SGS models in astrophysics is going to be the treatment of complex subresolution physics. The calculation of the turbulent burning speed in supernova simulations, which was pioneered by Niemeyer and Hillebrandt (1995), is just one example. Recently, Braun et al. (2014) demonstrated that feedback from supernovae in galaxy simulations can be implemented as a source term in the SGS turbulence energy equation in addition to heating. This offers an alternative to the often employed kinetic feedback, which produces resolved gas motions. Particularly in cosmological simulations, however, the production of turbulence by supernova blast waves is mostly an unresolved process. The application of LES in cosmological simulations therefore not only improves hydrodynamics and the consistency of AMR, but it allows for a treatment of feedback and mixing processes below the local grid scale.
A potential application of LES, which has not been exploited yet in astrophysics, are threedimensional simulations of convection in stars, including the convection zone of our Sun. Arnett et al. (2014) investigate the performance of RANS in comparison to ILES for the cores of massive stars. Their model, which encompasses nucleosynthesis, reproduces statistics from threedimensional simulations with high resolution. LES would be an intermediate approach that captures part of the turbulent convective flow, while explicitly modeling mixing processes on smaller scales. Many problems in astrophysics also involve magnetic fields. In this case, there is an analogy to feedback, because small scales can significantly contribute to the amplification of magnetic fields in turbulent plasmas. As demonstrated, for example, by Sur et al. (2010) and Latif et al. (2013d), this causes a pronounced dependence on resolution in ILES of collapsing gas clouds. Currently, it is completely open whether SGS models for MHD turbulence could amend this problem, but the potential of treating dynamo effects by an SGS model is an exciting prospect. Owing to the anisotropy of MHD turbulence and the potential influence of kinetic effects, however, this is indeed a very challenging problem.
Footnotes
 1.
In astrophysics, the term DNS is sometimes used in an improper sense, which actually corresponds to ILES (see below). We strictly distinguish between DNS and ILES here, which means that a DNS has to explicitly treat the physical viscosity of the fluid, i.e., it solves the NavierStokes equations.
 2.
For many applications, particularly in astrophysics, the definition used here is appropriate. In a broader sense, LES may include the case where microscopic dissipation is partially resolved. DNS can then be considered as limiting case of LES for Re(Δ) ∼ 1.
 3.
In astrophysics, the term subgridscale model may comprise models that capture subresolution physics other than turbulence. A typical example are starformation models in galaxy simulations.
 4.
For periodic boundary conditions with mean density ρ_{0}, the source term on the right hand side is 4πG(ρ − ρ_{0}).
 5.
In Garnier et al. (2009), 〈ρ〉Ẽ is identified with 〈ρE〉 and a different symbol is used for ẽ + ½ũ^{2}. However, we do not follow this nomenclature here.
 6.
Also called turbulence energy flux, although this is not a transport term. In the incompressible limit, Σ corresponds to the energy transfer in spectral space.
 7.
The definition of \({\tilde S_{ij}}\) is a consequence of integration by parts of ũ_{ i }∂_{ j }τ_{ ij }. The symbol \({\tilde S_{ij}}\) is used for convenience. It is important to keep in mind that \({\tilde S_{ij}} \neq \langle \rho {S_{ij}}\rangle/\langle \rho \rangle \) because ∂_{ j }ũ_{ i } = ∂_{ j }[〈ρu_{ i }〉/〈ρ〉] ≠ 〈ρ∂_{ j }u_{ i }〉/(ρ).
 8.
Since S_{ ij } is a velocity derivative, it is of the order of the velocity fluctuation at the smallest length scales. For incompressible turbulence, Kolmogorov scaling implies 〈σ〉〈S〉 ∼ ρε(Δ/ℓ_{K})^{−4/3} ∼ ρε/Re(Δ), where ℓ_{K} is the Kolmogorov length. For high Reynolds numbers, the ratio Δ/ℓ_{K} is typically very large. As a result, 〈σ〉〈S〉 is negligible compared to ρε ≃ 〈σS〉. From the same estimates follows τ ∼ (Δ/ℓ_{K})^{4/3}〈σ〉 ∼ Re(Δ)〈σ〉 (Röpke and Schmidt, 2009), which is applied to obtain Eq. (23) for the filtered momentum.
 9.
 10.
An effective filter length for isotropic turbulence simulations can be calculated from the second moment of the compensated energy spectrum as shown in Schmidt et al. (2006a).
 11.
For higherorder methods such as PPM (Colella and Woodward, 1984), the leadingorder truncation errors correspond to hyperviscosity terms proportional to ∇^{4}q.
 12.
This idea was originally proposed by Boussinesq in the 19th century (Boussinesq, 1877).
 13.
The expression for the turbulent viscosity is determined by the physical dimension of viscosity, the positivity in the incompressible limit, and the requirement that it must be a scalar, independent from the frame of reference. Scalars associated with the rateofstrain tensor S are d, ∣S*∣, and det S*.
 14.
The operator ⊗ signifies the tensor product, i.e., u_{ i,k } is the Jacobian matrix of the velocity.
 15.
In this simulation, a quasiisothermal equation of state is applied with an adiabatic exponent γ = 1.001.
 16.
The diffusion on numerically resolved length scales due to the turbulent stresses \(\tau _{ij}^{\ast}\) must not be confused with the subgridscale diffusion of K, which is given by Eq. (65).
 17.
In Gray and Scannapieco (2011), the model is extended with an eddyviscosity closure for \(\tau _{ij}^{\ast}\).
 18.
For filters with large stencils, test filtering can nevertheless become too inefficient because of the access to remote blocks of memory.
 19.
The simulation from Schmidt et al. (2009) was performed on a 768^{3} grid. In this case, the filter length is Δ_{4} = L/16 = 24Δ.
 20.
To be more precise, the notion of a flame front propagating with the turbulent flame speed s_{t} applies to the socalled flamelet regime, for which ℓ_{f} ≪ ℓ_{G}. When ℓ_{G} becomes comparable to ℓ_{f}, turbulence affects the internal structure of the flame and distributed burning sets in Niemeyer and Kerstein (1997); Röpke and Hillebrandt (2005a); Schmidt (2007).
 21.
The assumption ℓ_{*} = λ_{J} does not imply that a cloud of size ℓ_{*} collapses and is turned into stars as a whole. The internal structure of starforming clouds with strong density fluctuations due to supersonic turbulence is usually not resolved in galaxy simulations. For this reason, the Jeans length for the gas density in a grid cell serves only as a reference length scale for the mean properties of the numerically unresolved starforming clouds. If the clouds are partially resolved, however, this assumption needs revision.
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