A Compressible Atmospheric Model with a Cartesian Cut Cell Approach

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Geosci. Model Dev., 8, 317–340, 2015 www.geosci-model-dev.net/8/317/2015/ doi:10.5194/gmd-8-317-2015 © Author(s) 2015. CC Attribution 3.0 License.

ASAM v2.7: a compressible atmospheric model with a Cartesian cut cell approach

M. Jähn, O. Knoth, M. König, and U. Vogelsberg Leibniz Institute for Tropospheric Research, Permoserstrasse 15, 04318 Leipzig, Germany

Correspondence to: M. Jähn ([email protected])

Received: 16 June 2014 – Published in Geosci. Model Dev. Discuss.: 18 July 2014 Revised: 19 January 2015 – Accepted: 22 January 2015 – Published: 18 February 2015

Abstract. In this work, the fully compressible, three- Leipzig. ASAM was initially designed for CFD (compu- dimensional, nonhydrostatic atmospheric model called All tational fluid dynamics) simulations around buildings. The Scale Atmospheric Model (ASAM) is presented. A cut cell model can also be used with spherical or cylindrical grids. approach is used to include obstacles and orography into Stability problems with grid convergence in special points the Cartesian grid. Discretization is realized by a mixture (the pole problem) in both grids are handled through the im- of finite differences and finite volumes and a state limit- plicit time integration both for advection and the yet faster ing is applied. Necessary shifting and interpolation tech- gravity and acoustic waves. For simulating the flow around niques are outlined. The method can be generalized to any obstacles, buildings or orography, the cut (or shaved) cell other orthogonal grids, e.g., a lat–long grid. A linear im- approach is used. With this attempt one remains within the plicit Rosenbrock time integration scheme ensures numeri- Cartesian grid and the numerical pressure derivative in the cal stability in the presence of fast sound waves and around vicinity of a structure is zero if the cut cell geometry is not small cells. Analyses of five two-dimensional benchmark taken into account, which is not the case in terrain-following test cases from the literature are carried out to show that coordinate systems due to the slope of the lowest cells (Lock the described method produces meaningful results with re- et al., 2012). Since this skewness is also reproduced in upper spect to conservation properties and model accuracy. The levels, a cut cell model produces reduced or greatly reduced test cases are partly modified in a way that the flow field or errors in comparison models with terrain-following coor- scalars interact with cut cells. To make the model applicable dinates (Good et al., 2014). Several techniques have been for atmospheric problems, physical parameterizations like a developed to overcome these nonphysical errors associated Smagorinsky subgrid-scale model, a two-moment bulk mi- with terrain-following grids, especially when spatial scales crophysics scheme, and precipitation and surface fluxes us- of three-dimensional models become finer (which leads to ing a sophisticated multi-layer soil model are implemented a steepening of the model orography). Tripoli and Smith and described. Results of an idealized three-dimensional (2014a) introduced a variable-step topography (VST) sur- simulation are shown, where the flow field around an ide- face coordinate system within a nonhydrostatic host model. alized mountain with subsequent gravity wave generation, Unlike the traditional discrete-step approach, the depth of latent heat release, orographic clouds and precipitation are a grid box intersecting with a topographical structure is ad- modeled. justed to its height, which leads to straight cut cells. Nu- merical tests show that this technique produces better results than conventional approaches for different topography (severe and smooth) types (Tripoli and Smith, 2014b). In 1 Introduction their cases, also the computational costs with the VST approach are reduced because there is no need of extra func- In this paper we present the numerical solver ASAM (All tional transform calculations due to metric terms. Steppeler Scale Atmospheric Model) that has been developed at the et al.(2002) derived approximations for z coordinate non- Leibniz Institute for Tropospheric Research (TROPOS),

Published by Copernicus Publications on behalf of the European Geosciences Union. 318 M. Jähn et al.: ASAM v2.7 hydrostatic atmospheric models by using the shaved-element resolving resolution (König, 2013). There, the implementa- finite volume method. There, the dynamics are computed in tion of a dynamic Smagorinsky subgrid-scale model is tested the cut cell system, whereas the physics computation remains for a convective atmospheric boundary layer and an inflow in the terrain-following system. Using a z-coordinate system generation approach that produces a turbulent flow field is can also improve the prediction of meteorological parame- presented. ters like clouds and rainfall due to a better representation of A separately developed LES model at TROPOS is called the atmospheric flow near mountains in a numerical weather ASAMgpu (Horn, 2012). It includes some basic features prediction (NWP) model (Steppeler et al., 2006, 2013). The of the ASAM code and runs on graphics processing units cut cell method is also used in the Ocean–Land–Atmosphere (GPUs), which enables very time-efficient computations and Model (OLAM) (Walko and Avissar, 2008a), which extends post-processing. However, this model is not as adjustable the Regional Atmospheric Modeling System (RAMS) to as the original ASAM code and the inclusion of three- a global model domain. In OLAM, the shaved-cell method is dimensional orographical structures is not implemented so applied to an icosahedral mesh (Walko and Avissar, 2008b). far. ASAMgpu was applied for a study of heat island effects Yamazaki and Satomura(2008) simulated a two-dimensional on vertical mixing of aerosols by comparing the results of flow over different mountain slopes and compared the results large eddy simulations with wind and aerosol lidar observa- of their cut cell model with a model using terrain-following tions (Engelmann et al., 2011). coordinates. Especially for steep slopes, significant errors This paper is structured as follows. The next section deals were reported in the terrain-following model. A drawback is with a general description of the model. It includes the ba- the generation of low-volume cells when a cut cell method is sic equations that are solved numerically and the used energy used. To avoid instability problems around these small cells, variable. Also, the cut cell approach and spatial discretization the time integration scheme has to be adapted. This can be as well as the time integration scheme are described. This ap- achieved by using semi-implicit or semi-Langrangian meth- proach can be extended to other orthogonal grids like the lat– ods, for example. In ASAM, a linear-implicit Rosenbrock long grid. Section 3 deals with the model physics, including time integration scheme is used (Hairer and Wanner, 1996). a subgrid-scale model, a two-moment microphysics scheme Another option to handle the small cells problem is to merge and surface flux parameterization. Results of different ideal- small cut cells with neighboring cells in either the horizontal ized test cases are shown in Sect. 4. The first one is a dry ris- or vertical direction (Yamazaki and Satomura, 2010). How- ing heat bubble described by Wicker and Skamarock(1998) ever, this approach becomes more complicated when apply- followed by a flow past an idealized mountain ridge from ing it to three spatial dimensions, since a lot of special cases Schaer et al.(2002). Two other “classical” test examples have have to be considered. To achieve reasonable vertical solu- been chosen and modified so that cut cells are included and tions near the ground, the usage of local mesh refinement interaction within these cells are guaranteed. The first one is a techniques becomes interesting for large-scale models (Ya- falling cold bubble with a developing density current (Straka mazaki and Satomura, 2012). et al., 1993) but with a 1 km high mountain on the left part of The here-presented model is a developing research code the domain. The second case is the moist bubble benchmark with different options to choose from like different numeri- case reported by Bryan and Fritsch(2002). The bubble will cal methods (e.g., split-explicit Runge–Kutta or partially im- rise and interact with a zeppelin-shaped cut area in the center plicit peer schemes), number of prognostic variables, physi- of the domain. For both test cases, energy conservation tests cal parameterizations or the change to a spherical grid type. are carried out. Another two-dimensional case by Berger and Parallelization is realized by using the message passing inter- Helzel(2012) is presented to test the accuracy of the cut face (MPI) and the domain decomposition method. The code cell method by advecting a smooth bump in a radial wind is easily portable between different platforms like Linux, field in an annulus. For the last test case, a three-dimensional IBM or Mac OS. With these features, large eddy simulations mountain overflow with subsequent orographic cloud gener- (LES) with spatial resolutions of O(1–100 m) can be per- ation and precipitation is simulated (Kunz and Wassermann, formed with respect to a sufficiently resolved terrain struc- 2011). Concluding remarks and future work are in the final ture. In previous studies, the model was used to demonstrate section. the volume-of-fluid (VOF) method for nondissipative cloud transport (Hinneburg and Knoth, 2005). ASAM also took part in an intercomparison study of mountain-wave simula- 2 Description of the All Scale Atmospheric Model tions for idealized and real terrain profiles, where altogether 2.1 Governing equations 11 different nonhydrostatic numerical models were compared (Doyle et al., 2011). A partially implicit peer method The flux-form compressible Euler equations for the atmo- is presented in Jebens et al.(2011) in order to overcome the sphere are small cell problem around orography when using cut cells. The model was recently used for a study of dynamic flow structures in a turbulent urban environment of a building-

Geosci. Model Dev., 8, 317–340, 2015 www.geosci-model-dev.net/8/317/2015/ M. Jähn et al.: ASAM v2.7 319

Table 1. Physical constants.

Symbol Quantity Value 5 p0 Reference pressure 10 Pa −1 −1 Rd Gas constant for dry air 287 Jkg K −1 −1 Rv Gas constant for water vapor 461 Jkg K −1 −1 cpd Specific heat capacity at constant pressure for dry air 1004 Jkg K −1 −1 cpv Specific heat capacity at constant pressure for water vapor 1885 Jkg K −1 −1 cpl Specific heat capacity at constant pressure for liquid water 4186 Jkg K −1 −1 cvd Specific heat capacity at constant volume for dry air 717 Jkg K −1 −1 cvv Specific heat capacity at constant volume for water vapor 1424 Jkg K 6 −1 L00 Latent heat at 0 K 3.148 × 10 Jkg g Gravitational acceleration 9.81 ms−2 Cs Smagorinsky coefficient 0.2

2.2 Cut cells and spatial discretization ∂ρ + ∇ · (ρv) = 0, (1) ∂t 2.2.1 Definition of cut cells ∂(ρv) + ∇ · (ρvv) = −∇ · τ − ∇p − ρg − 2 × (ρv), (2) ∂t The spatial discretization is done on a Cartesian grid with ∂(ρφ) grid intervals of lengths 1xi,1yj ,1zk and can easily be + ∇ · (ρvφ) = −∇ · q + S , (3) ∂t φ φ extended to any orthogonal, logically rectangular structured grid (i.e., it has the same logical structure as a regular Carte- where ρ is the total air density, v = (u,v,w)T the three- sian grid) like spherical or cylindrical coordinates. First, it dimensional velocity vector, p the air pressure, g the grav- is described for the Cartesian case. Generalizations are dis- itational acceleration, the angular velocity vector of the cussed afterwards. Orography and other obstacles like build- earth, φ a scalar quantity and Sφ the sum of its corresponding ings are presented by cut cells, which are the result of the source terms. The subgrid-scale terms are τ for momentum intersection of the obstacle with the underlying Cartesian and qφ for a given scalar. grid. In Fig.1 different possible and excluded configurations The energy equation in the form of Eq. (3) is represented are shown for the three-dimensional case. For the spatial by the (dry) potential temperature θ. In the presence of wa- discretization only the six partial face areas and the partial ter vapor and cloud water, this quantity is replaced by the cell volume and the grid sizes of the underlying Cartesian density potential temperature θρ (Emanuel, 1994) as a more mesh are used. For a proper representation the orography is generalized form of the virtual potential temperature θv: smoothed in such a way that the intersection of a grid cell and the orography can be described by a single possible non- Rv θ = θ 1 + q − 1 − q , (4) planar polygon. Or in other words, a Cartesian cell is divided ρ v R c d in at most two parts, a free part and a solid part. For each where the equation of state can be expressed as follows: Cartesian cell, the free face area of the six faces and the free volume area of the cell are stored, which is the part outside p κm p = ρRdθρ . (5) of the obstacle. These values are denoted for the grid cell p0 i,j,k by FUi−1/2,j,k, FUi+1/2,j,k, FVi,j−1/2,k, FVi,j+1/2,k, κm FWi,j,k−1/2, FWi,j,k+1/2 and V i,j,k, respectively. In the fol- In the previous two equations θ = T (p0/p) is the potential lowing, the relative notations FUL and FUR are used, e.g., as temperature, qv = ρv/ρ is the mass ratio of water vapor in the shown in Fig.3. air (specific humidity), qc = ρc/ρ is the mass ratio of cloud water in the air, p0 a reference pressure and κm = (qdRd + qvRv)/(qdcpd +qvcpv +qccpl) the Poisson constant for the air 2.2.2 Spatial discretization mixture (dry air, water vapor, cloud water) with qd = ρd/ρ. Rd and Rv are the gas constants for dry air and water vapor, The spatial discretization is formulated in terms of the grid respectively. interval length and the face and volume areas. The variables The number of additional equations like Eq. (3) depends are arranged on a staggered grid with momentum on the complexity of the microphysical scheme used. Fur- V = (U,V,W) = (ρu,ρv,ρw) at the cell faces and all other thermore, tracer variables can also be included. The values variables at the cell center. The discretization is a mixture of all relevant physical constants are listed in Table1. of finite volumes and finite differences. In the finite volume www.geosci-model-dev.net/8/317/2015/ Geosci. Model Dev., 8, 317–340, 2015 icsinPpr|Dsuso ae icsinPpr|Dsuso ae | Paper Discussion | Paper Discussion | Paper Discussion | Paper Discussion

4 M. Jahn:¨ ASAM v2.7

3 markers 4 markers 5 markers 6 markers 6 markers 4 M. Jahn:¨ ASAM v2.7

320Fig. 1. Possible conﬁgurations for cut cell intersection. The last two cases are excluded. M. Jähn et al.: ASAM v2.7

FWR ...... . . . . . . . . ...... ...... . . ...... ...... . . VC ...... ...... | Paper Discussion | Paper Discussion | . Paper Discussion | Paper Discussion . . ...... ...... ...... ...... FUL ...... FC FUR . URC UF ULC ...... . . . ...... ...... ...... 3 markers .4 .markers...... 5 markers. 6 markers. . 6 markers ...... Figure 1. Possible configurations for cutFW cellL intersection (cases 1–3) for different numbers of face intersection points (markers). The last two casesPossible are excluded. configurations for cut cell intersection. The last two cases are excluded. Figure 1.Fig.Fig.Possible 1. 2. Cut cell with configurations face and volume area for information cut cell (left) intersection and arrangement of (casesface and cell 1–3) centered for momentum different (right). numbers of face intersection points (markers). The last two cases are excluded. φL φFWC R φR To achieve positivity in Eq. (6), we apply state limiting. For ...... U > 0 . where yn is. a given approximation. at y(t) at time tn and R . this task, Eq. (6. ) is rewritten in slope-ratio. formulation: icsinPpr|Dsuso ae icsinPpr|Dsuso ae | Paper Discussion | Paper . Discussion | Paper Discussion | Paper Discussion ...... y t =. t + τ J -...... subsequently. n+1 at time n+1 . n . In addition is an ...... VC ...... φ approximation= φ + K(φ. to− theφ ), Jacobian matrix. ∂F/∂y. A Rosenbrock(7) ...... FR C C L ...... FU ...... F. . . . . FU . UR U UL . L ...... C. . . . R . 240 methodC is thereforeF fullyC described. by the two matrices = ...... A ...... where ...... (αij), Γ = (. γij) and the parameter. γ. hL ...... h. C...... hR ...... AmongφR − theφ.C available methods. are a second order two stage Figure 2...... FigureStencil 2. forStencil third-order for approximation. third-order approximation...... K =methodα1 after...... Lanser+ α2. et al. (2001).. (8) FWL φC − φL

Sk1 =τF (yn), (17) Fig.Fig. 2. Cut 3. Stencil cell with for face third-order and volume approximation. area information (left) and arrangementThen ofK faceis and replaced cell centered by limiter momentum function (right).9 and Eq. (7) is FWR ...... rewritten as 2 4 ...... 245 ...... Sk2 =τF yn + k1 k1 , (18) ...... 3 3 VC ...... − φL...... φC . φR . φ − φ ...... R C ...... ...... where=y is+ a given approximation− at y(t) at time t and FU ...... F FU . U UR >U0 U . φ φn 9 (φ φ ), n (9) L ...... C. . . . R . RC F LC . FR C C L ...... 5 3 2.3 Time...... integration...... − ...... φ φ ...... y =y + kC + Lk , ...... -. . subsequentlyn+1 nyn+1 at1 time t2n+1 = tn + τ. In addition J is an (19) ...... 4 4 ...... approximation to the Jacobian matrix ∂F/∂y. A Rosenbrock After spatialFW discretization an ordinary differential equation S =I γτJ, J F 0(yn). (20) L 240 method is therefore− fully described≈ by the two matrices = 54 9(r) = max(0,min[r,min(δ,α1r + α2)]), δ = 2, A(10) Figure 3. Cut cell with facehL and volume area informationhC (left) and arrangementhR of face and cell(αij), Γ = (γij1) and1 the parameter γ. Figurey(t)0 = 3. CutF (y cell(t)) with face and volume area information (left) and(15) √ centered momentum (right). Amongwith γ the= available+ 3 methodsor in matrix are a form second in Table order (2).two stage arrangement of face- and cell-centered momentum (right). as proposed by2 Sweby6 (1984). This limiter has the property method after Lanser et al. (2001). is obtained that has to be integrated in time (method of lines). that the unlimited higher-order scheme (Eq.6) is used as much as possible0 and is utilized only then when it is needed. 230 To tackle the small time step problem connected with tiny Sk1 =τF (yn), (17) Fig. 3. Stencil for third-order approximation. 53 2/3 = 1 + 1 √3 contextcut cells, the main linear task implicit is the reconstruction Rosenbrock-W-methods of values and are gra- used In the case of 9 2 0, the scheme4 4/ degenerates3 2 to the6 simple 5/4 3/4 − dients(Jebens at cell et al., faces 2011). from cell-centered values. 245 first-orderSk2 =τF upwind−yn + scheme.k1 Thek1 , coefficients α1 and α2 can(18) be -Matrix3 − 3 Γ-Matrix γ TheA Rosenbrock discretization method of the has advection the form operator is performed computed in advanceA to minimize the overhead for a nonuni- 2.3 Time integration 5 3 for a generic cell-centered scalar variable φ. In the context yform= grid.y + In thek + casek of, a uniform grid the coefficients(19) are n+1Tablen 2. Coefficient4 1 4 table2 for ROS2. of a finite volume discretization,i 1 point valuesi 1 of the scalar constant, i.e., they are equal to 1/3 and 1/6. For a detailed After spatial discretization an− ordinary differential− equation value(I τγJφ are)k needed=τF ( aty the+ faces55α ofu this) + gridβ cell.k , Knowing i = 1,...,s discussionS =I γτJ, of the benefits J F 0 of(yn this). approach and numerical(20) ex- − i n ij j ij j − ≈ these face values, the advectionj=1 operatorj=1 in the x direction 250perimentsA second also see method Hundsdorfer was etconstructed al.(1995). This from procedure a low stor- X X 1 1 y(t)0 = F (y(t)) (15) withis appliedageγ = three in+ all stage√ three3 or second-orderin grid matrix directions, form Runge-Kutta in where Table the (2). virtualmethod, grid which is discretized by (FURUFRφR − FULUFLφL)/VC where U(16)F 2 6 is the discretized momentums at the corresponding faces. To sizesish usedare defined in split-explicit as time integration methods in the is obtained that has to be integrated in time (method of lines). 235 approximatey these=y values+ atα the faces,k , a biased upwind third- Weather Research and Forecasting (WRF) Model (Ska- n+1 n s+1j j = 0 230 To tackle the small time step problem connected with tiny hL VL/FL, (11) order procedure with additionalj=1 limiting is used (Van Leer, marock2/ et3 al., 2008) or in the Consortium1 + 1 √ for3 Small-scale cut cells, linear implicitX Rosenbrock-W-methods are used h = 0.5V /(F + F ), 4/3 2 6 (12) 1994). C 5C/4L 3/4 R − (Jebens et al., 2011). − Assuming a positive flow in the x direction, the third-order hR = VR/FR-Matrix. Γ-Matrix γ (13) A approximationA Rosenbrock at methodxi+1/2 hasis obtained the form by quadratic interpolation from the three values as shown in Fig.2. The interpola- Table2.2.3 2. Coefficient Momentum table for ROS2. i 1 i 1 tion condition is that the three− cell-averaged− values are fitted: (I τγJ)k =τF (y + α u ) + β k , i = 1,...,s − i n ij j ij j To solve the momentum equation, the nonlinear advection j=1 j=1 250 termA second is needed method on the was face. constructed This is achieved from by a low a shifting stor- h (hX+ h ) X φ =φ + C L C (φ − φ ) (16) agetechnique three stage introduced second-order by Hicken Runge-Kutta et al.(2005) method, for the incom- which FR C + + + R C (hC hRs)(hL hC hR) ispressible used in Navier–Stokes split-explicitequation. time integration For each methods cell, two in cell- the 235 yn+1 =yn + hChαRs+1jkj , Weather Research and Forecasting (WRF) Model (Ska- + (φC − φL) centered values of each of the three components of the Carte- (h + h )(hj=1 + h + h ) marock et al., 2008) or in the Consortium for Small-scale L C XL C R sian velocity vector are computed and transported with the =φC + α1(φR − φC) + α2(φC − φL). (6) above advection scheme for a cell-centered scalar value. The

Geosci. Model Dev., 8, 317–340, 2015 www.geosci-model-dev.net/8/317/2015/ M. Jähn et al.: ASAM v2.7 321 obtained tendencies are then interpolated back to the faces. Table 2. Coefficient table for ROS2. This approach avoids separate advection routines for the momentum components. For a normal cell the shifted values are 0 obtained from the six momentum face values, whereas for 2/3 √ − − 1 + 1 a cut cell the shift operation takes into account the weights of 5/4 3/4 4/3 2 6 3 the faces of the two opposite sides, compare with Fig.3 for A-Matrix 0-Matrix γ the used notation. Table 3. Coefficient table for ROSRK3. U if FU ≥ FU U = FL L R LC (U FU + U (FU − FU ))/FU else. FL L FR R L R 0 0 (14) 1/3 −11/27 1 11/54 1/2 17/27 −11/4 TU TU The interpolation of the cell tendencies LC and RC −17/27 11/4 1 back to a face tendency TUF is obtained by the arithmetic A-Matrix 0-Matrix γ mean of the two tendencies of the two shifted cell components originating from the same face. For a cut face the interpolation takes the form (see Fig.4) is added to the cut cell (cell 2 in Fig.5). The remaining TURCVL TULCVR part that has to be distributed is weighted by the face val- TUF = + / | − | · FUL + FUC FUR + FUC ues FUL FUR /Fsurf VC/(1x1y1z) to the neighbored cells in x direction (cell 1 in the example) and |FWB − VL VR + . (15) FWT|/Fsurf · VC/(1x1y1z) in z direction (cell 4 in the ex- FUL + FUC FUR + FUC ample), where Fsurf = |FUL−FUR|+|FWB−FWT|. With this The pressure gradient and the Buoyancy term are computed approach, only the available information of the considered for all faces with standard difference and interpolation for- cut cell (volume and common face areas with neighboring mulas with the grid sizes taken from the underlying Carte- cells) and not of its surrounding cells is needed. The exten- sian grid. To approximate the pressure gradient at the inter- sion of this method to the third spatial dimension is done face of two grid cells with only the pressure values of the analogously. two grid cells there is some freedom in choosing the grid size. Whereas in Adcroft et al.(1997) the grid size is chosen 2.3 Time integration to preserve energy in their model, we follow Ng et al.(2009) and do not take in to account the cut cell structure. Both ver- After spatial discretization, an ordinary differential equation sions are implemented in the ASAM code and it became ap- y0(t) = F (y(t)) (18) parent that the second one is more suitable to simulate flows in hydrostatic balance. is obtained that has to be integrated in time (method of lines). To tackle the small time step problem connected with tiny 2.2.4 Boundary flux distribution near cut cells cut cells, linear implicit Rosenbrock W methods are used (Knoth, 2006; Jebens et al., 2011; John and Rang, 2010). Due to small cell volumes around cut cells, boundary fluxes A Rosenbrock method has the form such as sensible and latent surface heat fluxes (cf. Sect. 3.4) have to be distributed to the surrounding cells to avoid insta- i−1 ! i−1 X X bility problems because of sharp gradients in the respective (I − τγ J)ki = τF yn + αij uj + γij kj , scalar fields. For simplicity we consider a two-dimensional j=1 j=1 case in x and z directions. An example configuration is dis- i = 1,...,s, (19) played in Fig.5. The common partial face area of two neigh- s X boring cells is greater than the opposite face (i.e., left vs. yn+1 = yn + αs+1j kj , right in x and y directions, bottom vs. top), e.g., in our case j=1 FUL > FUR and FWT > FWB. For the flux distribution we where yn is a given approximation at y(t) at time tn and first define a virtual volume over the cut cell face FC through subsequently yn+1 at time tn+1 = tn + τ. In addition, J is Vvirt = FChvirt , (16) an approximation to the Jacobian matrix ∂F/∂y. A Rosen- brock method is therefore fully described by the two matrices where A = (αij ), 0 = (γij ) and the parameter γ . hvirt = 1x|nx| + 1z|nz|. (17) Among the available methods are a second-order two- stage method after Lanser et al.(2001). T Here, (nx,nz) is the normal unit vector of the cut cell face FC. Then the flux fraction with a weight of VC/Vvirt Sk1 = τF (yn), (20) www.geosci-model-dev.net/8/317/2015/ Geosci. Model Dev., 8, 317–340, 2015 icsinPpr|Dsuso ae icsinPpr|Dsuso ae | Paper Discussion | Paper Discussion | Paper Discussion | Paper Discussion

322 M. Jähn et al.: ASAM v2.7

...... FUR ...... FU V ...... TU TU ...... C CR...... F LC...... VCL ...... TURC ...... FUL ......

Figure 4. TwoFigure neighbored 4. cut cells with face and volume area information (left) and arrangement of face- and cell-centered tendency of Two neighbored cut cells with | face Paper Discussion | and Paper Discussion | volume Paper Discussion | Paper Discussion area information (left) and arrangement of momentum (right). face and cell centered tendency of momentum (right).

Moreover, a new Rosenbrock method was constructed Vvirt = hvirt FC Δx from a low storage three-stage second-order Runge–Kutta Cell 3 Cell 4 Cell 5 method, which is used in split-explicit time integration methods in the Weather Research and Forecasting (WRF) Model VT Δz (Skamarock et al., 2008) or in the Consortium for Small- scale Modeling (COSMO) model (Doms et al., 2011). Its co- efﬁcients are given in Table3.

FWT The previously described Rosenbrock W methods allow a simplified solution of the linear systems without losing the VC order. When J = JA+JB the matrix S can be replaced by S = Qsurf − − VL FUL FUR (I γ τJA)(I γ τJB). Further simplification can be reached h by omitting some parts of the Jacobian or by replacing of the virt 56 FC derivatives by the same derivatives of a simplified operator n Cell 1 Cell 2 Fe (w ). For instance, higher-order interpolation formulae are FWB replaced by the first-order upwind method. The structure of the Jacobian is Figure 5.FigureExample 5. configurationExample for configuration surface flux distribution for surface around a cut flux cell. distribution Green shading represents the solid part of the cells, whereas blue shading is the "virtual volume" normal to the cut cell  ∂Fρ ∂Fρ  around a cut cell. Green shading represents the solid part | Paper Discussion of | the Paper Discussion | Paper Discussion | Paper Discussion 0 face. The total surface flux Qsurf is distributed within the dotted area. ∂ρ ∂V cells, whereas blue shading is the “virtual volume” normal to the J =  ∂FV ∂FV ∂FV . (24) 8cut cell face. The total surface flux Q is distributed within the  ∂ρ ∂V ∂2  M. Jahn:¨ ASAM v2.7 surf ∂F2 ∂F2 dotted area. 0 ∂V ∂2 57 A 0 block indicates that this block is not included in the Ja- (60) 0 cobian or is absent. The derivative with respect to ρ is only -1 taken for the buoyancy term in the vertical momentumR L equa- Aj Aj -2 tion. Note that this type of approximationmax is− the, standard0 Vmax approach∂(ρθρ in) the∂ derivationR of the Boussinesq approximationVmax V ) -3 + (ρθρuj ) = Sθρ − ,

-1 R starting∂t from the∂xj compressible Euler equations.Asurf The matrix JVmax -4 can be decomposed as (m s

f (61)

W -5  ∂Fρ   ∂F  515 where the superscripts0L and 0 R correspondρ to the left and -6 ∂ρ 0 ∂V 0 J =rightJ + neighborJ =  ∂FV cell,∂F respectively.V  +  The total surface∂FV  is com- -7 T P  ∂ρ ∂V 0  0 0 ∂2  puted by ∂F2 0 ∂F2 0 -8 0 0 ∂2 ∂V 1e-08 1e-07 1e-06 1e-05 0.0001 0.001 0.01 L R (25) -3 Asurf = Σ Aj Aj . (62) r (kg m ) | − | or 3.5 Soil model Figure 6.FigureTerminal 6. fallTerminal velocity of fall raindrops velocity after of Eq. raindrops (56). after Eq. (56).  ∂Fρ   ∂Fρ  Fig. 4. Terminal fall velocity of raindrops after Eq. (52). 520 In order to account∂ρ 0 for the 0 interaction0 between∂V 0 land and at- ∂F ∂F 2 4 = + =  ∂FV + V V  J mosphereJT JP and0 the high diurnal0 variability∂ρ 0 of the∂2 meteorolog-. Sk2 = τF yn + k1 − k1 , (21)  ∂V    3 3 ical variables0 in the 0 surface∂F2 layer, a0 soil∂F model2 has0 been im- 3 1 ∂2 ∂V Sv is the source5 term3 of water vapor in units of [kg m− s− ]. plemented into ASAM. In contrast to the constant(26) flux layer yn+1 = yn + k1 + k2 , 58 (22) 490 Considering Eq.4 (A33),4 adding the sensible heat flux and ne- model, the computation of the heat and moisture fluxes are 0 glectingS = I − phaseγ τJ, changesJ ≈ F (y leadsn), to (23) 525Thenow first dependentpart of the splitting on radiation,JT is called evaporation the transport/source and the transpira- √ parttion and of contains vegetated the area. advection, Phase diffusion changes and are notsource covered terms yet and ∂(ρθρ) = 1 ∂+ 1 with γ 2 6 3 or in matrix form in Table2. likeintercepted Coriolis, curvature, water is only buoyancy, considered latent in heat, liquid and state. so on. + (ρθρuj) = Sθρ (55) ∂t ∂xj Two different surface flux schemes are implemented, following the revised Louis scheme as integrated in the withGeosci. Model Dev., 8, 317–340, 2015 www.geosci-model-dev.net/8/317/2015/ 530 COSMO model (Doms et al., 2011) and the revised flux Sh Sv Rv Rv cpv scheme as used in the WRF model (Jimenez´ et al., 2012). Sθρ = ρθρ + lnπ (56) T ρd Rm − Rm − cpml The surface fluxes of momentum, heat and moisture are parameterized in the following way, respectively: 1 495 where Sh is the heat source in units of [K s− ], Rm = Rd + τ = ρC v u(h), (63a) rvRv and cpml = cpd +rvcpv +rlcpl are the gas constant and zx m| h| the specific heat capacity for the air mixture, respectively. 535 ρcpw0θ0 = ρcpCh vh (θ(h) θ(z0T )) , (63b) 2 − | | − The corresponding surface fluxes in [W m− ] are: ρLw0q0 = ρLCq vh (q(h) q(z0q)) . (63c) ρ c − | | − S = S d pml , (57) sens h ρA Cm, Ch and Cq are the bulk transfer coefficients and it is con- C = C V sidered that h q. As described in (Doms et al., 2011), 500 Slat = SvLv(T ) . (58) 540 the bulk transfer coefficients are defined as the product of the A n transfer coefficients under neutral conditions Cm,h and the Here, L = L + (c c )T is the latent heat of vaporiza- stability functions Fm,h depending on the Bulk-Richardson- v 00 pv − pl tion, A is the cell surface at the bottom boundary and V the Number RiB and roughness length z0.

cell volume. n Cm,h = C Fm,h (RiB,z/z0) . (64) 505 For the computation of the surface fluxes around cut cells, m,h an interpolation technique is used: 545 In Jimenez´ et al. (2012) the bulk transfer coefficients are defined as follows ∂(ρθρ) ∂ V + (ρθρuj) = Sθρ min , 1 (59) 2 ∂t ∂xj Vmax k Cm,h = (65) ΨM ΨM,H with the maximum cell volume Vmax = ∆x∆y∆z. For surrounding cells, the missing flux fraction is distributed de- with L R 510 pending on the left and right cut faces A and A in all z + z0 z + z0 z0 spatial directions: ΨM,H = ln φm,h + φm,h (66) z0 − L L AL AR max j − j , 0 550 and φm,h representing the integrated similarity functions. L Vmax ∂(ρθρ) ∂ L Vmax V stands for the Obukhov length and k is the von-Karm´ an-´ + L (ρθρuj ) = Sθρ − , ∂t ∂xj Asurf Vmax constant. In neutral to highly stable conditions φm,h follows M. Jähn et al.: ASAM v2.7 323

The second matrix JP is called the pressure part and in- the derivative of the pressure in the cell center with respect volves the derivatives of the pressure gradient, with respect to to the density-weighted potential temperature. Elimination of the density-weighted potential temperature, and of the diver- the momentum part gives a Helmholtz equation for the incre- gence, with respect to momentum of the density and potential ment of the potential temperature. This equation is solved by temperature equation. The difference between the two split- a CG-method with a multi-grid as a preconditioner. For the ting approaches is the insertion of the derivative of the grav- second splitting (Eq. 26), the resulting matrix is doubled in ity term in the transport or pressure matrix. The ﬁrst split- dimension and not symmetric anymore. ting (Eq. 25) damps sound waves. For this splitting the sec- As basic iterative solvers, BiCGStab is applied for the ond linear system with the pressure part of the Jacobian can transport/source system and GMRES for the pressure part be reduced to a Poisson-like equation. The second splitting (Dongarra et al., 1998). The number of iterations for the two (Eq. 26) damps sound and gravity waves but the dimension iterative methods are problem dependent. They increase with of the pressure system is doubled. Both systems are solved by increasing time step and are usually in the range of 2–5 iter- preconditioned conjugate-gradient (CG)-like methods (Don- ations. garra et al., 1998). For the transport/source system the Jacobian can be further split into 3 Physical parameterizations

JT = JAD + JS, (27) 3.1 Smagorinsky subgrid-scale model where the matrix JAD is the derivative of the advection and The set of coupled differential equations can be solved for diffusion operator where the unknowns are coupled between a given flow problem. For simulating turbulent flows with grid cells. The matrix JS assembles the source terms. Here large eddy simulations, the Euler equations mentioned at the coupling is between the unknowns of different compo- the beginning have to be modified. Within the technique of nents in each grid cell. With this additional splitting the linear LES it is necessary to characterize the unresolved motion. equation By solving Eqs. (1)–(3) numerically with a grid size which is larger than the size of the smallest turbulent scales, the (I − γ τJAD − γ τJS)1w = R (28) equations have to be filtered. Large eddy simulation employs a spatial filter to separate the large-scale motion from the is preconditioned from the right with the matrix small scales. Large eddies are resolved explicitly by the prog- − nostic Euler equations down to a predefined filter-scale 1, P = (I − γ τJ ) 1 (29) r AD while smaller scales have to be modeled. Due to the filtering and from the left with the matrix operation, additional terms that cannot be derived trivially occur in the set of Euler equations. −1 Pl = (I − γ τJS) . (30) Nevertheless, to solve the filtered set of equations, it is necessary to parameterize the additional subgrid-scale stress The matrix terms τij = uiuj −uiuj for momentum and qij = uiqj −uiqj for potential. Note that τ expresses the effect of subgrid- − − ij Pl(I γ τJAD γ τJS)Pr (31) scale motion on the resolved large scales and is often repre- can be written in the following form by using the Eisenstat sented as an additional viscosity νt with the following formu- trick (Eisenstat, 1981): lation: τ = −2ν S , (34) (I − γ τPlJAD)Pr = (I + Pl((I − γ τJAD) + I))Pr. (32) ij t ij

∂u Therefore, only the LU-decomposition of the matrix (I − where S = 1 ∂ui + j is the strain rate tensor and ν the ij 2 ∂xj ∂xi t γ τJS) has to be stored. The matrix (I − γ τJAD) is inverted turbulent eddy viscosity. To determine the additional eddy by a fixed number of Gauss–Seidel iterations. viscosity, the standard Smagorinsky subgrid-scale model The second matrix of the splitting approach is written in (Smagorinsky, 1963) is used: case of the first splitting (Eq. 25) as follows: ν = (C 1)2|S|, (35) I γ τGRADD t s (I − γ τJ ) = 2 , (33) P γ τDIVD I V where 1 is a length scale, Cs the Smagorinsky coefficient, and using the Einstein summation notation for standardiza- where DV and D2 are diagonal matrices. GRAD and DIV q are matrix representations of the discrete gradient and diver- tion |S| = 2Sij Sij . The grid spacing is used as a measure gence operator. The entries of the matrix DV are the potential for the length scale. This standard Smagorinsky subgrid- temperature at cell faces and the entries of the matrix D2 are scale model is widely used in atmospheric and engineering www.geosci-model-dev.net/8/317/2015/ Geosci. Model Dev., 8, 317–340, 2015 324 M. Jähn et al.: ASAM v2.7 applications. The Smagorinsky coefficient Cs has a theoreti- 3.2 Two-moment warm cloud microphysics scheme cal value of about 0.2, as estimated by Lilly(1967). Applying this value to a turbulence-driven flow with thermal convec- The implemented microphysics scheme is based on the work tion fields results in a good agreement with observations as of Seifert and Beheng(2006). This scheme explicitly rep- shown by Deardorff(1972). resents two moments (mass and number density) of the hy- To take stratification effects into account, Lilly(1962) drometeor classes cloud droplets and rain drops. Ice phase modified the standard Smagorinsky formulation by changing hydrometeors are currently not implemented in the model. the eddy viscosity to Altogether, seven microphysical processes are included: condensation/evaporation (COND), cloud condensation nuclei Ri 1/2 (CCN) activation to cloud droplets at supersaturated con- ν = (C 1)2max 0, |S|2 1 − (36) t s Pr ditions (ACT), autoconversion (AUTO), self-collection of cloud droplets (SCC), self-collection of rain drops (SCR), with accretion (ACC) and evaporation of rain (EVAP):

g ∂θρ ∂(ρqv) θ ∂z + ∇ · (ρvq ) = −S − S + S , (41) Ri = ρ . (37) ∂t v COND ACT EVAP | |2 S ∂(ρqc) + ∇ · (ρvqc) = +SCOND + SACT − SAUTO − SACC , (42) Here, Ri is the Richardson number and Pr is the turbulent ∂t ∂(ρqr) Prandtl number. In a stable boundary layer the vertical gra- + ∇ · (ρvqr) = +SAUTO + SACC − SEVAP , (43) dient of the potential temperature is greater than zero (posi- ∂t ∂N tive), which leads to a positive Richardson number and, thus, CCN + ∇ · = − − + (vNCCN) SCONDN SACTN SEVAPN , (44) the additional term Ri/Pr reduces the square of the strain rate ∂t ∂N tensor and decreases the turbulent eddy viscosity. Therefore, c + ∇ · = + + − (vNc) SCONDN SACTN SAUTON less turbulent vertical mixing takes place. ∂t − − The implementation in the ASAM code is accomplished SACCN SSCC , (45) in the main diffusion routine of the model. It develops the ∂Nr + ∇ · (vNr) = +SAUTO + SACC − SEVAP − SSCR . (46) whole term of ∂/∂xj ρDSij for every time step. The coef- ∂t N N N ficient D represents Dmom for the momentum and Dpot for Details on the conversion rates can be found in Seifert and the potential subgrid-scale stress. Further routines describe Beheng(2006). Additionally, a limiter function is used to the computation of Dmom and Dpot in the following way: ensure numerical stability and avoid nonphysical negative values (Horn, 2012). Since there is no saturation adjustment = 2| | Dmom (Cs1) S . (38) technique in ASAM, the condensation process is taken as an example to demonstrate the physical meaning of the limiter The potential subgrid-scale stress is related to the Prandtl functions. Considering the available water vapor density ρ similarity and can be developed by dividing the subgrid-scale v and the cloud water density ρ , the process of condensation stress tensor for momentum by the turbulent Prandtl number c (or evaporation of cloud water, respectively) is forced by the Pr that typically has a value of 1/3 (Deardorff, 1972). The water vapor density deficit and limited by the available cloud length scale 1 in the standard Smagorinsky formulation is set water. to the value of grid spacing. However, the cut cell approach makes it difficult because of tiny and/or anisotropic cells. To FOR = ρv − (pvsT /Rv) (47) overcome this deficit the value is defined after Scotti et al. LIM = ρc (48) (1993): FOR − LIM + (FOR2 + LIM2)1/2 1/3 SCOND = (49) 1 = (111213) f (a1,a2). (39) τCOND Here, p is the saturation vapor pressure and the relaxation 1 is the grid spacing in orthogonal directions, and a correc- vs time is set to τ = 1 s. The numerator term is called the tion function f is applied as follows: COND Fischer–Burmeister function and has originally been used in 4 1/2 the optimization of complementary problems (cf. Kong et al., f (a ,a ) = cosh ln2a − lna lna + ln2a 2010). A simple model after Horn(2012) is applied to deter- 1 2 27 1 1 2 2 mine the corresponding changes in the number concentra- 11 12 with a1 = , a2 = . (40) tions and to ensure a reduction of the cloud droplet number 13 13 density to zero if there is no cloud water present. This means that Nc decreases when droplets are getting too small, Here, a1 and a2 are the ratios of grid spacing in different directions with the assumption that 1 ≤ 1 ≤ 1 . For an ρ 1 2 3 = c − = SCONDN min 0,C Nc , (50) isotropic grid f 1. xmin

Geosci. Model Dev., 8, 317–340, 2015 www.geosci-model-dev.net/8/317/2015/ M. Jähn et al.: ASAM v2.7 325 and increases when droplets are getting too large, 3.4 Surface ﬂuxes

ρ A simple way to parameterize surface heat fluxes is the us- = c − SCONDN max 0,C Nc , (51) age of a constant flux layer. There, the energy flux is di- xmax rectly given and does not depend on other variables. With the where xmin and xmax are limiting parameters for cloud wa- density potential temperature formulation (Eq.3), the source ter. This ensures that the cloud droplet number concentra- term for this quantity has to be calculated: tion is within a certain range defined by distribution param- ∂(ρθρ) ∂ ∂θρ ∂ρ ∂ρuj ∂θρ eters in Seifert and Beheng(2006) if condensate is present. + (ρθ u ) =ρ + θ + θ + ρu ∂t ∂x ρ j ∂t ρ ∂t ρ ∂x j ∂x A timescale factor of C = 0.01 s−1 controls the speed of this j j j correction and appears to be reasonable for this particular ∂θρ ∂θρ ∂ρ ∂ρuj =ρ + uj + θρ + process. ∂t ∂xj ∂t ∂xj dθρ =ρ + θρS . (58) 3.3 Precipitation dt v −3 −1 The sedimentation velocity of raindrops is derived as in Sv is the source term of water vapor (in kgm s ). Consid- the operationally used COSMO model from the German ering Eq. (A33), adding the sensible heat flux and neglecting Weather Service (Doms et al., 2011). There, the following phase changes leads to assumptions are made. The precipitation particles are ex- ∂(ρθρ) ∂ ponentially distributed with respect to their drop diameter + (ρθρuj ) = Sθρ (59) (Marshall–Palmer distribution): ∂t ∂xj with = r − fr(D) N0 exp λrD. (52) S S R R cpv S = ρθ h + v v − lnπ v − , (60) θρ ρ T ρ R R c Here, λr is the slope parameter of the distribution function d m m pml and N r = 8 × 106 m−4 is an empirically determined distri- 0 where S is the heat source (in Ks−1), R = R + r R and bution parameter. The terminal fall velocity of raindrops is h m d v v c = c + r c + r c are the gas constant and the spe- then assumed to be uniquely related to drop size, which is pml pd v pv l pl cific heat capacity for the air mixture, respectively. The cor- expressed by the following empirical function: responding surface fluxes (in Wm−2) are

W (D) = c D1/2 (53) ρdcpml f r S = S , (61) sens h ρA 1/2 −1 with cr = 130m s . Finally, the precipitation flux of rain- V S = S L (T ) . (62) water can be calculated as lat v v A ∞ Here, L = L + (c − c )T is the latent heat of vaporiza- Z v 00 pv pl tion, A is the cell surface at the bottom boundary and V the Pr = ρrWf(ρr) = m(D)Wf(D)fr(D)dD (54) cell volume. 0 3.5 Soil model with the raindrop mass In order to account for the interaction between land and at- 3 m(D) = πρWD /6, (55) mosphere and the high diurnal variability of the meteorological variables in the surface layer, a soil model has been im- −3 where ρW = 1000kgm is the mass density of water. This plemented into ASAM. In contrast to the constant flux layer leads to an expression for the terminal fall velocity of rain- model, the computation of the heat and moisture fluxes are drops in dependence on their density: now dependent on radiation, evaporation and the transpiration of vegetated area. Phase changes are not covered yet and 1/8 0(4,5) ρr intercepted water is only considered in liquid state. Wf(ρr) = −cr . (56) 6 πρWN0r The implemented surface flux scheme follows the description of Jiménez et al. (2012), which is the revised flux scheme This takes place at the tendency equation for the rain water used in the WRF model. The surface fluxes of momentum, density: heat and moisture are parameterized in the following way, respectively: ∂(ρqr) ∂ + ∇h · (ρvhqr) + (ρqr [w + Wf]) = Sq . (57) = | | ∂t ∂z r τzx ρCm vh u(h), (63a) www.geosci-model-dev.net/8/317/2015/ Geosci. Model Dev., 8, 317–340, 2015 326 M. Jähn et al.: ASAM v2.7

0 0 − ρcpw θ = ρcpCh|vh|(θ(h) − θ(z0T )) , (63b) Planton(1989). The variation of the soil temperature is a re- 0 0 sult of heat conductivity depending on the soil texture and − ρLw q = ρLC |v | q(h) − q(z q ) . (63c) q h 0 the soil water content of the respective soil layer:

Cm, Ch and Cq are the bulk transfer coefficients and it is con- ∂Tsoil 1 ∂ ∂Tsoil sidered that Ch = Cq. In Jiménez et al.(2012), the bulk trans- = λ + Eq ρwcwT soil . (70) fer coefficients are defined as follows: ∂t ρc ∂z ∂z

k2 Tsoil is the absolute temperature in the kth soil layer (in K), C = (64) T is the mean soil temperature of two neighboring soil m, h 9 9 soil M M, H layers. The change in internal energy due to changes in mois- with ture by the inner soil water flux, evapotranspiration and evaporation from the upper soil layer and the interception reser- z + z z + z z voir is treated by the second term in brackets. The heat con- = 0 − 0 + 0 9M ,H ln φm, h φm, h (65) ductivity λ and the volumetric heat capacity ρc are variables z0 L L that depend on the soil texture. The heat capacity of the soil and φm, h representing the integrated similarity functions. L ρc formulated by Chen and Dudhia(2001) is the sum of the stands for the Obukhov length and k is the von Kármán heat capacity of dry soil (ρ0c0, see Tables B1 and B2), the constant. In neutral to highly stable conditions φm, h follows heat capacity of wet soil (ρwcw) and the heat capacity of the Cheng and Brutsaert(2005) and in unstable situations the φ- air within the soil pores (ρaca). functions follow Fairall et al.(1996). For further details con- cerning limitations and restrictions see Jiménez et al.(2012). ρc = Wsoilρwcw + 1 − Wpv ρ0c0 + Wpv − Wsoil ρaca (71) The transport of the soil water as a result of hydraulic pressure due to diffusion and gravity within the soil layers is de- with Wpv corresponding to the soil pores, ρwcw = 4.18 × 6 −3 −1 −3 −1 scribed by Richard’s equation: 10 Jm K and ρaca = 1298 Jm K . The heat conductivity λ is defined after Pielke(1984): ∂Wsoil,k ∂ ∂Wsoil,k = Diff + κsoil,k (66) ( ∂t ∂z ∂z 418exp −9log − 2.7 if 9log ≤ 5.1 λ = (72) 0.172 if 9log > 5.1 with the diffusion coefficient with 9 = log |1009 |. ∂9soil,k log 10 soil Diff = κsoil,k . (67) The topmost layer is exposed to the incoming radiation ∂W soil,k and thus has the strongest variation in temperature in comparison to the other soil layers within the ground. The tem- Wsoil,k is the volumetric water content in the kth soil layer. perature equation of the first layer is, in addition to the in- 9soil stands for the matric potential and κsoil is the hy- coming radiation, determined by the latent and sensible heat draulic conductivity. 9soil and κsoil are parameterized based on Van Genuchten(1980): flux. 0 ∂Tsoil,1 1 ∂ ∂Tsoil,1 p h 1 im2 = λ + 1Q (73) κsoil = κsat Weff 1 − 1 − (Weff) m (68) ∂t ρc ∂z ∂z

1 h − 1 i n with 9soil = 9sat (Weff) m − 1 . (69) 4 1Q = Qdir + Qdif − σ Tsfc − cpQSH − LvQLH. (74) Weff describes the effective soil wetness, which takes a resid- ual water content W into account, restricting the soil from res Here, QLH is the latent heat flux, describing the moisture flux complete desiccation. κsat and 9sat are the hydraulic conduc- between soil and atmosphere as the sum of evaporation and tivity and the matric potential at saturated conditions, respec- transpiration and QSH is the sensible heat flux. Qdir and Qdif tively. The parameters m and n describe the pore distribution represent the direct and diffusive irradiation, respectively. (Braun, 2002) with m = 1−1/n (also see Tables B1 and B2). Further addition/extraction of soil water is controlled by the percolation of intercepted water into the ground and the 4 Test cases evaporation and transpiration of water from bare soil and vegetation. The mechanisms implemented are based on the In this section, we present six example test cases. In five of Multi-Layer Soil and Vegetation Model TERRA_ML as de- these cases, orography or obstacles are included to test con- scribed in Doms et al.(2011). The evaporation of bare soil servation properties and model accuracy. The first test case is is adjusted to the parameterization proposed by Noilhan and the rising heat bubble prescribed in Wicker and Skamarock

M. Jähn et al.: ASAM v2.7 327

0 -6 -4 -2 0 2 4 6 -6 -4 -2 0 2 4 6

Figure 7. Results for the dry bubble simulation after Wicker and Skamarock(1998) at t = 1000 s: contours of perturbation potential tem- peratureFigure (left 7. panel)Results and vertical for thevelocity dry (right bubble panel). simulation Solid (dashed) after lines indicateWicker positive and Skamarock(negative) values. (1998) Contour at intervalst = 1000 are 0.25 s: K (zerocontours contour omitted) of perturbation and 2 ms−1, respectively. potential temperature (left panel) and vertical velocity (right panel). Solid (dashed) lines indicate positive (negative) values. Contour intervals are 0.25 K (zero contour omitted) 1 (1998and). 2 Thems bubble− , respectively. is initially defined by a radial tempera- applied. After 1000 s the bubble has been transported through ture perturbation, which leads to rising motion due to buoy- the lateral periodic boundaries and is again located at the cen- ant forces. Results of a two-dimensional gravity wave simu- ter of the domain. The perturbation field takes the following lation (Schaer et al., 2002) are shown in the second test case. form: There, the gravity waves are induced by an idealized moun- πL tain ridge. Two subcases with different atmospheric stabili- θ 0 = 2cos2 (75) ties and reference temperatures are examined. The third case 2 is a sinking cold bubble in a dry environment, from which with a density current develops (Straka et al., 1993). A 1 km tall s 2 2 hill is added at the left side of the domain so that the resulting x − xc z − zc current overflows over the mountain. Considering moisture L = + ≤ 1 . (76) 59 xr zr effects and phase changes, the moist bubble case by Bryan and Fritsch(2002) with the addition of a mid-air zeppelin The parameters xc = 10 km, zc = 2 km and xr = zr = 2 km (Klein et al., 2009; Jebens et al., 2011) is performed. Be- determine the position and radius of the heat bubble. The at- sides analyzing the flow field in the vicinity of the obstacles, mosphere is in hydrostatic balance and neutrally stable with conservation studies regarding total energy are performed a surface pressure p0 = 1000 hPa and a constant potential for both cases. Another idealized benchmark case is carried temperature θ = 300 K. Results of this simulation are dis- out to analyze the accuracy of the presented discretization played in Fig.7. The overall shape is reproduced and com- method for cut cells. There, a scalar field is advected by a ra- parable to the reference solution. Slight asymmetries are ob- dial wind field in an annulus (Berger and Helzel, 2012). This served due to lateral transport. However, it becomes apparent is also a suitable test for convergence studies by calculating that third-order Runge–Kutta time integration together with a L1 and L∞ error norms since an analytical solution can be fifth-order advection scheme produces better results in terms used for comparison. The last test case is a three-dimensional of minimum/maximum values and symmetry (cf. Bryan and simulation study regarding flow dynamics around an ide- Fritsch, 2002) than the third-order advection scheme that is alized mountain and orographic precipitation by Kunz and used here. Wassermann(2011). 4.2 2-D mountain gravity waves 4.1 Dry bubble In this test case, a flow over a mountain ridge is simulated A two-dimensional simulation of a rising thermal is pre- (Schaer et al., 2002). Skamarock et al.(2012) A dry sta- sented in Wicker and Skamarock(1998). This test case is also ble atmosphere is defined by a constant Brunt–Väisälä fre- used as a dry reference case for the moist bubble simulation quency N, upstream surface temperature T0, surface pres- in Bryan and Fritsch(2002). The domain is 20 km in the hori- sure p0 and a uniform inflow velocity U. The domain ex- zontal direction and 10 km in the vertical direction with a uni- tends 200 km horizontally and 19.5 km vertically with grid form grid spacing of 125 m. A mean flow of U = 20 ms−1 is spacings of 1x = 500 m and 1z = 300 m. The structure of www.geosci-model-dev.net/8/317/2015/ Geosci. Model Dev., 8, 317–340, 2015 icsinPpr|Dsuso ae icsinPpr|Dsuso ae | Paper Discussion | Paper Discussion | Paper Discussion | Paper Discussion

328 M. Jähn et al.: ASAM v2.7

10 a) b) 9

0 -20-30 -10 0 10 20 30 -30 -10-20 0 10 20 30

Figure 8. Steady-state solution for the simulation of the Schaer et al.(2002) test case: contours of vertical velocity for (a) N = 0.01 s−1 and (b)FigureN = 0.01871 8. Steady-state s−1. Solid (dashed) solution lines indicate for the positive simulation (negative) of values. the Contour Schaer intervals et al. are (2002)(a) 0.05 test ms− case:1 and (b) contours0.09355 ms of−1, 1 1 respectively.vertical velocity for a) N = 0.01 s− and b) N = 0.01871 s− . Solid (dashed) lines indicate positive

5 4.3 Cold bubble with orography interaction

A nonlinear test problem is the density current simulation 0 study documented in Straka et al.(1993). In this case, the computational domain extends from −18 to 18 km in the hor- -5 %) izontal direction and from 0 to 6.4 km in the vertical direction -4 with isotropic grid spacing of 1x = 1z = 100 m. Boundary

-10 conditions are periodic in the x direction and the free-slip condition is applied for the top and bottom model boundaries. The total integration time is t = 1800 s. The initial atmo- -15 60 sphere is in a dry and hydrostatically balanced state and there is a horizontally homogeneous environment with θ = 300 K

-20 (i.e., neutrally stratified). The perturbation (cold bubble with Relative total energy error (10 negative buoyancy) is defined by a temperature perturbation of -25 ( Hill 0.0 ◦C if L > 1.0 Flat 0 = Difference T ◦ , (78) -30 −15.0 C(cos[πL] + 1.0)/2 if L ≤ 1.0 0 200 400 600 800 1000 1200 1400 1600 1800 Time (s) where 2 20.5 Figure 9.FigureTime series 9. Time of total series energy of error total for the energy density error current for test the case density with and current without the hill. h −1i h −1i 4 L = (x − x )x + (z − z )z (79) The errortest is expressed case with as and10− without% of the thetotal hill. energy The at the error beginning is expressed of the simulation. as 10−4 % c r c r of the total energy at the beginning of the simulation. 61 and xc = 0.0 km, xr = 4.0 km, zc = 3.0 km and zr = 2.0 km. At first, there is no fixed physical viscosity turned on as in the original test case (with ν = 75 m2 s−1) since a conservation the mountain ridge is represented by a bell curve shape with test regarding total energy is carried out. For this test, two superposed variations: simulation runs are performed with (a) the above described standard setup and (b) a modified setup where a mountain is 2 2 h(x) = h0 exp −[x/a] cos (πx/λ), (77) added at the left part of the domain. The mountain follows the “Witch of Agnesi” curve: = = = ( 2 where h0 250 m, a 5 km and λ 4 km. The simulation H/(1 + [(x − xM)/a1] ) if x < xM result for the steady state is shown in Fig.8. There are no h(x) = , (80) H/(1 + [(x − x )/a ]2) if x ≥ x nonphysical distorted wave patterns and the result agrees M 2 M well with the analytical and reference solutions shown in with half-width lengths a1 = a2 = 1 km, mountain peak cen- Schaer et al.(2002). ter position xM = −6 km and mountain height H = 1 km.

M. Jähn et al.: ASAM v2.7 329

0 -12-18 0-6 6 12 18

= | Paper Discussion | Paper Discussion | Paper Discussion | Paper Discussion Figure 10. PotentialFigure 10. temperaturePotential ﬁeld temperature at t 900 s forﬁeld the at densityt = 900 current s for test the case density with an current Agnesi test hill caseon the with left side an of’Agnesi’ the domain and for different grid spacings 1x = 1z = 200, 100, 50, and 25 m (top to bottom). hill on the left side of the domain and for different grid spacings ∆x = ∆z = 200, 100, 50, 25 m (top to bottom). ulations in the order of 10−3 % at the end of the integration 10 time. However, this is still acceptable due to the fact that in the test case there are very sharp gradients in potential tem- 8 62 perature and wind speeds. Also, the difference of the total energy error between the two cases is very small (10−4 %).

6 This means that, in this case, cut cells do not affect the conservation properties in the model at all. Total mass is always conserved within machine precision. 4 Another analysis is carried out by switching on the physical viscosity of ν = 75 m2 s−1 as in Straka et al.(1993). 2 Four simulations are performed with different isotropic grid spacings of 200, 100, 50 and 25 m, respectively. The poten- 0 tial temperature field after 900 s of integration time for these -6-8-10 -4 -2 0 2 4 6 8 10 spatial resolutions is shown in Fig. 10. Table4 shows minimum/maximum values of horizontal wind speed and poten- Figure 11. Equivalent potential temperature field for the moist rising bubble test case with back- Figure 11. Equivalent1 potential temperature field for the moist ris- tial temperature at this time. Skamarock et al.(2012) pointed ground wind of U = 20 m s− . Snapshot taken at t = 1000 s simulation time. −1 ing bubble test case with background wind of U = 20 ms . Snap- out that their solutions show convergence at the 50 m spacing shot taken at t = 1000 s simulation time. for this test case (without hill) with a fully compressible nonhydrostatic model. The same behavior can be observed with ASAM simulations (not shown here), which does also not Figure9 shows the temporal evolution63 of the total energy error for both simulations. In a dry atmosphere, the total energy change when the mountain is added to the domain. Despite is that there is a slight change in maximum wind speed, the potential temperature field for the 25 and 50 m resolutions are 2 Ed = ρ(qdcvdT + gz + 0.5|v| ). (81) nearly identical. Some notable differences in the field can be observed for the 100 m resolution, which are even more pro- Since exact energy conservation is not expected due to the nounced for the 200 m simulation. model design, there is some kind of energy loss for both sim-

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Table 4. Convergence study for the density current test case with 0.5 a 1 km tall hill. Minimum/maximum values of horizontal velocity and potential temperature for different grid spacings. 0

−1 −1 -0.5 1x umin (ms ) umax (ms ) θmin (K) θmax (K) %) 200 −25.93 35.64 291.89 300.01 -4 -1

100 −28.87 38.52 290.85 300 | Paper Discussion .01 | Paper Discussion | Paper Discussion | Paper Discussion 50 −28.90 38.31 290.71 300.00 -1.5 25 −28.91 37.89 290.70 300.00 -2

-2.5 10 Relative total energy error (10 -3 8

-3.5 Zeppelin U = 20 m s-1 6 Difference -4 0 200 400 600 800 1000 1200 1400 1600 1800 2000

4 Time (s)

Figure 13.FigureSame as 13. Fig.Same 9, but as for inthe Fig. zeppelin9, but and for the thelateral zeppelin transported and moist the bubble lateral test cases. 2 transported moist bubble test cases.

65 0 -6-8-10 -4 -2 0 2 4 6 8 10 duced in Klein et al.(2009) and Jebens et al.(2011). How- ever, their tests were carried out with the dry bubble, which Figure 12.FigureEquivalent 12. potentialEquivalent temperature potential field temperature for the moist rising field bubble for thetest including moist ris- a zeppelin-was also shifted 1 km to the left. The result for the first case is shaped cut area in the center of the domain. Snapshot taken at t = 1250 s simulation time. ing bubble test including a zeppelin-shaped cut area in the center of shown in Fig. 11. The equivalent potential temperature field the domain. Snapshot taken at t = 1250 s simulation time. is very close to the benchmark simulation, despite that the maximum value of θe is a little bit lower in our case compared to the literature values and that there is a slight asymmetry at 4.4 Moist bubble with mid-air64 zeppelin the top of the thermal due to lateral transport. The position of the rising thermal for the zeppelin case after t = 1250 s is The moist bubble benchmark case after Bryan and Fritsch shown in Fig. 12. Because of the centered obstacle, the bub- (2002) is based on its dry counterpart described in Wicker ble is split into two parts and deformed but, still, two typical and Skamarock(1998). There, a hydrostatic and neutrally rotors are formed by each bubble and the result remains sym- balanced initial state is realized by a constant potential tem- metric. When moisture and liquid water are present, the total perature. A warm perturbation in the center of the domain energy takes the form leads to the rising thermal. For the present test case, a moist neutral state can be expressed with the equivalent potential 2 Et = ρ qdcvd + qvcvv + qlcpl T + qvL00 + gz + 0.5|v| . temperature θe and two assumptions: the total water mixing ratio rt = rv+rl remains constant and phase changes between (82) water vapor and liquid water are exactly reversible. The perturbation field is identical to the dry bubble test case (Eqs. 75 Again, energy is not fully conserved but the total relative and 76). The domain is 20 km long in the x direction and energy error after 2000 s of simulation time (there, in both the vertical extent is 10 km. Grid spacing is again isotropic cases, the bubbles reach the top boundary resulting in zonal with 1x = 1z = 100 m. Periodic boundary conditions are divergence) stays in an acceptable range of 10−4 % (Fig. 13), applied in the lateral direction, whereas free-slip conditions which is 1 order of magnitude smaller than in the cold bubble are used for the top and bottom boundaries. Again, a total test case. The difference of the error in total energy between energy test is performed by comparing two modifications of the zeppelin and the classical case is again very small. So, the present test case: (a) a uniform horizontal wind speed of even with very small cut cells (≈ 1 % of full cell volume) and U = 20 ms−1 is applied. With that, the center of the bubble is microphysical conversions there is no indication that conser- again located at x = 0 m at t = 1000 s after passing through vation properties are deteriorated. For all cases, total mass the periodic boundaries. (b) In the center of the domain, a is conserved within the numerical accuracy. After Bryan and zeppelin-shaped region is cut out and acts as an obstacle for Fritsch(2002), both mass and energy conservation are re- the rising bubble. A similar test like this was already intro- quired to obtain the benchmark result.

Geosci. Model Dev., 8, 317–340, 2015 www.geosci-model-dev.net/8/317/2015/ M. Jähn et al.: ASAM v2.7 331

Table 5. Convergence study for the annulus advection test. L1 error norm (full domain), experimental order of convergence (EOC), L∞ error norm, minimum and maximum tracer values for different meshes.

N Domain L1 error EOC L∞ error φmin φmax 50 1.6377 × 10−2 – 1.9176 × 10−1 −2.0022 × 10−3 1.00048 −3 −1 −4 100 4.9439 × 10 1.73 1.1860 × 10 −8.3884 × 10 1.00066 | Paper Discussion | Paper Discussion | Paper Discussion | Paper Discussion 200 1.3653 × 10−3 1.86 6.3112 × 10−2 −8.5424 × 10−13 0.99977 400 3.7196 × 10−4 1.88 3.1822 × 10−2 −2.6209 × 10−16 0.99977 800 9.7302 × 10−5 1.93 1.9877 × 10−2 −1.0274 × 10−12 0.99978

Figure 14. Computational meshes and difference scalar fields of φ for (a) N = 50, (b) N = 100, (c) N = 200, (d) N = 400, (e) N = 800, and (f) the scalar field for N = 400 after one rotation. Figure 14. Computational meshes and difference scalar fields of φ for a) N = 50, b) N = 100, c) N =4.5 200 Annulus, d) N advection= 400, e) testN = 800, f) scalar field for Nshows= 400 theafter difference one fieldsrotation. between the analytical and numerical solutions (1φ) for different mesh sizes, where N is The test problem reported in Berger and Helzel(2012) de- the amount of grid cells in each spatial direction. Figure 14f scribes the advection of a smooth bump by a radial wind field shows the final field after 5 s of integration time for N = 400. in an annulus. It is described by the radius of the inner cir- With greater N, the order of magnitude of the error decreases cle R1 = 0.75 and the radius of the outer circle R2 = 1.25 for the inner and outer boundaries and the intermediate part [− ] × [− ] within a rectangular domain 1.5, 1.5 1.5, 1.5 . The66 of the annulus is less affected. Table5 shows the results of initial scalar field takes the following form: the convergence study, including error norms, experimental orders of convergence (EOC) as well as minimum and max- φ = . {ϑ − π/ } + { π/ − ϑ} , 0 5 erf 5 3 erf 5 2 3 (83) imum values of the tracer field for different mesh sizes. For where ϑ = arctan(y/x). Deriving the velocity field from a fixed time step (0.01, 0.005, 0.0025 and 0.00125 s, respec- the stream function ψ(x,y) = π(R2 − r2)/5 with r = (x2 + tively) the advection scheme used in ASAM together with the 2 Koren limiter shows almost second-order convergence in the y2)1/2, one full rotation is reached at t = 5 s. Figure 14a–e 1 www.geosci-model-dev.net/8/317/2015/ Geosci. Model Dev., 8, 317–340, 2015 332 M. Jähn et al.: ASAM v2.7 icsinPpr|Dsuso ae icsinPpr|Dsuso ae | Paper Discussion | Paper Discussion | Paper Discussion | Paper Discussion icsinPpr|Dsuso ae icsinPpr|Dsuso ae | Paper Discussion | Paper Discussion | Paper Discussion | Paper Discussion

1 L1 error 310 L-inf error N-2 0.1 300

290 0.01 280 0.001 Error norms 270

0.0001 260

1e-05 250 100 | 1000 Paper Discussion | Paper Discussion | Paper Discussion | Paper Discussion N 250 260 270 280 290 300 310 x in km FigureFigure 15. Convergence 15. Convergence study for the study annulus for advection the annulus test. L1 (red) advection and L-inﬁnity test. (green) L1 error norms for the full domain and reference line (blue dotted) for "perfect" 2nd order convergence. (red) and L∞ (green) error norms for the full domain and referenceFigure 17.FigureHorizontal 17. Horizontal cross-section cross of horizontal section wind of horizontalvectors at z = wind 200 m vectors height forat the RH95 case (black)z = and200 the m RH50 height case for (grey). the SurfaceRH95 gridcase cells (black) around and the mountain the RH50 in green, case circle lines line (blue dotted) for “perfect” second-order67 convergence. | Paper Discussion | Paper Discussion | Paper Discussion | Paper Discussion represent(grey). 200 m Surfaceorography grid intervals. cells around the mountain in green, circle lines represent 200 m orography intervals. 69 1

0 250 270260 280 290 300 310 x in km

Figure 16.FigureComputational 16. Computational grid around the mountain grid around for an thex-z cut mountain plane at y for=1.38 an x km–z (cellcut center). plane at y = 1.38 km (cell center).

L1 norm, whereas the L∞ norm is nearly ﬁrst-order accurate (see Fig. 15).

4.6 3-D mountain flow in a moist atmosphere 68 In this section, a test case described in Kunz and WassermannFigure 18.FigureVertical 18. cross-sectionVertical cross (x-z plane) section of vertical (x–z plane) wind speed of vertical for the RH95 wind case speed (black) and the 1 (2011) is chosen. It includes forced lifting around a 1 km highRH50 casefor (grey). the RH95 Updrafts case in solid (black) lines(0.2 and m the s− RH50contour case interval, (grey). zero Updrafts line included), in downdrafts in dashed lines (0.2 m s 1 contour− interval, zero line excluded). mountain (see Fig. 16), latent heat release and orographic solid lines (0.2− ms 1 contour interval, zero line included), down- −1 precipitation. Compared to the first three test cases, this case drafts in dashed lines (0.2 ms contour interval, zero line ex- is now three-dimensional and uses a more realistic initial pro- cluded). file, which mimics atmospheric conditions when it comes to 70 orographically dominated precipitation in the mountainous A bell-shaped mountain is located at the center of the do- area of southwestern Germany. In their work, they used the main: three-dimensional, nonhydrostatic weather prediction model H COSMO with terrain-following coordinates to describe the h(x,y) = 1.5 (84) orography of the idealized mountain. The model setup for x2+y2 2 + 1 the ASAM simulations is as follows: the domain extends a 553 km × 553 km with a horizontal grid spacing of 2.765 km with the mountain peak height H = 1 km and the half-width and 70 vertical layers with uniform spacing of 1z = 200 m. length a = 11 km. Inflow and outflow boundary conditions

M. Jähn et al.: ASAM v2.7 333

LWC qc qr (g/kg) (g/kg) (g/kg) 7

0 240 260250 280270 290 300 310 320 330 340

x in km

x z Figure 19. VerticalFigure cross 19. Vertical section ( cross-section– plane) of microphysical (x-z plane) ofproperties microphysical for the RH95 properties case. Liquid for the water RH95 content case. (shaded), Liquid contours of specific cloud water content qc (red–yellow) and specific rain water content qr (blue). water content (shaded), contours of specific cloud water content qc (red-yellow) and specific rain water content qr (blue). are set according to the initial conditions. A Rayleigh damp- following coordinates in COSMO). Overall, the amplitude of ing layer above 11 km is applied to suppress gravity wave vertical wind is higher for the resulting gravity waves. Typi- reflections from the top boundary. Surface heat fluxes and cal patterns of orographic clouds (one cloud upstream of the Coriolis force are turned off. For turbulence parameteriza- mountain and a larger cloud with a high amount of liquid wa- tion, the standard Smagorinsky subgrid-scale model is used. ter content (LWC) and precipitation that reaches the ground Microphysics are parameterized by the warm (i.e., no ice in the lee of the mountain) are also reproduced (Fig. 19). In phase present) two-moment scheme described in Sect. 3.2. 71 this particular case the resulting patterns as well as the cloud Initial profiles are obtained by assuming hydrostatic equi- and rain water contents are comparable to the literature re- librium, a near-surface temperature Ts = 283.15 K, a con- sults, despite using different coordinate systems and cloud stant mean flow U = 10 ms−1, a constant dry static stability microphysical schemes. −3 −1 Nd = 11 × 10 s and a relative humidity profile, which is constant up to zm = 5 km and rapidly decreases above this level according to 5 Conclusions and future work −1 z − zm RH(z) = RHS 0.5 + π arctan (85) A detailed description of the three-dimensional, fully com- 500 pressible, nonhydrostatic All Scale Atmospheric Model with the near-surface humidity RHS = 95 % (RH95 case). To (ASAM) was presented. Main focus of this work was the compare the results with its dry counterpart, another simula- description of the cut cell method within a Cartesian grid tion with RHS = 50 % is performed (RH50 case). Figure 17 structure. With this method there is no accuracy loss near shows the wind field at 200 m height around the mountain steep slopes, which can occur around mountains using a high for both cases. In the nearly saturated atmosphere, there is spatial resolution or when obstacles or buildings are embed- a more direct overflow over the mountain, which is caused ded. The concepts of the spatial discretization of the advec- by the reduced stability due to high moisture. These dif- tion operator and a nonlinear term in the momentum equa- ferent flow characteristics also affect gravity wave structure tion were outlined. A technique to distribute surface fluxes (Fig. 18). The resulting waves are steeper and have a greater around cut cells was described. An implicit Rosenbrock time wavelength, which is in agreement with gravity wave theory integration scheme with two splitting approaches of the Ja- and the results from Kunz and Wassermann(2011). Most no- cobian were presented, which is particularly useful to by- table differences in the numerical results are discrepancies in pass the small cell problem. With the described scheme, rel- vertical wind strength in the lowest model layer at the wind- atively large time steps are possible. Physical parameteriza- ward side the mountain (w ≈ 0.6 ms−1 in ASAM vs. tions (Smagorinsky subgrid-scale model, two-moment warm w ≈ 0.2 ms−1 in COSMO), which can be explained by the microphysics scheme, multi-layer soil model) which are nec- different surface coordinate systems of the models (Carte- essary for performing particular atmospheric simulations, sian grid with cut cells in ASAM and generalized terrain- e.g., large eddy simulations of marine boundary layers, are www.geosci-model-dev.net/8/317/2015/ Geosci. Model Dev., 8, 317–340, 2015 334 M. Jähn et al.: ASAM v2.7 implemented in ASAM. The model produces good results Another focus on future model development lies in the when comparing scalar and velocity fields for typical bench- model physics, which includes further testing of current im- mark test cases from the literature. It was shown that energy plementations as well as adding new parameterizations, e.g., conservation is not affected when it comes to interaction with an ice microphysics scheme. For the description of turbu- the flow in the vicinity of cut cells. However, perfect energy lence, other (dynamic) Smagorinsky models (e.g., Kleissl conservation cannot be expected by design. Accuracy tests et al., 2006; Porté-Agel et al., 2000) might be better suited show that the EOC is almost second-order for the annulus for particular simulations compared to the present model ver- advection test. sion. Also, a so-called implicit LES will be tested and veri- Other model features that could not be presented in the fied. There, no turbulence model is used and the numerics framework of this paper are local mesh refinement and par- of the discretization generate unresolved turbulent motions allel usage of the model. They will be part of future studies. themselves. In this type of LES, the sensitivity of the thermo- There, performance tests for highly parallel computing with dynamical formulation (especially in the energy equation) on a large number of processors will be conducted. Furthermore, the resulting motions has to be analyzed. high-frequency output is desired for statistical data analysis. ASAM already was and will further be applied for large For this reason, efficient techniques like adaption of the out- eddy simulations of urban and marine boundary layers. An- put on modern parallel visualization software will be devel- other ongoing study deals with island effects on boundary oped. There are no operator splitting techniques used, which layer modification in the trade wind area exemplified by the leads to a consistent treatment of new processes with respect island of Barbados, where the island topography plays a sig- to time and to a simple programming style for the most part. nificant role and can be well described by the cut cell method.

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Appendix A: Derivation of tendency equations The term Se is related to the water vapor source term:

In this section, a straightforward derivation of the density po- Se = hvSv . (A11) tential temperature tendency equation is given to get the necessary source terms for microphysics, surface fluxes and pre- Transforming Eq. (A7) into a tendency equation for the ab- cipitation. Therefore, phase changes are allowed and a water solute temperature: vapor source term Sv and sedimentation velocity Wf for rain ∂ρ ∂e drops are added to the system. e + ∇ · (ρv) + ρ + v · ∇e = RHS(A7), ∂t ∂t dρ de = −ρ∇ · v + S − S , (A1) e(Sv − Sfall) + ρ = RHS(A7) dt v fall dt dρv d(ρe) dρ = −ρ ∇ · v + S + S , (A2) e(Sv − Sfall) + − e = RHS(A7). (A12) dt v v ph dt dt dρ l = −ρ ∇ · v − S − S , (A3) With Eq. (A9), this leads to dt c ph fall dρd d(ρe) d = −ρ ∇ · v . (A4) = [ρdcvd + ρvcvv + ρlcpl]T + ρvL00 dt d dt dt dρd dρv dρl The precipitation term is Sfall = ∂/∂z(ρrWf) with the sed- =T c + c + c vd dt vv dt pl dt imentation velocity Wf after Eq. (56). One can rewrite dT dρ Eqs. (A2) and (A3) with the mixing ratios rv = ρv/ρd and v + (ρdcvd + ρvcvv + ρlcpl) + L00 rl = ρl/ρd: dt dt − (ρdcvd + ρvcvv + ρlcpl)T ∇ · v drv 1 = (Sv + Sph), (A5) + T cvv(Sv + Sph) − T cpl(Sph + Sfall) dt ρd dT drl 1 + (ρdcvd + ρvcvv + ρlcpl) = − (Sph + Sfall). (A6) dt dt ρd − L00(ρv∇ · v − Sv − Sph) For the sake of simplicity (regarding the following deriva- dT = − eρ∇ · v + (ρ c + ρ c + ρ c ) tions) the liquid water density and mixing ratio are used with d vd v vv l pl dt ρl = ρc + ρr or rl = rc + rr. The model however solves the + T (cvv[Sv + Sph] − cpl[Sph + Sfall]) prognostic equations for the cloud water density ρc and rain + L00(Sv + Sph), (A13) water density ρr separately. dρ e =e(S − S ) − eρ∇ · v . (A14) A1 Internal energy and absolute temperature dt v fall A prognostic equation for the internal energy e is derived Inserting Eqs. (A13) and (A14) in Eq. (A12): from the first law of thermodynamics; cf. Bott(2008, Eq. 31) dT and Satoh et al.(2008, Eq. B.13), ρ c = − T (c [S + S ] − c [S + S ]) d vml dt vv v ph pl ph fall ∂(ρe) ∂ − L00(Sv + Sph) − p∇ · v + Svhv + ∇ · (ρev) = −p∇ · v + S − (ρ W e ) − ρ W g , ∂t e ∂z r f l r f ∂ − (ρrWfel) − ρrWfg (A7) ∂z = − p∇ · v + (hv − cvvT − L00)Sv and alternatively with the specific enthalpy h in the advection + (c T − c T − L )S part, pl vv 00 ph ∂ − (ρ W e ) − ρ W g . (A15) ∂(ρe) ∂ ∂z r f l r f +∇·(ρhv) = v·∇p+S − (ρ W e )−ρ W g . (A8) ∂t e ∂z r f l r f Here we define There, the total specific internal energy is cvml ≡ cvd + rvcvv + rlcpl . (A16) p e = h − = (q c + q c + q c )T + q L , (A9) ρ d vd v vv l pl v 00 Rewriting the pressure and elimination of the velocity divergence: and the specific internal energy for liquid water 1 dρd = = −p∇ · v = − (ρdRd + ρvRv)T − el hl cplT. (A10) ρd dt www.geosci-model-dev.net/8/317/2015/ Geosci. Model Dev., 8, 317–340, 2015 336 M. Jähn et al.: ASAM v2.7

dρ =(R + r R )T d , (A17) The moist potential temperature is d v v dt dρd 1 dp pRdT drv Rm = − c r T p pml t + v t rv 2 t θ = with π = . (A25) d RdT 1 ε d ε RdT 1 + d e e ε eπ p0 p dT − 2 + rv Taking the logarithm of the Exner function π leads to RdT 1 ε dt e ρd dp ρd drv ρd dT = − − Rm p (A18) π = . p dt ε + rv dt T dt lne ln (A26) cpml p0 dp dr dT ⇒ −p∇ · v = − ρ R T v − (ρ R + ρ R ) dt d v dt d d v v dt The time derivative is dp dT = − RvT (Sv + Sph) − (ρdRd + ρvRv) . dlnπ R dr lnπ dr dr dt dt e = v v − e v + l lneπ cpv cpl (A19) dt Rm dt cpml dt dt R dlnp This now leads to the temperature equation: + m cpml dt dT dT Rv Sv + Sph ρd(cvml + Rm) =ρdcpml = dt dt lneπ Rm ρd dp = − + c S + S c S + S RvT (Sv Sph) − pv v ph − pl ph fall dt lneπ cpml ρd cpml ρd + (hv − cvvT − L00)Sv Rm dlnp + (cplT − cvvT − L00)Sph + cpml dt ∂ − (ρ W e ) − ρ W g . R c S + S c S + S r f l r f (A20) = v − pv v ph + pl ph fall ∂z lneπ Rm cpml ρd cpml ρd With cpv −cvv = Rv, the water vapor source term disappears: R dlnp + m cpml dt (hv − cvvT − L00 − RvT )Sv − = (c T + L − c T − L − R T )S = . lneπ Rv cpv lneπ Rv cpl cpv pv 00 vv 00 v v 0 (A21) = − Sv + + Sph ρd Rm cpml ρd Rm cpml Further simplifying, lnπ cpl R dlnp + e S + m , (A27) ρ c fall c dt (cplT − cvvT − RvT − L00)Sph d pml pml = − − − + − (cplT cvvT RvT Lv (cpv cpl)T )Sph which leads us to the moist potential temperature equation: = −LvSph . (A22) dlneθ dlnT dlnπ Rearranging ﬁnally leads to the temperature equation, = − e dt dt dt R dlnp L 1 ∂ dT dp ∂ = m − v − ρdcpml = − LvSph − (ρrWfel) − ρrWfg , (A23) Sph (ρrWfel) dt dt ∂z cpml dt ρdcpmlT ρdcpmlT ∂z ρrWfg lneπ Rv cpv and its logarithmic derivative − − − Sv ρdcpmlT ρd Rm cpml dlnT R dlnp L = m − v lnπ R c − c lnπ c Sph − e v + pl pv − e pl dt cpml dt ρdcpmlT Sph Sfall ρd Rm cpml ρd cpml 1 ∂ ρ W g − − r f R dlnp (ρrWfel) . (A24) − m ρdcpmlT ∂z ρdcpmlT cpml dt A2 Potential temperature lneπ Rv cpv = − − Sv ρd Rm cpml A prognostic equation for the (moist) potential temperature is 1 L R c − c derived here. This is necessary because it appears in the den- − v + v + pl pv lneπ Sph sity potential temperature equation later on. Quantities that ρd cpmlT Rm cpml contain water vapor and liquid water are marked with a tilde lneπ cpl 1 ∂ ρrWfg − Sfall − (ρrWfel) − . to distinguish them from their dry equivalents (e.g., dry po- ρd cpml ρdcpmlT ∂z ρdcpmlT tential temperature θ). (A28)

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A3 Density potential temperature With the relation 1 1 R R With the deﬁnition of the density potential temperature = = v = v R , (A32) ε + rv d + r Rd + rvRv Rm Rv v 1 + rv/ θρ = eθ (A29) 1 + rv + rl we get the density and potential temperature equation sorted by source terms: and by using the product rule, dθ θ R ρ R c ρ = ρ v − d − v − pv dθρ 1 + rv/ deθ eθ 1 drv lneπ Sv = + dt ρd Rm ρ Rm cpml dt 1 + rv + rl dt 1 + rv + rl dt θ R R cpl − cpv L 1 + r / dr dr + ρ v − lnπ v + − v S − θ v v + l e ph e 2 ρd Rm Rm cpml cpmlT (1 + rv + rl) dt dt θ ρ c θ ∂ + ρ d − pl − ρ θρ deθ 1 1 drv lneπ Sfall (ρrWfel) = + θρ − ρd ρ cpml ρdcpmlT ∂z eθ dt ε + rv 1 + rv + rl dt θρρrWfg θρ drl − . (A33) − . (A30) ρdcpmlT 1 + rv + rl dt Inserting Eqs. (A28), (A5) and (A6) in Eq. (A30): dlnθ dlnθ 1 1 dr ρ = e+ − v dt dt ε + rv 1 + rv + rl dt 1 dr − l 1 + rv + rl dt lneπ Rv cpv = − − Sv ρd Rm cpml 1 L R c − c − v + v + pl pv lneπ Sph ρd cpmlT Rm cpml lneπ cpl 1 ∂ ρrWfg − Sfall − (ρrWfel) − ρd cpml ρdcpmlT ∂z ρdcpmlT 1 1 S + S + − v ph ε + rv 1 + rv + rl ρd 1 Sph + Sfall + . (A31) 1 + rv + rl ρd

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Appendix B: Soil and land use parameters Code availability and visualization

Varying ratios of silt, clay and sand significantly change the The ASAM code is managed with Git, a distributed revi- properties of soil and thus determine the heat and moisture sion control and source code management (SCM) system. To fluxes of the surface. Accordingly, these different ratios are get access to the source code and additional scripts for pre- referred to specifically defined soil types. In Tables B1 and and postprocessing, registration at the TROPOS Git host- B2, parameters describing the physical properties of the ap- ing website https://gitorious.tropos.de/ is mandatory. Addi- propriate soil type are listed. Wpv stands for the pore volume tional information can be found at the ASAM webpage (http: of the soil, Wfc is the field capacity describing a threshold //asam.tropos.de). value for runoff in the soil layers. κsat and 9sat define the As visualization tool, the free and open source software hydraulic conductivity and the matric potential at saturation, VisIt (https://wci.llnl.gov/codes/visit/) is used. VisIt can read respectively (Eqs. 68, 69). ρ0c0 is the heat capacity of dry over 120 scientific file formats and offers the opportunity to soil as used in Eq. (71) and b∗ is a parameter for the soil include own scripts, if necessary. It is available for Unix, porosity. Windows and Mac workstations.

Table B1. Soil parameters from Doms et al.(2011).

3 −3 3 −3 −1 3 −1 ∗ a Soil type Wpv (m m ) Wfc (m m ) κsat (ms ) ρ0c0 (W(m K) ) b Sand 0.364 0.196 4970 × 10−8 1.28 × 106 3.5 Sandy loam 0.445 0.260 943 × 10−8 1.35 × 106 4.8 Loam 0.455 0.340 531 × 10−8 1.42 × 106 6.1 Clay loam 0.475 0.370 764 × 10−8 1.50 × 106 8.6 Clay 0.507 0.463 1.7 × 10−8 1.63 × 106 10.0 Peat 0.863 0.763 5.8 × 10−8 0.58 × 106 9.0

a With n = 1/b∗ + 1, see Eqs. (68) and (69).

Table B2. Soil parameters as used in Pielke(1984) (adapted from McCumber, 1980).

3 −3 −1 3 −1 ∗ a Soil type Wpv (m m ) 9sat (m) κsat (ms ) ρ0c0 (W(m K) ) b Sand 0.395 −0.121 1760 × 10−8 1.47 × 106 4.05 Sandy Loam 0.435 −0.218 341 × 10−8 1.34 × 106 4.90 Loam 0.451 −0.478 70 × 10−8 1.21 × 106 5.39 Clay loam 0.476 −0.630 25 × 10−8 1.23 × 106 8.52 Clay 0.482 −0.405 13 × 10−8 1.09 × 106 11.40 Peat 0.863 −0.356 80 × 10−8 0.84 × 106 7.75

a With n = 1/b∗ + 1, see Eqs. (68) and (69).

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