An inherently mass-conserving semi-implicit semi-lagrangian discretization of the deep-atmosphere global non-hydrostatic equations

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1 Quarterly Journalof the RoyalMeteorologicalSociety Q. J. R. Meteorol. Soc. : July DOI:./qj. n inherently mass-conserving semi-implicit semi-lagrangian discretization of the deep-atmosphere global non-hydrostatic equations Nigel Wood a * ndrew Staniforth a ndy White a Thomas llen a Michail Diamantakis a Markus Gross a Thomas Melvin a Chris Smith a Simon Vosper a Mohamed Zerroukat a and John Thuburn b a Met Office Exeter UK b University of Exeter UK *Correspondence to: Nigel Wood Met Office FitzRoy Road Exeter EX PB UK. Nigel.Wood@metoffice.gov.uk This article is published with the permission of the Controller of HMSO and the Queen s Printer for Scotland. Following previous work on an inherently mass-conserving semi-implicit SI semi- Lagrangian SL discretization of the two-dimensional D shallow-water equations and D vertical slice equations that approach is here extended to the D deep-atmosphere non-hydrostatic global equations. s with the reduced-dimension versions of this model an advantage of the approach is that it preserves the same basic structure as a standard nonmass-conserving SISL version of the model. dditionally the model is simply switchable to hydrostatic and/or shallow-atmosphere forms. It is also designed to allow simple switching between various geometries Cartesian spherical spheroidal. The resulting mass-conserving model is applied to a standard set of test problems for such models in spherical geometry and compared with results from the standard SISL version of the model. Key Words: C-grid; Charney Phillips; SLICE; spatial discretization; spheroidal coordinates; temporal discretization Received February ; Revised 8 July ; ccepted ugsust ; Published online in Wiley Online Library December. Introduction The semi-implicit semi-lagrangian SISL method as first proposed by Robert 98 98; see Staniforth and Côté 99 for a review has contributed significantly to the success of operational numerical weather prediction over the last few decades. It allows the use of time steps considerably larger than would otherwise be possible because of stability limitations without compromising on accuracy. Increasingly however SISL models are also being used for long-term climate simulations for which conservation of constituents such as dry mass water species and chemical species is important. disadvantage of the semi-lagrangian scheme is that typically the interpolation required to evaluate a quantity at the departure point does not conserve the mass of the quantity being interpolated. Over the last decade or so some groups have therefore been developing transport schemes that although based on the semi-lagrangian approach do conserve the mass of the transported quantity see for example the review by Machenhauer et al. 8. However to apply these schemes in a SISL model to achieve conservation of the model s mass requires the transport schemes to be coupled to the semi-implicit aspects of the model. Lauritzen et al. 8 present one such coupling approach in the context of a limited-area model and Kaas 8 presents a related one for a Cartesian shallow-water model. Current address: ECMWF Reading UK. c Royal Meteorological Society and Crown Copyright the Met Office Quarterly Journal of the Royal Meteorological Society c Royal Meteorological Society Here an alternative coupling method proposed by Zerroukat et al. 9b in the context of the global shallow-water equations and subsequently applied in a Cartesian vertical-slice model by Melvin et al. is extended to the global three-dimensional D dry and unforced Euler equations. The efficacy of the method is demonstrated by comparing it with a standard nonmass-conserving SISL implementation which is identical to the mass-conserving model in all respects other than its handling of the density field. This is the next natural step on the way to implementing such a scheme in an operational weather and climate prediction model. In principle any SL mass conserving scheme based on a conservative remapping approach can be employed within the formulation. Here the SLICE scheme of Zerroukat et al. 9a and Zerroukat and llen is used. The D formulation has two significant differences from the reduced-dimension cases previously presented. Firstly the reference-state-free scheme of Thuburn et al. is applied; this means that at convergence of the iterative scheme the results are independent of the choice of reference profile. However the choice of reference profile does determine whether the scheme converges at all and also the rate of convergence. Hence to obtain satisfactory convergence the second modification to the schemes presented in Zerroukat et al. 9b and Melvin et al. is to use the thermodynamic fields from the previous time step modified to ensure that they are statically stable as the reference states for evaluation of the next time step fields.

2 N. Wood et al. dditionally the discretization has been specifically designed to allow the model to be straightforwardly switchable from nonhydrostatic to quasi-hydrostatic and from deep-atmosphere to shallow-atmosphere. lso a general terrain-following coordinate system is employed which allows the model to be run in Cartesian spherical or spheroidal geometries. In section the continuous governing equations are presented. The mass-conserving SISL discretization of these equations is developed in section followed by the spatial discretization in section. The method of solving the resulting nonlinear coupled set of equations is presented in section with a non-mass-conserving variant of the model given in section. The mass-conserving and non-mass-conserving schemes are compared in spherical geometry for a variety of test cases in section 7. summary is given in section 8 with various definitions and specific details of the scheme documented in the appendices.. Continuous equations The starting point for the discretization is the continuous governing equations. These are first given in vector form. Then a general geopotential curvilinear orthogonal coordinate system is introduced before transformation to a terrain-following vertical coordinate... Vector form In vector form the continuous governing equations for a perfect gas are Du Dt + u = c pθ π + g Dθ = Dt D ρ dv = Dt V R π κ/κ = ρθ together with the kinematic equation p Dx = u Dt required in anticipation of the semi-lagrangian discretization. Here D Dt + u t is the material derivative u is the velocity vector θ is potential temperature ρ is density V is an elemental fluid volume π p/p κ is the Exner pressure with p denoting pressure and p a constant reference pressure g is the apparent gravitational vector being the sum of actual gravity and the centrifugal force per unit mass is Earth s rotation vector R is the gas constant per unit mass c p is the specific heat at constant pressure and κ R/c p. Equations are respectively the momentum thermodynamic continuity and state equations. Equation is the integral equivalent of the more usual material derivative form Dρ + ρ u = 7 Dt of the continuity equation used in traditional SISL discretizations... Geopotential curvilinear orthogonal coordinates curvilinear orthogonal coordinate system is considered in which the vertical coordinate is aligned everywhere with the direction of g i.e. the surfaces of the coordinate system orthogonal to the vertical coordinate are geopotential surfaces. This system can then be approximated in different ways see ppendix and these choices define the degree of geopotential approximation made. The coordinates are denoted by ξ ξ and ξ where the direction associated with ξ is aligned with g and the directions associated with ξ and ξ form a right-handed system typically they are associated with the zonal and meridional directions respectively. The corresponding metric factors for this coordinate system are h h and h respectively. The components of the wind associated with each coordinate are denoted interchangeably by u u u u v wwhere Dξ i u i h i i =. 8 Dt.. terrain-following vertical coordinate Consider now a transformation of the vertical coordinate η = η ξ ξ ξ which is assumed to be independent of time. The governing equations are transformed from the curvilinear orthogonal coordinates ξ ξ ξ t to the non-orthogonal coordinates ξ ξ η t but with all velocity components i.e. u v and w retained in their original form. The relevant transforms of the vector operators are given in ppendix B Staniforth and Wood give further details for the case of spherical polar coordinates. In particular the material rate of change of the transformed vertical coordinate ηisgivenby η Dη Dt = ξ w u h h ξ ξ v ξ. 9 h ξ There is a great deal of flexibility available for the choice of functional form for η the simplest being the linear form 7 used for the integrations of the idealized test cases presented in section 7.. Semi-implicit semi-lagrangian discretization.. The vector form SISL discretization is applied to Eqs. This is achieved by integrating Eqs along a trajectory over a time interval of t and approximating any residual integrals using a possibly off-centred trapezoidal rule. s an example consider the generic equation where F and G could be either both scalars or both vector fields DF = G. Dt This is first integrated along the trajectory to give F x t+ t t + t F x t t = t+ t t G dt and then the right-hand side of Eq. is approximated to give F x t+ t t + t F x t t = α tg x t+ t t + t + α tg x t t. Here x t+ t is the position along any particular trajectory at time t + t and is termed the arrival point x ; it is chosen to be a grid point and is therefore known. x t is the position along the same trajectory at time t and is termed the departure point x D. c Royal Meteorological Society and Crown Copyright the Met Office Quarterly Journal of the Royal Meteorological Society c Royal Meteorological Society Q. J. R. Meteorol. Soc. :

3 Mass-Conserving SISL Non-Hydrostatic Global Discretization 7 The departure points are evaluated by solving the discrete form of the kinematic equation as described in section.. The quantity α is an off-centring parameter and takes the value. for a centred scheme. To avoid spurious orographic resonance Rivest et al. 99 operational models typically use a value of α between. and.7. Equation is written more compactly as F α tg n+ = F + β tg n D where β α. The superscripts n and n + denote the time level of a variable and subscripts and D denote evaluation at the arrival and departure points respectively. pplying this discretization to Eqs gives where u α t n+ = u + β t n D u c p θ π + g θ n+ = θ n D and n+ n ρ dv = ρ dv. 7 V V D.. The SISL discrete set in component form geopotential coordinate system is now assumed see sections. and. and the equations are written in the appropriate component form. The scalar equations Eqs 7 are unchanged in form by the choice of coordinate system. However extracting the three components of the velocity requires the evaluation of a vector field at the departure point i.e. the right-hand side of Eq. for which in general the unit base vectors do not coincide with those at the arrival point. Developing the shallow-atmosphere rotation matrix approaches of Côté 988 and Temperton et al. Staniforth et al. show how within a deep-atmosphere context this issue can be addressed by defining a rotation matrix M the elements of which depend on the particular coordinate system. Using this approach and accepting a slight abuse of vector notation Eq. becomes u α u t n+ = M u + β u t n D L 8 where the subscript D L denotes evaluation at the departure point in terms of the local basis vectors at that departure point the usual subscript D denotes evaluation at the departure point but in terms of the basis vectors at the arrival point. Subscripts on the off-centring parameters α and β indicate which prognostic quantity they are associated with. Writing = u v w the components of the momentum equation Eq. 8 are therefore u α u t u n+ = M u + β u t u n D u L + M v + β v t v n D u L + M δ V w + β w t w n D u L R n u 9 v α v t v n+ = M u + β u t u n D v L + M v + β v t v n D v L + M δ V w + β w t w n D v L R n v where δ V + μ t w n+ α w t w n+ = M u + β u t u n D w L u u u c pθ ξ h { ξ v u v c pθ ξ h { ξ w u w ξ c pθ h ξ ξ + M v + β v t v n D w L + M δ V w + β w t w n D w L R n w π π ξ ξ ξ ξ } π } π π g the subscripts u v and w indicate the ξ ξ and ξ components respectively and the superscripts on D L indicate the departure pointtobeused. The fact that the coordinate system is a geopotential one has been used to write the components of apparent gravity g as g = g. The functional form of g is given in ppendix. non-hydrostatic switch δ V has been introduced into the vertical momentum equation Eq. such that the deepatmosphere equation set is fully non-hydrostatic when δ V but reduces to the quasi-hydrostatic equation set White et al. when δ V. n optional Rayleigh damping term has been added to the w equation. This has been done as in Melvin et al. ; a single term μw has been added to the right-hand side of the w equation which has then been discretized fully implicitly in time. This leads to the appearance of the term involving μw n+ on the left-hand side of Eq.. The specification of the spatially varying parameter μ is discussed in section 7. No further damping or filtering mechanisms have been added to the model... Kinematic equation In principle and as proposed by Wood et al. the kinematic equation can be discretized in time in the same way as the momentum equation. This has the advantage that it is simply switchable between various geometries Cartesian spherical and spheroidal. However Thuburn and White showed that this approach can lead to a numerical instability in spherical geometry. Therefore for the kinematic equation only an algorithmic switch is made between the Cartesian and spherical both deep- and shallow-atmosphere options. In the Cartesian case the same method as outlined in Melvin et al. is used but extended in the natural way to D. In the spherical case the method proposed by Thuburn and White is used. This is a For the shallow-atmosphere equation set it is again non-hydrostatic when δ V but hydrostatic rather than quasi-hydrostatic when δ V. c Royal Meteorological Society and Crown Copyright the Met Office Quarterly Journal of the Royal Meteorological Society c Royal Meteorological Society Q. J. R. Meteorol. Soc. :

4 8 N. Wood et al. deep-atmosphere extension of the shallow-atmosphere methods described by for example Côté 988 and Wood et al.. Let r be a position vector with r r the radial distance. Then k r/r defines the horizontal position of a point r defines its vertical position and Eq. can be decomposed as and Dr Dt = w Dk Dt = v r 7 where v u k k u is the horizontal part of the velocity vector. In the terrain-following coordinate system Eq. is replaced by Dη = η. 8 Dt Equation 7 has the same form as it would have for a shallowatmosphere model with Earth s mean radius a generalized here to r. This is therefore solved using the shallow-atmosphere version of the scheme described in Wood et al. and also in Zerroukat et al. 9b. Equation 8 is solved using the same method as for the Cartesian case. The adjustment of Wood et al. 9 is applied to keep the departure points within the model domain. Equations 7 and 8 are solved together iteratively. This procedure can be repeated to convergence. In all simulations presented here two iterations were used. Separate departure points are computed for each dependent variable according to where it is carried on the staggered grids introduced below.. Spatial discretization.. Grid and storage aspects The equations are discretized on an rakawa C grid in the horizontal and a Charney Phillips grid in the vertical. Thus with respect to a cell-centred point where ρ and are stored u is staggered half a grid length in the ξ direction v is staggered half a grid length in the ξ direction and w and θ are staggered half a grid length in the vertical ξ or η direction. In the vertical the lowest w points lie on the model s bottom surface and the highest ones lie on the model s top. In the ξ direction the first and last v points lie respectively on the southern and northern extremities of the model i.e. at the Poles for spherical and spheroidal geometry; no other variables are stored at the Poles. This follows the recommendation of Thuburn and Staniforth and also avoids having to solve for pressure at a coordinate singularity. The averaging and differencing operators are defined in ppendix C. The elements of M are defined such that they do not need averaging specifically: M M and M are evaluated on u points; M M and M are evaluated on v points; and M M and M are evaluated on w points. The h values are evaluated where they are needed and this position relative to a ρ point has been indicated by a superscript. For example h indicates the value of h on a ρ point; h ξ indicates the value of h on a u point; and h ξ η indicates the value of h on a v θ point i.e. the point obtained by moving vertically from a v point or horizontally from a θ point. This approach ensures that the value of h used at the Poles is identically zero. Consequently v at the polar points only enters the discretization through the departure points and interpolation to departure points near the Poles. In the same way the various gradients of ξ are evaluated where they are needed. The position at which any particular δ ψ ξ for generic ψ is needed relative to a ρ point is indicated by a superscript. However the orography is only defined at surface θ points and is linearly averaged horizontally to other points as required... Discrete equations The spatially discretized forms of each of the prognostic equations are now given.... Momentum equations 9 Equations 9 are unchanged in form by the spatial discretization. Equations become u h ξ ξ v w v ξ f w η f ξ c p θ ξ η h ξ δη ξ ξ {δ ξ πδη ξ δη π ξ η δ ξ ξ ξ η } 9 h ξ ξ w η f u ξ f ξ c p θ ξ η ξ δη ξ h ξ {δ ξ πδη ξ δη π ξ η } ξ η δ ξ ξ h η u ξ η f δη ξ v ξ f η h η c p θ η δ η π g. δη ξ Thuburn and Staniforth showed that an inappropriate averaging of the Coriolis terms on a spherical C grid leads to a misrepresentation of the Rossby modes with the shortest meridional scale. They also showed that an energy-conserving averaging of the Coriolis terms is not only compatible with the constraints required for conservation of mass angular momentum and energy of the linearized equations but also improves the dispersion properties of Rossby modes. Thus the Coriolis terms have been evaluated here following an extension of Thuburn and Staniforth to allow additionally for the Coriolis terms arising from the deep atmosphere and possible rotation of the coordinate system. s given the Coriolis terms in the absence of advection preserve a global discrete energy principle. They also give good Rossby-mode dispersion properties in the sense of Thuburn and Staniforth discussed above. The mass-flux variables are given by u ξ ηh ξ hξ ξ δη ξ ρ ξ u v ξ ηh ξ hξ ξ δη ξ ρ ξ v w ξ ξ h η hη ρη w cf. eqs. and.7 of Thuburn and Staniforth ; the starred Coriolis terms are defined by f f ρh ξ f f ρh ξ f f ρh δη ξ η. 7 The Coriolis parameters f f and f are evaluated and stored on ρ points their definitions for the different model options are given in ppendix. Finally the δ differencing operators are defined in ppendix C and the averaging operator denotes a simple equal-weighted / / average in the direction indicated by the superscripts see ppendix C. η c Royal Meteorological Society and Crown Copyright the Met Office Quarterly Journal of the Royal Meteorological Society c Royal Meteorological Society Q. J. R. Meteorol. Soc. :

5 Mass-Conserving SISL Non-Hydrostatic Global Discretization 9... Vertical motion equation 9 η n+ w n+ η = η δη ξ h η un+ξ η h η δξ ξ η δξ ξ 8 vn+ξ η h η except at the lower and upper surfaces where zero mass-flux boundary conditions require η =. 9 The superscript n + on the wind components in Eq. 8 indicates evaluation at the latest available estimate of their new time-level values see section. for details.... Thermodynamic equation θ n+ θ ref = R n θ θ n D θ ref. For convenience of later notation see section. a reference profile has been subtracted from both sides.... Continuity equation 7 ρ n+ ρ ref = R n ρ n ρ dv ρ ref V V D where V is the volume of the grid box centred on each arrival ρ point. gain for convenience of later notation a reference profile has been subtracted from both sides. ny conservative remapping algorithm can be used for the evaluation of R n ρ ; here the SLICE algorithm of Zerroukat and llen using the piecewise parabolic method of Colella and Woodward 98 is adopted without monotonicity.... Equation of state Thuburn showed that the dispersion properties and structures of some vertical discretizations of the compressible Euler equations are sensitive to the form in which the pressure-gradient term is expressed and identified discretizations that have optimal dispersion properties when compared with linear analytical dispersion relations. Thus following the recommendation of Thuburn the spatial discretization of the equation of state is.. Boundary conditions R π κ/κ = p ρθ η. To close the discrete problem boundary conditions are needed at the bottom and top levels in addition to requiring that η = there. Interpolation of quantities stored at u v and ρ points to departure points that are in the lowest and uppermost half-layers is achieved by assuming the interpolants are constant in those half-layers. The assumption of zero vertical shear in the horizontal winds is also used in the evaluation of Eq. 8. Computation of the departure points requires a value of the horizontal wind at the surface. To obtain this the horizontal wind speed scaled by r in the deep-atmosphere spherical case is assumed to be constant in the bottom and top half-layers. This then allows Eq. to be evaluated at the top and bottom surfaces i.e. θ is advected along the top and bottom surfaces. n estimate of w n+ at the surface is obtained within the iterative procedure by applying Eq. at the surface and eliminating the surface value of w n+ using Eq. 8 with η = and assuming zero horizontal shear in the wind. t the initial time step the surface value of w is set to zero. The same procedure is followed to obtain an estimate of w n+ at the model s top. In terrain-following coordinates the evaluation of the pressure gradient term in Eqs 9 requires an estimate of the surface pressure. This is obtained by requiring Eq. to be satisfied at the surface with w given by its surface estimate and applying a first-order one-sided derivative for the vertical pressure gradient. For spherical and spheroidal geometry for which ξ λ the values of v at the Poles are required for evaluation of departure points in the vicinity of the Poles. dditionally values of v at the Poles are required for the interpolation of v + β v t v to those departure points. Since when required at v points h is stored and evaluated at those points the value of v at the Poles is never required elsewhere within the dynamics. The values of v and v at both South and North Poles are obtained following the procedure outlined in appendix B of McDonald and Bates 989. There is one difference however: here the values of u and u surrounding the Poles are used to compute the associated polar vector quantities and hence polar values of v and v ; in contrast because they store u at the Poles McDonald and Bates 989 use the surrounding values of v to compute the polar vector wind.. Solution procedure.. Setting the scene Equations 9 and 9 together with the kinematic equations Eqs 7 and 8 form a coupled nonlinear system of equations for the prognostic variables at the new time step. The method used here to solve these equations is a nested iterative approach similar to that used by Côté et al. 998 and within the present setting by Zerroukat et al. 9b and Melvin et al.. The departure-point equations are solved in an outer loop using the latest available estimates for the wind components. Interpolation of required quantities to these departure points is also done within the outer loop. Then within an inner loop a linear Helmholtz equation is solved with the Coriolis terms and all nonlinear terms updated using the latest available estimates for the constituent fields. Both the outer and inner loops can be iterated to convergence but in all simulations presented here two outer and two inner iterations are used. key aspect of the approach is the derivation of a Helmholtz problem. This is achieved using the reference-state-free scheme of Thuburn et al. which reduces the impact of the choice of reference profile. In this approach appropriate linear terms are added to both sides of each equation with the terms that are added to the right-hand side of the equations lagged within the iteration. These quantities are denoted by superscript n +. For the θ and ρ equations outer-loop estimates of these quantities are used rather than the more iterated inner-loop estimates. This is motivated by considerations of convergence as suggested by linear analysis and discussed in Thuburn et al.. In the non-mass-conserving scheme a mix of inner- and outer-loop estimates is used see section for details. For all other variables the latest inner-loop estimate is used. c Royal Meteorological Society and Crown Copyright the Met Office Quarterly Journal of the Royal Meteorological Society c Royal Meteorological Society Q. J. R. Meteorol. Soc. :

6 N. Wood et al. For known right-hand sides the equations now constitute by construction a linear set of equations. Elimination of variables leads to a Helmholtz equation which once solved gives via back-substitution an updated estimate of the model variables. The advantage of the reference-state-free scheme is that at convergence of the encompassing iterative procedure the added linear terms cancel from both sides. s such there is some flexibility regarding their precise form: analogous to the semiimplicit scheme the closer they are to the corresponding nonlinear terms the faster convergence is likely to be. The choice of what linear terms to add to both sides of the equations is inspired by what would result from linearizing the equations about a stationary reference profile that is a function only of η. Note though that in contrast to Melvin et al. and because of the large variations of temperature from Pole to Equator the thermodynamic reference profiles used here are not onedimensional but functions of the model state at the previous time step. For the terms added to both sides the α factor associated with the term could be left unchanged. However it is advantageous to replace α in these terms but not elsewhere by a new factor τ.theτ factors are then just relaxation parameters for the encompassing iterative procedure. The values of the α offcentring parameters are primarily chosen for accuracy changing the α values changes the discretization and therefore the accuracy of the model. However the values of the τ relaxation parameters do not at convergence change the discretization and are chosen empirically to optimize convergence of the iterative procedure. The specific values used for the test cases presented herein are giveninsection7. Equations 9 and 8 are therefore modified as follows in which F F n+ where u ref : F ref and F n+ F n+ F ref u + τ u tc p θ ref ξ η ξ δ ξ π δ η ξ = R u + R n u δη ξ h ξ v + τ v tc p θ ref ξ η ξ δ ξ π δ η ξ = R v + R n v δη ξ h ξ δ V + μ t w τw tc p θ ref δη + η π ref θ δη ξ where h η h η θ ref τw tc p θ ref + η δ η π = R w δη ξ + Rn w h η η δη ξ η w = R η θ + τ θ tδ η θ ref η η = R θ + Rn θ 7 ρ + τ ρ t. ρ ref u = R ρ + Rn ρ 8 κ π ρ η θ + = κ π ref ρ ref θ ref R π 9 R u α u t n+ u + τ u tc p θ ref ξ η ξ δ ξ π n+ δ η ξ δη ξ h ξ R v α v t n+ v + τ v tc p θ ref ξ η ξ δ ξ π n+ δ η ξ δη ξ h ξ R w α w t n+ w τw tc p θ ref δη + η π ref θ n+ δη ξ h η θ ref τw tc p θ ref + η δ η π n+ δη ξ h η R η h η h η h η h η δξ ξ η u n+ξ η δξ ξ η v n+ξ η R θ τ θ tδ η θ ref η η n+ R ρ τ ρ t. ρ ref u n+ with the discrete divergence operator defined by Eq. C and R π p π n+ κ/κ Rρ ref θ n+η κ π n+ η θ n+ +. κ π ref θ ref Defining a number of H coefficients as given in ppendix D and using Eq. C for this set of equations can be rewritten more succinctly as u + H u δ ξ π δ η ξ = R u + R n u 7 v + H v δ ξ π δ η ξ = R v + R n v 8 δ V + μ t w + H w δη π ref θ θ ref + H w δ η π = R w + Rn w 9 H η η w = R η with η = at the bottom and top of the model θ + H θ η = R θ + Rn θ ρ + H V δξ Hρx u + δ ξ Hρy v + δ η Hρz η = R ρ + Rn ρ π ρ η θ H ππ + = π ref ρ ref θ ref R π. Note that the H coefficients are evaluated at the points of the staggered grids appropriate to the terms they multiply... Helmholtz equation lgebraically eliminating u v w ρ θ and η from Eqs 7 leads to the discrete Helmholtz equation: H V δ ξ Hρx H u δ ξ π δ η ξ + H V δ ξ Hρy H v δ ξ π δ η ξ where + H V D D π ρref H ππ π ref π = R + R n D X δ η Hρz X + ρref H V Hθ X θ ref η c Royal Meteorological Society and Crown Copyright the Met Office Quarterly Journal of the Royal Meteorological Society c Royal Meteorological Society Q. J. R. Meteorol. Soc. :

7 Mass-Conserving SISL Non-Hydrostatic Global Discretization and D X H C H w δ η X H C δ V + μ t H η H wh θ δη π ref 7 θ ref R R R η ρ θ ρref + θ ref R π + H V δ ξ Hρx R u + HV δ ξ Hρy R v { + H V D HC R w + δ V + μ t R η } HC H w δη π ref H V D 8 θ ref R R n R n n η ρ θ ρref θ ref + H V δ ξ Hρx R n u + HV δ ξ Hρy R n v + H V D HC R n w HC H w δη π ref H V D. 9 θ ref The right-hand side of Eq. contains both explicitly known terms i.e. the R n and implicitly defined nonlinear and Coriolis terms i.e. the R leading to a nonlinear coupling with Eqs 7... Iterative solution The nonlinear coupled equations 7 and are solved using an iterative approach: for details see ppendix E. t each inner iteration a Helmholtz problem is solved for π the latest estimate of π n+ is updated and the latest estimates of the prognostic variables are obtained from the discretized equations by a process of back-substitution. s part of this process η is obtained from π using η + D π R θ R n θ = H C R w + R n w + δ V + μ t R η H CH w δη π ref θ ref R θ + R n θ. 7 This equation results from the algebraic elimination of w and θ from Eqs 9 to. ny suitable solver may be used for the Helmholtz problem with the precise choice constrained by computational efficiency and dependent on computer architecture. The current choice is Bruaset 99 s post-conditioning adaptation of van der Vorst 99 s bi-conjugate gradient stabilized Bi-CGSTB method.. Standard SISL version of the model To facilitate validation of the model a version has been created that differs from that described above only in its use of a standard interpolating SISL discretization of the continuity equation. pplying a standard SISL and spatial discretization to Eq. 7 results in ρ n+ ρ ref = ρ β ρ tρ u n D ρref α ρ t ρ u n+. 7 This is the SISL version of Eq.. pplying the same approach to Eq. 7 as that used to obtain Eq. 8 results in ρ + τ ρ t ρ ref u = R ρ + Rn ρ 7 but with R n ρ and R ρ redefined respectively as and R n ρ ρ β ρ tρ u n D ρref 7 R ρ τ ρ t ρ ref u n+ α ρ t ρ u n+. 7 Since by construction Eq. 7 has the same form as Eq. 8 the solution procedure described above goes through virtually unchanged. The first term on the right-hand side of Eq. 7 includes the advection of ρ ref and is therefore updated in the outer loop of the iterative procedure. The last term on the right-hand side of Eq. 7 represents the nonlinear divergence term and this is updated in the inner loop. t convergence the resulting version of the model corresponds to a two-time-level fully implicit semi-lagrangian discretization of the fully compressible inviscid equations referred to herein as the standard SISL scheme. 7. Computational examples The test cases presented are taken from a standard set of test problems for dynamical cores used in the Dynamical Core Model Intercomparison Project DCMIP: see org/projects/dcmip-/ and media/docs/dcmip-testcasedocument v.7.pdf for detailed descriptions of the test problems together with the deepatmosphere baroclinic instability test of Ullrich et al.. The DCMIP suite of problems tests a variety of salient aspects of a dynamical core: advection; handling of orographic forcing and dry and moist dynamics with simple physics parametrizations. Here only the dry dynamics tests are used. This test suite makes extensive use of the small-earth framework also used in Wedi and Smolarkiewicz 9 to simulate flow in both the hydrostatic and non-hydrostatic regimes at low computational cost. This involves defining a new small-earth radius a a/x where a is the mean radius of the unscaled Earth and an increased rotation rate = X along with a reduced time step t t/x where X is the small-earth scaling factor. These test cases are all in spherical geometry. ppendix. gives details of the coordinates metric factors etc. for the deep-atmosphere cases and ppendix. gives them for the shallow-atmosphere ones. In particular ξ = λ ξ = φ and ξ = r. In cases where orography is present the transformation between the physical height z r a and the terrain-following coordinate η used here is assumed linear with the form η = z z S λ φ z T z S λ φ 7 where z S λ φ is the orographic height and z T = constant is the domain depth in the absence of orography. The great circle distance r c from the point λ φ is r c a arccos sin φ sin φ + cos φ cos φ cos λ λ. 7 The upper boundary condition implies a rigid reflecting lid. Where this causes problems for orographically forced flow a damping layer is used to reduce the impacts of reflections from the upper boundary. The application of this damping layer is implemented gradually from the base of the layer η B < to the top of the domain using the formula { η<ηb μ η = μ sin π η ηb η B η B η 77 where μ is a constant that is set appropriately for each application. The reference profile θ ref π ref ρ ref is taken to be the atmospheric state at the previous time level except that the c Royal Meteorological Society and Crown Copyright the Met Office Quarterly Journal of the Royal Meteorological Society c Royal Meteorological Society Q. J. R. Meteorol. Soc. :

8 N. Wood et al. θ ref profile is adjusted by reordering the values using a bubblesort algorithm to ensure static stability θ ref / z > and ρ ref is recomputed from the equation of state. For all simulations a time-centred α = β = / discretization is used including time-centred computation of the departure points α x = β x = /. The relaxation parameters are chosen to be equal to their off-centring counterparts i.e. τ = α = / except that it was found beneficial for convergence to set τ ρ =. Two iterations of both the inner loop and the outer loop are also used. In addition except where noted all tests are run in the non-hydrostatic δ V = shallow-atmosphere as prescribed for the DCMIP tests configuration see ppendix. and use the mass-conserving form of the continuity equation. Evaluation of fields at departure points is achieved using tri-cubic Lagrange interpolation. The monotone limiter of Bermejo and Staniforth 99 is applied to potential temperature as a precautionary measure to avoid spurious introduction of negative static stability due to possible overshoots and undershoots. Common settings of various parameters for all the illustrative examples are listed in Table. The initial reference profiles π ref ρ ref θ ref for each test case are set equal to the initial states. Note that generally in the figures of this section following other authors only the regions of interest are shown. 7.. Mountain waves over a Schär-type mountain on a small planet To test the response to orographically induced gravity waves a non-rotating test based upon a D version of the test case of Schär et al. is used Paul. Ullrich personal communication. Two forms of the test are used that are almost identical except for the presence or absence of shear in the initial horizontal wind field. The mountain profile has the form z s λ φ = h exp r c d cos πrc 78 ξ where the mountain height is h = m and the half-width and wavelength are given by d ξ m m. The great circle distance r c from the mountain centre at λ φ π/ is given by Eq. 7. The initial atmospheric state is taken to be in hydrostatic and cyclostrophic balance with the pressure given by p λ φ z = p eq exp u eq sin φ gz. 79 R d T eq R d T φ The temperature field is specified to be T λ φ z T φ = T eq and the zonal velocity field is cu eq g sin φ 8 T eq T φ u λ φ z = u eq cos φ cz + 8 T φ T eq with zero initial meridional and vertical wind fields. In the above the surface values at the Equator are peq T eq u eq hpa K m s 8 and c is the shear parameter such that c = gives an initial wind field that is uniform with height and c gives a sheared flow. This test uses a damping layer with μ given by Eq. 77 in the top third of the computational domain η B = / η. Compared with the original DCMIP test specification which requires a damping layer on u v and w with a damping time-scale of s here the damping layer is applied to just w but with a reduced damping time-scale of. s which is equivalent to μ =.8 s in Eq No shear c = When c = in Eq. 8 the initial wind profile is constant with height. Figure a shows the equatorial cross-section of the vertical velocity. The characteristic downstream tilt with height of the non-hydrostatic waves is evident. The reduction in activity in the top km of the model is due to the action of the damping layer. Repeating this test but in hydrostatic mode δ V = the downstream tilt of the waves is removed and they propagate almost vertically not shown. The difference in the vertical velocity between the standard SISL w SISL and massconserving SLICE w SLICE schemesisshowninfigureb.these differences are small and isolated to the regions of strongest wave activity With shear c =. m When c =. m in Eq. 8 the initial zonal wind field increases from m s at the surface to 8 m s at km. n equatorial cross-section of the flow over the mountain is shown in Figure c for the vertical velocity. Compared with the results of the previous section the waves produced in this case are of larger scale and larger amplitude. The differences in the vertical velocity between the standard SISL and mass-conserving SLICE schemes are again shown in Figure d. s in the case without shear in hydrostatic mode the waves propagate almost vertically. In both the no-shear and sheared cases there is at least at a broad visual level reasonable agreement between these results and those presented by other modelling groups within DCMIP. 7.. Non-orographic gravity waves on a small planet The model s handling of non-orographic gravity waves is investigated using a test based upon the work of Skamarock et al. 99 Tomita and Satoh and Jablonowski et al. 8 Christiane Jablonowski personal communication. non-rotating small Earth X = is used. t the initial time an unbalanced potential temperature perturbation is overlaid on a hydrostatically and cyclostrophically balanced state with constant buoyancy frequency N g/θ θ/ z=constant. The Table. Model parameters for each test. Identifiers: 7.. mountain waves over a Schär-type mountain on a small planet; 7.. mountain waves over a Schär-type mountain on a small planet with wind shear; 7. non-orographic gravity waves on a small planet; and 7. deep-atmosphere baroclinic instability. Section Resolution Number of layers Model depth t Small-Earth Earth rotation λ φ degrees and spacing z T km s factor X rate s uniform uniform 7... uniform 7. quadratic 9 or 7.9 c Royal Meteorological Society and Crown Copyright the Met Office Quarterly Journal of the Royal Meteorological Society c Royal Meteorological Society Q. J. R. Meteorol. Soc. :

9 Mass-Conserving SISL Non-Hydrostatic Global Discretization a s. b s..8. Height km.. Height km c s d s..8. Height km Height km Figure. Mountain waves over a Schär-type mountain on a small planet. Equatorial cross sections of w for the mass-conserving scheme: a no shear and c with shear with contour interval.ms. Corresponding differences between the standard SISL and mass-conserving schemes w SISL w SLICE : b no shear and d with shear with contour interval. m s. ll plots are after s of scaled small-earth time corresponding to h of unscaled simulation time. background temperature field is defined via a surface temperature T s λ φ = g + T N eq g c p N c p exp u eq N g cos φ 8 which is used to obtain the full D temperature field T λ φ z = g N N exp c p g z N + T s λ φ exp g z. 8 The pressure field is defined to be u eq N c p p λ φ z = p eq exp R g cos φ T λ φ z /κ exp N T eq gκ z 8 and the initial zonal wind field is u = u eq cos φ 8 with zero initial meridional and vertical velocity. In the above the values at the Equator are Teq p eq u eq K hpa m s. 87 The overlaid potential temperature perturbation is of the form θ d πz = θ d + rc sin 88 L z where θ = K the horizontal width and vertical wavelength are d = m L z = m and r c is the great-circle distance Eq. 7 from the point λ φ π/. Equatorial cross-sections of the potential temperature perturbation from the initial background state before the perturbation Eq. 88 is applied are shown in Figure for both the non-hydrostatic δ V = and hydrostatic δ V = modes along with the differences between the standard SISL θ SISL and mass-conserving θ SLICE schemes. There is again reasonable agreement with other nonhydrostatic DCMIP model results particularly for example with the ICON IP model. The evolution of the gravity waves from the initial perturbation can be seen as an increasing sequence of positive and negative perturbations on the background state. long the Equator the gravity waves propagate with a phase speed given approximately in the limit of long horizontal wavelength by c λ = u ± NL z π. 89 The numerical phase speed in the model integration is compared with this analytical value in Figure. This is a Hovmöller diagram in which the potential temperature perturbations along the Equator at a height z = km are shown as a function of time. The analytic phase speeds Eq. 89 are overlaid and there is good agreement between the analytic and numerical phase speeds. Figure shows that the differences between the results of the standard SISL and mass-conserving schemes are relatively small. c Royal Meteorological Society and Crown Copyright the Met Office Quarterly Journal of the Royal Meteorological Society c Royal Meteorological Society Q. J. R. Meteorol. Soc. :

10 N. Wood et al. a Height s... b Height s c 9 8 s. d 9 8 s.. Height 7.. Height Figure. Non-orographic gravity waves on a small planet. Equatorial cross-sections of the θ perturbation from the basic state after s of scaled small-earth time for the mass-conserving scheme: a non-hydrostatic and c hydrostatic with a contour interval of. K. Corresponding differences between the standard SISL and mass-conserving schemes θ SISL θ SLICE : b non-hydrostatic and d hydrostatic with a contour interval of. K. a Time.... b Time x Figure. Hovmöller diagram of the non-hydrostatic θ perturbation along the Equator at height z = km: a mass-conserving scheme with a contour interval of. K and b corresponding differences between the standard SISL and mass-conserving schemes θ SISL θ SLICE with a contour interval of. K. The approximate long horizontal wavelength analytic phase speeds given by Eq. 89 are shown by the thick black lines. 7.. Deep-atmosphere baroclinic instability The baroclinic wave test of Jablonowski and Williamson ab has been widely used to test the response of dynamical cores to a mid-latitude baroclinic instability and results for the standard SISL formulation presented here can be viewed on the DCMIP website denoted there as ENDGame. However as the Jablonowski and Williamson a test is formulated using the shallow-atmosphere approximation to test the deep-atmosphere formulation the switchable shallow- and deep-atmosphere baroclinic wave test of Ullrich et al. is used instead. This test is inspired by the Jablonowski and Williamson a test but is reformulated for deep-atmosphere dynamical cores by using the basic state proposed in Staniforth and White and Staniforth and Wood. This test further differs from the Jablonowski and Williamson test in setting a zero geopotential lower boundary and also using a reformulated stream-function perturbation to selectively perturb the vortical component of the flow but not the divergent component. dditionally a vertically varying tapering function is applied to localize the perturbation to the lower atmosphere only. c Royal Meteorological Society and Crown Copyright the Met Office Quarterly Journal of the Royal Meteorological Society c Royal Meteorological Society Q. J. R. Meteorol. Soc. :

11 Mass-Conserving SISL Non-Hydrostatic Global Discretization The same vertical grid is used as that suggested for the Jablonowski and Williamson a test as part of the DCMIP project. Higher resolution is obtained near the lower boundary by using a quadratically stretched vertical grid in which the model levels η k are calculated as η k = ε k/ + ε + 9 using a stretching parameter ε =. Once η k is obtained from Eq. 9 the physical heights z k are then obtained from Eq. 7. The Northern Hemisphere surface pressure fields at days 8 andareshowninfiguresandrespectively.theseagree well with the results in Ullrich et al. their figure. However it should be noted that in contrast to the results in Ullrich et al. the results here are obtained from running the mass-conserving scheme with no off-centring with a reduced time step and to ensure exact mass conservation even when the model is not iterated to convergence ρ n+ is obtained via back-substitution from the continuity equation instead of the equation of state. The development of the pressure systems in this test is similar to that in the Jablonowski and Williamson a test with an initial phase of slow growth followed by a rapid intensification and finally wave breaking and overturning albeit at a different rate. s noted in Ullrich et al. on an unscaled Earth X = there is very little difference between shallow- and deep-atmosphere results for this test which is expected since the ratio r/a only varies by.% across the depth of the atmosphere. However on a small Earth X = the ratio r/a changes by % and so the differences between a shallow and deep atmosphere are more pronounced. Panels c and e of Figures and show the small-earth shallowand deep-atmosphere surface pressure fields respectively. The shallow-atmosphere integration has hardly changed from the unscaled Earth one whilst in common with the results seen in Ullrich et al. the deep-atmosphere integration develops much stronger pressure systems at a faster rate than the unscaled integration. Panels b d and f of Figures and show the differences between the standard SISL and mass-conserving schemes. s with the previous tests the differences are small with increased differences in the day plots; this is to be expected as at this point wave breaking has begun and even small differences in model formulations can lead to noticeable differences in observed behaviour. fter days of integration the mass-conserving scheme preserves the total mass of the atmosphere with a relative error of about i.e. effectively machine precision. For the standard SISL scheme the relative error in the total mass increases to about. 8. Summary n inherently mass-conserving semi-implicit semi-lagrangian discretization of the deep-atmosphere non-hydrostatic Euler equations has been presented. Mass conservation can be achieved using any conservative remapping scheme; here the SLICE scheme of Zerroukat et al. 9a and Zerroukat and llen has been used. a 8 Surface Pressure hpa Day b 8 SL SLICE Surface Pressure hpa Day 8 9 c 8 Surface Pressure hpa Day d 8 SL SLICE Surface Pressure hpa Day 8 9 e 8 Surface Pressure hpa Day f 8 SL SLICE Surface Pressure hpa Day 8 9 Figure. Surface pressure at day 8 for the mass-conserving scheme: a X = deep-atmosphere c X = shallow-atmosphere and e X = deep-atmosphere all with contour interval 8 hpa. Corresponding surface pressure differences between the standard SISL and mass-conserving schemes: b X = deep-atmosphere d X = shallow-atmosphere and f X = deep-atmosphere all with contour interval hpa. c Royal Meteorological Society and Crown Copyright the Met Office Quarterly Journal of the Royal Meteorological Society c Royal Meteorological Society Q. J. R. Meteorol. Soc. :

12 N. Wood et al. a 8 Surface Pressure hpa Day b 8 SL SLICE Surface Pressure hpa Day c 8 Surface Pressure hpa Day d 8 SL SLICE Surface Pressure hpa Day e 8 Surface Pressure hpa Day f 8 SL SLICE Surface Pressure hpa Day Figure. Surface pressure at day for the mass-conserving scheme: a X = deep-atmosphere c X = shallow-atmosphere and e X = deep-atmosphere all with contour interval 8 hpa. Corresponding surface pressure differences between the standard SISL and mass-conserving schemes: b X = deep-atmosphere d X = shallow-atmosphere and f X = deep-atmosphere all with contour interval hpa. standard non-mass-conserving SISL version of the scheme has also been developed which requires only a small localized modification to the model. comparison of the mass-conserving and non-mass-conserving schemes on a variety of published tests in spherical geometry shows little significant difference between them though somewhat larger differences were found in the deep-atmosphere small-earth baroclinic wave test. The semiimplicit scheme employs a reference profile to obtain a linear Helmholtz problem. particular feature of the discretization is that it has been implemented such that at convergence of the iterative scheme all terms involving the reference profile cancel and the results are independent of the choice of the profile other than its influence on whether the scheme converges and at what rate. The model has been designed and coded in a flexible way that allows it to be switched straightforwardly from being nonhydrostatic to hydrostatic and also from deep-atmosphere to shallow-atmosphere. This opens the way to perform controlled experiments with exactly the same numerics to measure the importance of various terms and approximations as a function of any given application. dditionally the model employs a general terrain-following geopotential coordinate system which allows application of the model in Cartesian spherical and also spheroidal geometries. Here only the unforced dry Euler equations have been considered. However the model has been successfully coupled to a suite of physics parametrizations a data assimilation scheme and an ocean model. Results for numerical weather prediction and climate simulations will be the subject of future publications. cknowledgement We thank two anonymous reviewers for their helpful comments on an earlier version of this article. ppendix : Coordinate systems For a given coordinate system the parameters needed to specify the equations are the coordinates ξ ξ ξ metric factors h h h elements of the rotation matrix M Coriolis parameters f f f and gravitational acceleration g. These parameters are now given for a selection of coordinate systems... Deep-atmosphere spherical ξ ξ ξ = λ φ r h h h = r cos φ r f = cos φ P sin λ f = sin φ P cos φ cos φ P sin φ cos λ f = sin φ P sin φ + cos φ P cos φ cos λ where r is the spherical radius and is the Earth s rotation rate. Here a rotated coordinate system has been used in which the North Pole of the rotated spherical polar system is placed at the point that has latitude φ P and longitude λ P in the geographical non-rotated system. λ in the rotated system is chosen so that the geographical North Pole has λ =. in the c Royal Meteorological Society and Crown Copyright the Met Office Quarterly Journal of the Royal Meteorological Society c Royal Meteorological Society Q. J. R. Meteorol. Soc. :