Lagrangian time derivatives in the prognostic equations given in Sect.  are approximated in time as dψ/dt=(ψn+1-ψ*n)/Δt, where the superscript indicates time level tn=nΔt; subscript * indicates that quantity ψ is evaluated at the departure position of the Lagrangian parcel (at tn). Under the SL approach, each point of a fixed computational grid is the arrival position of some Lagrangian particle. The advection equation dψ/dt=0 is discretized in time as ψn+1=ψ*n. SL approximation is not subject to the Courant–Friedrichs–Lewy stability limitation, so Δt can be chosen from accuracy considerations that greatly contribute to the computational efficiency.

The departure position of the Lagrangian parcel arriving at some grid point can be approximately found by integrating the equation drdt=V one step back in time. Following the SETTLS scheme , we approximate this equation as rn+1-r*nΔt=12(Vn+V*(n+1)e), where V(n+1)e=2Vn-Vn-1. The iterative approach is used to solve Eq. (): r*m+1n=rn+1-Δt2(Vn+V*m(n+1)e), where *m indicates the departure point position found at mth iteration. Additional geometric approximations are needed iterating Eq. () to account for spherical geometry and shallow atmosphere approximation see.

The quantities at the departure positions of Lagrangian particles are evaluated using interpolation. We use 3-D tensor-product cubic Hermite interpolation for advective terms of equations (the terms that arise from SL approximation of d/dt). Trilinear interpolation is used for components of V in iterative process () and non-advective terms of equations. The change in the direction of coordinate system unit vectors between departure and arrival positions should be taken into account when interpolating the vector components: dψdt=1Δtψλψφ-Rψλ*ψφ*, where R is the rotation matrix (see , , detailed derivation in , ).

The disadvantage of a semi-Lagrangian approach as formulated in Sect.  is the lack of tracer mass conservation. To address this problem, SL-AV20 incorporates the finite-volume SL Conservative Cascade Scheme CCS-3D,. Finite-volume SL discretization starts with the advection equation in form () where the arrival volume V(tn+1) coincides with grid cell Vijl (l is the vertical index): (∂p∂ηq)Vijln+1SijΔηl=∫Vijl*(∂p∂η(q+FqΔt))ndV. (.)Vijl is quantity averaged over Vijl; Sij and SijΔηl are the cell horizontal square and volume, respectively. Cell-averaged tracer density ∂p∂ηq is chosen as a prognostic variable.

The departure cell V(tn)=Vijl* is defined by its vertices, i.e., the departure positions of Lagrangian parcels arriving at the vertices of Vijl at tn+1. Departure cell vertices are found with interpolation of known coordinates of departure cell centers. The latter are departure positions of the Lagrangian parcels arriving at the grid points computed using Eq. ().

To approximate the integrals over the departure cells in Eq. (), we extend the 2-D Conservative Cascade Scheme by to 3-D space. Following the ideas of , we split the 3-D integration problem into the sequence of three 1-D integrations that greatly contributes to the computational efficiency. The resulting locally mass-conserving SL advection scheme uses a piecewise parabolic subgrid tracer density distribution and incorporates several monotonicity and positivity preserving options.

The non-advective terms of prognostic equations (from Sect. ) are approximated using the combination of the Crank–Nicolson scheme with the pseudo-second-order decentering for linear terms and the SETTLS scheme for nonlinear terms. For the equation dψdt=Lψ+N(ψ), where ψ is an arbitrary scalar or vector variable the scheme writes as ψn+1-ψ*nΔt=12N(ψ)*(n+1)e+N(ψ)n+1+ϵ2Lψn+1+1+ϵ2Lψ*n-ϵ2Lψ*(n+1)e+Lψn.

The absolute vorticity Eq. () is discretized in time as follows: ζn+1+Δt2fDn+1=Rζ, where Rζ is a combination of time level n and extrapolated in time ((n+1)e) quantities (see Appendix  for details on the right-hand-side terms of the SI system of equations); no decentering is applied to term fD. We use the background state with constant reference temperature T¯ and reference pressure p¯=A(η)p0+B(η)p¯s (p¯s is a constant) to extract the linear terms in Eqs. (), (), (), and (). First, the pressure gradient term of momentum equations () is split into linear and nonlinear contributions: ∇(Φs-∫1ηRdTvdln⁡p)+B(η)psA(η)p0+B(η)psRdTv∇ln⁡ps=(∇G)︸ linear term-∇∫1ηRd(Tv-T¯)dln⁡pp¯+B(η)psA(η)p0+B(η)psRd(Tv-T¯)∇ln⁡ps︸ nonlinear term, G=Φs-∫1ηRd(Tv-T¯)dln⁡p¯+RdT¯ln⁡ps. Here we use the fact that ∇∫1ηRdT¯dln⁡p=RdT¯[∇p(η)/p(η)-∇ln⁡ps]=RdT¯[B(η)ps/(A(η)p0+B(η)ps)-1]∇ln⁡ps. A similar relation is derived for ∇2 of pressure in the analytic divergence Eq. (). The approximation of vertical integrals is described in Sect. .

The time-discretized divergence equation is written as Dn+1+(1+ϵ)2Δt∇2Gn+1=∇⋅RV,Dn+1-Δt2fζn+1+(1+ϵ)2Δt∇2Gn+1=RD. Equation () is the divergence equation derived from the momentum equation () discretized in time via Eq. (); RV is the right-hand side (RHS) of the corresponding vector equation. The RHS term ∇⋅RV is calculated with the discrete divergence operator from Sect. . Equation () is based on the analytic divergence Eq. (). This equation allows us to use the same Crank–Nicolson discretization in time for -fζ and ∇2G, the two terms representing geostrophic balance.

In the thermodynamic equation (), the Lagrangian time derivative of ln⁡ps is discretized in the SL manner in the linear part of the energy conversion term, whereas it is replaced by the RHS of continuity Eq. () in the nonlinear part. The resulting time-discrete equation is Tvn+1-RdT¯cpdp¯sA(η)p0+B(η)p¯s(B(η)ln⁡psn+1+1+ϵ2Δts˙n+1)=RT.

The continuity equation () is discretized as follows: ∂B∂ηln⁡psn+1+1+ϵ2Δt(∂A/∂η)p0+(∂B/∂η)p¯sp¯sDn+1+1+ϵ2Δt∂s˙n+1∂η=RP. Integration of Eq. () from model top to the ground provides the expression for ln⁡psn+1 independent of s˙n+1, whereas the integration to the arbitrary level η gives the s˙n+1: (1-B(ηtop))ln⁡psn+1=-1+ϵ2Δt∫η=ηtopη=1((∂A/∂η)p0+(∂B/∂η)p¯sp¯sDn+1+RP)dη,1+ϵ2Δts˙n+1(η)=-(B(η)-B(ηtop))ln⁡psn+1-1+ϵ2Δt∫η=ηtopη=η((∂A/∂η)p0+(∂B/∂η)p¯sp¯sDn+1+RP)dη. The RHS of Eq. () can easily be substituted for the ln⁡psn+1 term in Eq. (), providing an equation for s˙n+1 which depends only on the Dn+1 unknown quantity.

Equations (), (), (), (), and () and the definition of linear geopotential G () form the system of linear equations for time-level n+1 variables (ζ,D,Tv,ln⁡ps,s˙,G). The solution procedure for this system is as follows. First, if an analytically derived divergence equation () is used, the relative vorticity ζn+1 is excluded by substitution from Eq. (). Then, Eqs. () and () are substituted in Eq. () to obtain an expression for Tvn+1 that depends only on Dn+1. This expression and Eq. () are used to eliminate Tvn+1 and ln⁡psn+1 in the Gn+1 definition Eq. (). Therefore, we obtain the pair of equations that contains only Gn+1 and Dn+1 unknowns. With the aid of vertical discretization (see Sect. ), this pair can be written as G+1+ϵ2ΔtMD=H,(1+αf2Δt24)D+1+ϵ2Δt∇2G=RD+αfΔt2Rζ, where G, D and Rψ are columns of Nlev (number of model levels) components, with the lth component representing the corresponding horizontal field at the level l; H=Φs+RdART+M′RP, M, M′, and A are the matrices of discrete vertical operators (resulting from vertical integration; see details in Appendix ). The coefficient α=0 is used for the standard divergence equation () and α=1 for the analytically derived divergence equation (); RD corresponds to the RHS term of the relevant divergence equation.

Substituting D from Eq. () in Eq. () and using eigenvalue decomposition M=PΛP-1 results in Nlev 2-D Helmholtz problems for components of P-1G vector. The solution procedure for Helmholtz problems is described in Sect. . Once G is found, we can find the divergence D with Eq. () and use it to calculate ln⁡psn+1, s˙n+1, Tvn+1 and ζn+1 (see Eqs. , , , ). The n+1th time-step calculations are finalized by reconstruction of the horizontal wind Vn+1 as described in Sect.  and calculation of η˙ and recalculation of s˙ (Sect. ).

We have also implemented an iterative time-integration scheme in addition to the time discretization described above. The basic discretization is used at first iteration to obtain first-guess n+1 time-step fields (ζ,D,T,ln⁡ps,V). At the second iteration, we replace the time-extrapolated nonlinear terms N(n+1)e of the equations and time-extrapolated wind velocity (u,v,η˙)(n+1)e involved in the upstream trajectory computations with those obtained using n+1 time-step first-guess values. Using this iterative approach allows us to improve stability and use larger time steps without degrading accuracy.

The inherently mass-conserving (IMC) version of the model uses a continuity equation in the finite-volume form (). The nonlinearity of this equation is hidden in the trajectory computation or equivalently in the evolution of the Lagrangian cell V(t). The equation is linearized using the orography-dependent reference pressure profile pref=Ap0+Bpsref, where psref=1013.25hPa×exp⁡(-Φs/(RdT¯)). The following equation is ddt∫V(t)(∂p′∂η)dV=-∫V(t)(∇⋅(∂pref∂ηV)+∂∂η(η˙∂pref∂η))dV, where p′=p-pref. The equation for mass-conservative update of ps is obtained with discretization of Eq. () using the scheme () and its integration from model top to bottom: (1-B(ηtop))Sij(psn+1-psref)Sij=∑l=1l=Nlev(-1+ϵ2Δt∇⋅(∂pref∂ηVn+1)+ϵ2Δt∇⋅(∂pref∂ηVn))VijlSijΔηl+∑l=1l=Nlev∫Vijl*((∂p′∂η)-1+ϵ2Δt∇⋅(∂pref∂ηVn)+ϵ2Δt∇⋅(∂pref∂ηV(n+1)e))dV, where (ps)Sij is the surface pressure averaged over Sij. The departure volume integrals are evaluated using CCS-3D .

Using an orography-dependent reference pressure profile is crucial for stability in mountainous regions. The term ∇⋅(∂pref∂ηVn+1), however, cannot be expanded in terms of model prognostic variables, so it is very difficult to solve the system of equations similar to those formulated in Sect. , but with a mass-conservative equation for psn+1 () instead of Eq. (). Therefore, the following procedure is implemented. First, we run a time step of the non-mass-conserving model version, solve the semi-implicit system as it is formulated in Sect. , and obtain the n+1 time-step horizontal wind Vn+1. Then the mass-conservative update of ps, Eq. (), is calculated.

We achieve inherent mass conservation, however, at the cost of introducing some inconsistency between the surface pressure and horizontal wind fields. This results in a slight noise in the regions with steep orography, but does not significantly affect the model accuracy. The greater problem not solved yet is the consistency between dry air and tracer mass, so the IMC model version now is rather a research option.

The SL-AV20 model incorporates a reduced latitude–longitude grid with variable resolution in latitude. The grid consists of points located on the latitudes φ=φj, j∈[0,Nlat] with longitude spacing Δλj, (λ,φ)ij=(iΔλj,φj). The number of points on a grid latitude is reduced polewards of the Equator. The pole points are assumed to be the grid latitudes. The regular latitude–longitude grid is the special case of the grid described above with constant latitude spacing and the same number of points on each grid latitude.

Formally, the model algorithms can work with any set of φj and Δλj. However, the choice of these crucially affects accuracy. The grids are constructed with the algorithm of . This algorithm generates the set of φj that satisfy grid smoothness constraints (to avoid spurious wave reflection at the regions of abrupt spacing change) and match the desired grid spacing latitude dependence as close as possible. Then, given φj and the total number of grid points, the algorithm optimizes Δλj to minimize the error of interpolation of an analytic test function to some finer grid.

The following fourth-order accurate formula is applied to evaluate the first derivative: (∂ψ∂x)i+1/2=ψi-1-27ψi+27ψi+1-ψi+224Δx+O(Δx4), where x can be either φ or λ; ψ is a scalar quantity or vector component. However, we use an unstaggered grid and need the values of ∂ψ∂x to be located at the same points as ψ; thus, we interpolate (∂ψ∂x)i+1/2 to the integer nodes of the grid using fourth-order Lagrangian interpolation. To calculate divergence and vorticity, fourth-order Lagrangian interpolation is used to obtain vector components at the grid half-nodes. Then a second-order locally conservative formula is applied to calculate the meridional derivative: 1acos⁡φ∂ψcos⁡φ∂φ=ψj+1/2cos⁡φj+1/2-ψj-1/2cos⁡φj-1/2a(sin⁡φj+1/2-sin⁡φj-1/2)+O(Δφ2). The longitudinal derivative is evaluated with Eq. () as in the gradient operator.

To account for the variable resolution in latitude, we introduce pseudo-latitude φ′, such that the grid is equally spaced in φ′ following. Then the meridional derivative ∂ψ∂φ=∂ψ∂φ′∂φ′∂φ. The derivative ∂ψ∂φ′ and inverse mapping factor M=∂φ∂φ′ are calculated using Eq. ().

The longitudinal derivatives are calculated in the grid-point space, whereas Fourier representation in longitude ψ(φj,λ)=A^0(φj)/2+∑k(A^k(φj)cos⁡(kλ)+B^k(φj)sin⁡(kλ)) is used to calculate meridional derivatives on the reduced lat–lon grid. The meridional derivatives of A^k, B^k are calculated and then the inverse Fourier transform is applied to obtain the values of derivatives in grid-point space. If Nj is the number of points at the grid latitude φj, the wavenumbers k>Nj/2-1 can not be represented, so it is natural to set A^k, B^k and their derivatives to zero at this latitude.

The disadvantage of Fourier transform is the necessity of global communications in parallel implementation, which could lead to poor scalability. The calculation of meridional derivatives in the grid point space is implemented as an option. The points in the reduced grid are not aligned along meridian lines. Therefore, the interpolation along longitude is carried out first to obtain the values of ψ at the needed longitude and then Eq. () is applied to obtain half-integer values of derivatives. The similar approach was applied in . We have tested this grid point algorithm and there is no impact on the results for the baroclinic test case presented in Sect.  as compared with the computations in Fourier space as described above.

Calculation of meridional derivatives near the poles using Eq. () requires the values of ψ at virtual latitudes φ-1=-π/2-(φ1+π/2) and φNlat+1=π/2+(π/2-φNlat-1). If one continues the meridional line λ beyond the pole, it will coincide with λ+π. Thus ψ(φ-1,λ)=(-1)νψ(φ1,λ+π), where ν is 0 for scalar quantities and 1 for vector components (the latter is due to change of basis vectors orientation after π phase shift). It is easy to show that A^k(φ-1)=(-1)k+νA^k(φ1), the similar relations take place for B^k(φ-1), A^k(φNlat+1), B^k(φNlat+1).

To evaluate divergence and vorticity at the pole point, one uses the fact that the scalar quantities can have only A^0 non-zero Fourier coefficients at the pole point to be uniquely defined. The fourth-order accurate Lagrangian interpolation is applied to obtain A^0 polar values from known A^0 at adjacent latitudes (including virtual latitudes). Similarly, vector components can have only A^1, B^1 non-zero Fourier coefficients at the pole point (this corresponds to the unique value of the vector quantity at the pole point and basis vectors dependent on longitude). Fourier coefficients of gradient components are also interpolated to the pole point.

The IMC version of the model also requires calculation of a flux-divergence operator ∇⋅(ψV), where ψ is some scalar function. For mass conservation ∑i,j∇⋅(ψV)ijSij=0 over the whole sphere must hold up to machine precision. This operator is discretized as described in in the flux form with fourth-order accuracy. In the IMC version, this operator with ψ=1 replaces the divergence operator described above.

The Laplace operator arises in the time-discrete divergence equation (). In the case of Eq. () derived in a discrete way from momentum equations () (which is considered standard in SL-AV20) ∇2 appears in the implicit part, whereas in the explicit part it stands as the product of discrete divergence and gradient operators (see Sect.  for details). If analytical divergence equation () is used, ∇2 is present in both implicit and explicit parts of equation.

If the discrete-derived divergence equation is used, the compact finite-difference (CompFD) approximation of the meridional derivatives in the Laplace operator is applied, for their smaller truncation errors. CompFDs discretization require matrix inversion that generaly causes the parallel scalability problems. However, using CompFDs to discretize Laplace operator in the implicit part of divergence equation is not a computational burden, since it is already involved in the matrix inversion problem (see Sects. , ).

The Laplace operator is discretized using Fourier representation in longitude. The meridional part of Laplacian is discretized with double application of compact formula for the first derivative: 124(∂ψ∂φ)i-1+1112(∂ψ∂φ)i+124(∂ψ∂φ)i+1=ψi+1/2-ψi-1/2Δφ+O(Δφ4). The longitudinal part of ∇2 is the Fourier image of product of two longitudinal derivatives defined by Eq. (). 1a2cos⁡2φ∂2exp⁡(ikλ)∂λ2=-k̃2exp⁡(ikλ)=-(261sin⁡(kΔλ)+sin⁡(3kΔλ)-36sin⁡(2kΔλ)192acos⁡φΔλ)2exp⁡(ikλ). Further details about compact finite-difference approximation of ∇2 operator are in Sect. .

The main objective of using the analytically derived divergence equation () is time-symmetric discretization, which theoretically leads to better inertia-gravity wave dispersion properties . Temporal symmetry means that the same discretization of ∇2 is used in implicit and explicit parts of the divergence equation. We do not use CompFDs for approximation of the Laplace operator to avoid matrix inversion in the explicit part of the equation. The following approximation to ∇2 is used: (∇2ψ)i,j=112a2cos⁡2φjΔλ2(ψi-2,j+16ψi-1,j-30ψi,j+16ψi+1,j-ψi+2,j)+1a2(sin⁡φj+1/2-sin⁡φj-1/2)Δφ((∂ψ∂φ)i,j+1/2cos⁡φj+1/2-(∂ψ∂φ)i,j-1/2cos⁡φj-1/2). At the pole points, Eq. () is transformed to (∇2ψ)Nlat=-1a2(1-sin⁡φNlat-1/2)Δφ(∂ψ∂φ)‾i,Nlat-1/2cos⁡φNlat-1/2,(∇2ψ)0=1a2(sin⁡φ1/2-1)Δφ(∂ψ∂φ)‾i,1/2cos⁡φ1/2, which preserves the formal order of accuracy and local conservation of the whole ∇2 discretization. (∂ψ/∂φ)i,j±1/2 in the above equations are evaluated using Eq. (); the overbar indicates the zonal average.

The calculation of nonlinear terms can give rise to nonlinear instability, when the interaction of the shortest scales leads to the exaggerated contribution to the largest scales. We use spatial averaging to suppress this kind of instability. The following formulae do not have a strict theoretical basis, but they are rather a result of experience in optimizing model performance.

The first type of nonlinear term is a scalar-by-scalar product, namely ζD, D2 in vorticity and analytic divergence equations () and (): (ζD)i,j=116∑m=±1[(ζi,j+ζi+m,j)(Di,j+Di+m,j)+(ζi,j+ζi,j+m)(Di,j+Di,j+m)]. The term D2 is computed in a similar way. Surprisingly, this leads to better medium-range forecast scores as compared to the formula with the increased weight of the central term ζi,jDi,j.

Before computing nonlinear terms involving first-order derivatives in λ or φ (components of ∇ln⁡ps, ∂η˙/∂x, ∂Tv/∂x), these derivatives are averaged in the direction perpendicular to the direction of differentiation; e.g., ∂Tv/∂λ is averaged in latitude. The averaging formula is (∂ψ∂x)‾j=12+c((∂ψ∂x)j-1+c(∂ψ∂x)j+(∂ψ∂x)j+1), where j is index in the direction perpendicular to x. The constant c=3 for Tv and η˙ derivatives, c=4 for ∇ln⁡ps components. The horizontal wind components and temperature are averaged in 2-D for computation of Jζ Eq. (), JD Eq. () and terms involving Tv in momentum () and thermodynamic () equations: u‾i,j=112(8ui,j+∑m=±1(ui+m,j+m+ui+m,j-m)),Tv‾i,j=112(8Tvi,j+∑m=±1(Tvi+m,j+Tvi,j+m)).

Equations ()–() are valid in the case of constant latitudinal resolution. To account for the variable resolution, we mention that ∫ψdφ=∫ψMdφ′ and multiply all (ζi,j+ζi,j±1)(Di,j+Di,j±1)-type combinations in Eq. () and ψi,j±1 terms of Eqs. () and () by 2Mj±1/2/(Mj+1/2+Mj-1/2).

No averaging is applied for computations of terms involving the coefficients of hybrid coordinate system A, B, and their derivatives. Also, no averaging is applied for computation of Rmoist/cp term in thermodynamic equation () and hydrostatic equation () terms.

We use a Lorenz-staggered grid in the vertical where all variables, except vertical velocity η˙, are located at integer levels ηl (cell centers) and η˙ is located at half-levels ηl+1/2 (cell interfaces). Integer grid levels are set to ηl=(ηl+1/2+ηl-1/2)/2, η1/2 corresponds to the model top, and ηNlev+1/2 is at the Earth's surface. Similarly to η, the hybrid coordinate system coefficient Al=(Al+1/2+Al-1/2)/2, and the same holds for Bl, Δηl=ηl+1/2-ηl-1/2 and the same is implied for ΔAl, ΔBl and other variables. The vertical derivatives are approximated with a second order of accuracy (∂ψ∂η)l=ΔψlΔηl+O(Δηl2).

It is almost standard for hydrostatic atmospheric models to use vertical discretization, where the geopotential is calculated at half-levels ηl+1/2 by integrating the hydrostatic Eq. () with the mid-point quadrature rule and then interpolated to the integer levels. It was found that the trapezoidal rule allowing us to calculate geopotential directly at the integer levels is more accurate. Equation () is integrated in the following way: ΦNlev=Φs+RdTvNlevln⁡pspNlev,Φl-1=Φl+Rd2(Tvl+Tvl-1)ln⁡plpl-1.

The vertical integrals in the time-discrete continuity equation () and also Eqs. () and () are calculated using the mid-point rule. At the end of each time step, the vertical velocity is recalculated via the diagnostic relation derived from the Eulerian form of the continuity equation (): ∂B∂η∂ps∂t=-∇⋅(∂p∂ηV)-∂∂η(∂p∂ηη˙). Integration of Eq. () from η1/2 to ηL+1/2 using mid-point rule and vertical boundary conditions gives (BL+1/2-B1/2)∂ps∂t=-∑l=1l=L(Alp0Dl+Bl∇⋅(psV)l)Δηl-(∂p∂ηη˙)L+1/2. The expression for ∂ps∂t can be obtained by setting L=Nlev in Eq. () and using η˙Nlev+1/2=0. After substituting this expression back to Eq. () for arbitrary L, (∂p/∂η⋅η˙)L+1/2 can be evaluated. The term ∇⋅(psV) is calculated using standard divergence discretization (see Sect. ), second-order averaging is applied to ps in latitude (longitude) before computing psu (psv). To retrieve the vertical velocity η˙ for calculation of upstream trajectories (see Sect. ) and terms of Eq. (), (∂p/∂η⋅η˙) is interpolated to integer levels with second-order accuracy and divided by (p0ΔAl+psΔBl)/Δηl which is proxy for ∂p/∂η.

Following , the definitions of ζ=k⋅∇×V and D=∇⋅V are applied to reconstruct the horizontal wind from the known vertical component of relative vorticity and horizontal divergence. Fourier representation in longitude Eq. () is used. The relations for the u and v 0th Fourier components are ∂A^0ucos⁡φ∂φ=A^0ζacos⁡φ,∂A^0vcos⁡φ∂φ=A^0Dacos⁡φ. These relations are integrated in latitude with the compact formula : 124(∂ψ∂φ)j-1+1112(∂ψ∂φ)j+124(∂ψ∂φ)j+1=ψj+1/2-ψj-1/2Δφ+O(Δφ4), where A^0ζcos⁡φ (A^0Dcos⁡φ) is substituted for ∂ψ/∂φ on the left-hand side and A^oucos⁡φ (A^ovcos⁡φ) is substituted for ψ on the right-hand side. The resulting Fourier coefficients for horizontal wind components at half-grid latitudes are interpolated to the integer grid latitudes using sixth-order compact interpolation .

The kth Fourier coefficients of horizontal wind components are determined solving -kA^kv-∂B^kucos⁡φ∂φ=B^kζacos⁡φ,kB^ku+∂A^kvcos⁡φ∂φ=A^kDacos⁡φ,-kA^ku+∂B^kvcos⁡φ∂φ=B^kDacos⁡φ,kB^kv-∂A^kucos⁡φ∂φ=A^kζacos⁡φ, where ∂ψ/∂φ is approximated using Numerov's scheme: 16(∂ψ∂φ)j-1+23(∂ψ∂φ)j+16(∂ψ∂φ)j+1=ψj+1-ψj-12Δφ+O(Δφ4).

If A^kψ, B^kψ are Nlat+1-element columns, with the jth element representing A^kψ, B^kψ at the jth grid latitude, and then system () can be written as -kA^kv-12ΔφM-1δCB^ku=aCB^kζ,kB^ku+12ΔφM-1δCA^kv=aCA^kD, where C is the diagonal matrix with Cjj=cos⁡φj, jth component of column δA^ is A^j+1-A^j-1, M is matrix with diagonals (1/6,2/3,1/6). We multiply the system of Eqs. () by matrix M from the left and rewrite it for (A^kv,B^ku)jT pairs, j∈[0,Nlat] that results in 2×2 block tri-diagonal system of equations. The same operations repeated for system () yields 2×2 block tri-diagonal system of equations for (A^ku,B^kv)jT. The details of this solver are given in . As shown in , this solver alleviates the problems associated with non-symmetric discretization in time. Also, it avoids the solution of Poisson problem on the sphere which requires certain care due to non-trivial kernel of Laplace operator.

The discrete Helmholtz problem for ψ – the kth Fourier harmonic of arbitrary quantity can be written in the column-matrix notation of Sect.  as Lψ+μ2ψ=R, where L is a discrete Laplace operator, μ2 is some scalar, R is kth Fourier harmonic of RHS. When divergence equation () derived from time-discrete momentum equation () is used, the meridional part of ∇2 operator is approximated by double application of compact finite-difference formula for the first derivative Eqs. () and () is used for the approximation of longitudinal part of Laplacian. The resulting 4th order accurate Helmholtz problem discretization is 1a2C-1M-1δ1/2(C1/2M1/2-1δψ)+(μ2-1a2C-2k̃2)ψ=R, where operator δ1/2 acts on quantities defined at half-integer grid points, the jth component of δ1/2ψ is ψj+1/2-ψj-1/2, δ acts on quantities defined at integer grid points, the jth component of δψ is ψj+1-ψj, and M and M1/2 are tri-diagonal matrices with diagonals (1/24,11/12,1/24) acting on quantities defined at integer and half-integer points, respectively. Matrix C is the same as in Eq. (), and matrix C1/2 is diagonal with a jth diagonal element cos⁡φj+1/2.