Following the vector-invariant formulation based on the Lamb transformation (Gassmann and Herzog, 2008), together with the deep-atmosphere formulations of Wood and Staniforth (2003), Borchert et al. (2019), and Skamarock et al. (2021), GMCORE solves the fully compressible, nonhydrostatic deep-atmosphere equations in a dry terrain-following mass-based coordinate in vector-invariant form. Compared to the shallow-atmosphere equations, the deep extension introduces three important changes: the constant planetary radius a is replaced by the local radial distance r=a+z, the constant gravity g0 becomes altitude-dependent gr=g0ar2, and the full Coriolis and spherical curvature terms are retained in the momentum equations. The resulting GMCORE governing equations are written as follows: 1∂V∂t=-ξak×V-∇ηdK-η˙∂V∂η-L1ρd∇ηdp+ra2∂p∂π∇ηdϕ-2Ωcosφwi-wrV+SV,2∂w∂t=-V⋅∇ηdw-η˙∂w∂η-gr+ρdρg0ra2∂p∂π+2Ωcosφu+u2+v2r+Sw,3∂∂t∂π∂η=-∇ηd⋅∂π∂ηV-∂∂η∂π∂ηη˙,4∂∂t∂π∂ηθm=-∇ηd⋅∂π∂ηVθm-∂∂η∂π∂ηη˙θm+Sθm,5∂∂t∂π∂ηqi=-∇ηd⋅∂π∂ηVqi-∂∂η∂π∂ηη˙qi+Sqi,i=1,…,N,6∂ϕ∂t=-V⋅∇ηdϕ-η˙∂ϕ∂η+wgr+Sϕ. Here V=u,v is the wind vector, η˙ is the vertical coordinate velocity, ∇ηd is the deep-atmosphere horizontal differential operator, K=u2+v2/2 is the horizontal kinetic energy, π is the dry-air mass (actually mass per unit area), expressed in pressure units through multiplication by the constant reference gravitational acceleration g0, Ω is the planetary angular rotation velocity, L=ρdρ=11+qm is the moist contribution where the qm represent total moist dry mixing ratio, ρd and ρ denote the dry air and total density, ϕ is the geopotential and S represents the unresolved physical tendencies. The deep-atmosphere absolute vorticity ξa is correspondingly written as 7ξa=f+1r2cosφ∂rv∂λ-∂rucosφ∂φ.

The layer-averaged mass continuity and thermodynamic equations are then expressed in conservative form as 8∂∂tδπ=-∇ηd⋅δπV-δ∂π∂ηη˙,9∂∂tδπθm=-∇ηd⋅δπVθm-δ∂π∂ηη˙θm+Sθm, where θm is the modified potential temperature including the thermodynamic effects of moisture species (Lauritzen et al., 2018) and δ(⋅)=(⋅)k-12-(⋅)k+12 denotes the vertical increment. In this form it is already evident that the deep-atmosphere extension changes not only the explicit dynamical terms, such as gr and the curvature contributions, but also the geometric structure of the transport operators that must be discretized.

For the numerical discretization, the operator modifications follow naturally from the replacement of the constant radius by the local radial distance on the generalized η-surfaces. The horizontal gradient and pseudo-horizontal divergence therefore become 10∇ηdα=1acosφar∂α∂λi+1aar∂α∂φj,11∇ηd⋅B≡1acosφ∂∂λarBλ+∂∂ϕarcosφBϕ, where α and B are arbitrary scalar and vector. Here the factor a/r is part of the local horizontal metric on the η-surface. It cannot, in general, be treated as a purely layer-dependent multiplicative factor, because the terrain-following η-surface may still carry horizontal variations of r. Consequently, the deep metric does not generally commute with the horizontal derivatives, and the metric factor must remain attached to the face fluxes or, equivalently, inside the deep divergence operator used by the discrete scheme.

It is also necessary to distinguish the pseudo-horizontal divergence appearing in the transport operators from the true three-dimensional divergence D3 of the velocity field. Following Wood and Staniforth (2003), the exact relationship is 12D3=∇rd⋅V+∂w∂r+2wr.

In the continuous equations, the factor 1/r may be understood either as part of the modified horizontal mass flux or as part of the deep horizontal divergence operator (Wood and Staniforth, 2003). These two understandings are equivalent only when the metric factor remains inside the horizontal derivative and is evaluated at the same face location as the flux. For the GMCORE mass-based discretization, this means that the local radius must be diagnosed and interpolated consistently to the staggered positions required by the horizontal mass flux, pressure-gradient force and advection process.

The vertical transport operator requires a different geometric correction. Because the upper and lower interfaces of a control volume lie on spherical shells of different radii, the vertical mass and tracer fluxes must be weighted by the shell-area factor r/a2 before calculating the vertical flux divergence. This shell-area weighting belongs specifically to the vertical transport, whereas the horizontal transport uses a local edge modification, i.e., the radius-dependent scaling of each horizontal grid-edge length by the local factor r/a, together with the deep-atmosphere horizontal divergence operator on the staggered grid. It is crucial for both conservation and numerical stability to consistently use appropriate and consistent discrete deep-atmosphere geometric extensions in calculations involving pressure gradients, horizontal divergence, vertical flux divergence, and advection.

As demonstrated by Wood and Staniforth (2003), a mass-based vertical coordinate requires adjustment for the deep-atmosphere. They denoted s as the generalized vertical coordinate. In contrast to Laprise (1992), the deep-atmosphere mass coordinate chooses ρr2∂r∂s as a constant instead of ρa2∂r∂s . This makes the definition of hydrostatic pressure coordinate turn to a more physical description of the atmosphere mass above a cell.

We use the dry mass π to substitute the generalized coordinate s and introduce terrain-following coordinate η (Kasahara, 1974; Simmons and Burridge, 1981): 13πx,y,η,t=ηAπ0+ηBπsx,y,t, where π0 is a constant (usually, 1000 hPa) and πs represents the surface mass. The coefficients ηA and ηB prescribe a set of hybrid coordinate defined at the model interfaces. The uppermost interface defines the model top, whereas the lowest interface follows the lower boundary. The specific vertical resolution and model-top setting used in each experiment are given in Sect. 3.

An obvious difference in this deep-atmosphere mass-coordinate dycore is the diagnostic calculation of the radial distance r. In the η-coordinate, to determine the radial distance r without relying on complex iterative solvers, GMCORE directly integrates the differential relation defining the vertical coordinate. The definition is given by: 14∂π∂η=-ρdg0ra2∂r∂η.

By rearranging the terms to isolate the geometric dependence, we obtain: 15r2∂r∂η=-a2ρdg0∂π∂η.

Recognizing the exact differential form r2dr=13dr3, this relation can be analytically integrated from the planetary surface (where r=rs and π=πs) upwards to any given model level k: 16rk3-rs3=3a2g0∫πkπs1ρddπ.

In the numerical implementation of the dynamical core, this integral is evaluated layer-by-layer from the surface upward. The radial distance rk+12 at a specific half model level is diagnosed through a discrete cumulative summation of the density-weighted layers: 17rk+12=rs3+3a2g0∑l=kksfcΔπlρd,l‾1/3, where Δπk=πk+12-πk-12 is the dry-air mass between two-layer interfaces, the ρd,l‾ is the dry density at the layer center (evaluated at full level).

The present deep-atmosphere geometry uses a spherical reference planet. Orography can enter through the lower-boundary surface geopotential and the terrain-following η coordinate, but the idealized experiments reported here use zero orography. Deviations of the geoid from a perfect sphere, such as oblateness, are not represented in the present version.

The regular latitude–longitude grid remains widely used in several established atmospheric models (Williamson, 2007; Staniforth and Thuburn, 2012) and is still under continuous development (Lima, 2024). Its orthogonal geometry aligns the coordinate directions with the zonal and meridional flow components, facilitating the formulation of fluxes and differential operators. GMCORE retains this grid structure as a practical choice that is consistent with its existing numerical framework. However, the convergence of meridians toward the poles leads to increasingly small zonal grid spacing and restrictive stability constraints, requiring the polar filtering treatment described in Sect. 2.5.

GMCORE uses an Arakawa C-grid staggering in the horizontal and a Lorenz staggering in the vertical, as illustrated in Fig. 1a. The full levels are indexed by k=1,…,nlev from the model top downward. Dry mass and thermodynamic variables are defined at the full levels. The Arakawa C-grid is well suited for representing geostrophic adjustment and inertia–gravity waves, while the Lorenz vertical staggering has a relatively simple structure and facilitates conservative discretization (Arakawa and Lamb, 1977; Adcroft et al., 1997; Simmons and Burridge, 1981). The horizontal velocity components u and v are also defined at the full levels, while retaining their horizontal C-grid staggering. The vertical velocity w and vertical coordinate velocity η˙ are defined at the half levels.

Following Ringler et al. (2010), the subscripts c, e, and v denote a scalar-cell center, a velocity edge, and a vorticity vertex (the center of the corresponding dual control volume), respectively. In Fig. 1b, Ac and Av are the primal- and dual-control-volume areas, le is the primal edge length, and de is the distance between the two neighboring cell centers sharing edge e. The sets Ec, Ce, Ev, Cv, and E⟂e denote the edges around cell c, the two cells sharing edge e, the edges around vertex v, the cells surrounding vertex v, and the neighboring transverse edges of a target edge, respectively. The orientation factors ne,c=±1 and te,v=±1 give the outward flux sign for a cell and the positive circulation sign around a vertex.

Let a be the reference planetary radius and rc, re, and rv the local radial distances at cell, edge, and vertex locations. A superscript a denotes the corresponding horizontal geometry on the reference sphere. The deep-atmosphere metric is therefore introduced directly through the local lengths and areas: 18le=realea,de=readea,Ac=rca2Aca,Av=rva2Ava.

The shallow-atmosphere form is recovered when rc=re=rv=a. Vertical transport uses the analogous shell-area factor r/a2 at layer interfaces.

For the basic finite-volume operators, Fe below denotes a generic edge-normal flux per unit edge length (distinct from the subgrid tendencies above). Its divergence at cell c is 19∇⋅Fc=1Ac∑e∈Ecne,cFele.

For a scalar qc defined at cell centers, the edge-normal gradient is 20∇qe=1de∑c∈Ce-ne,cqc=qc2-qc1de, where c1 and c2 are the two cells in Ce, ordered along the oriented edge normal from c1 to c2. With Ve denoting the edge-normal wind component (ue or ve according to the edge orientation), the relative vorticity at vertex v is obtained from the circulation around the dual control volume: 21ξv=k^⋅∇×Vv=1Av∑e∈Evte,vVede, where k^ is the local vertical unit vector. At the exact poles, GMCORE evaluates the same Stokes circulation over the complete polar cap (Li et al., 2020).

At a full model level, ue and ve denote the zonal and meridional wind components on their respective C-grid edges, we is the vertical velocity horizontally collocated with that edge, φe is the latitude. The semi-discrete horizontal momentum equations are: 22∂ue∂t=Reu-∇ηKeu-η˙∂u∂η-LPGFue-2Ωcosφe+uerewe+S,23∂ve∂t=Rev-∇ηKev-η˙∂v∂η-LPGFve-vewere+S. Here Reu and Rev are the PV-flux rotational tendencies; K is the horizontal kinetic energy and ∇η denotes the horizontal gradient on an η surface; PGFue and PGFve are the pressure-gradient forces; and S represents subgrid tendencies. The nontraditional Coriolis and curvature terms are kept explicitly in Eqs. (22) and (23).

The PV-flux rotational tendencies Reu and Rev in Eqs. (22) and (23) are evaluated as follows. Let δπc be the layer dry-air mass at cell c. The mass is interpolated to vertex v with the primal-dual overlap area weight Av,c. The vertex layer mass and layer-mass potential vorticity are 24δπv=1Av∑c∈CvAv,cδπc,ξp,v=ξv+fvδπv,fv=2Ωsinφv. Thus δπv is the layer mass collocated with the vorticity, fv is the traditional Coriolis parameter, and ξp,v is the layer-mass potential vorticity.

At edge e, δπe is the mean layer mass of the two cells in Ce, Ve is the edge-normal wind, and Fe is the normal dry-mass flux. The transverse flux Fe⟂ at the same target edge is reconstructed as 25Fe=δπeVe,δπe=12∑c∈Ceδπc,Fe⟂=1de∑e′∈E⟂eωe,e′le′Fe′, where e′ runs over the four transverse neighbors in E⟂e, and ωe,e′ is the corresponding geometric interpolation weight and we take ωe,e′=14. This is the Ringler-type normal-to-transverse flux mapping used to collocate the mass flux with the rotational tendency.

GMCORE then reconstructs PV from the vertices to the target edge with its default third-order upwind stencil. Let v-1 and v+1 be the two nearest vertices on the two sides of edge e in the transverse direction and v-2, v+2 the next outward vertices. The reconstructed edge PV is 26ξp,e3=712ξp,v+1+ξp,v-1-112ξp,v+2+ξp,v-2+βpv12sgnFe⟂[ξp,v+2-ξp,v-2-3ξp,v+1-ξp,v-1], where βpv controls the upwind contribution and sgnFe⟂ selects the upstream side from the transverse mass flux. The PV-weighted rotational tendency is then 27Re=σe1de∑e′∈E⟂eωe,e′le′Fe′ξp,e3+ξp,e′32, where σe=±1 accounts for the orientation of the zonal or meridional momentum equation, and ξp,e3 and ξp,e′3 are the third-order reconstructed PV values at the target and neighboring transverse edges. According to the edge orientation, Re in Eq. (27) supplies Reu or Rev in Eqs. (22) and (23).

The kinetic energy is diagnosed at cell centers. Following Ringler et al. (2010) and the kinetic-energy reconstruction used by Gassmann (2013) and Skamarock et al. (2012), we define the edge-associated area Ae=12lede, the edge-based estimate Kce, the optional vertex-based estimate Kcv, and the prescribed blending coefficient αK: 28Kce=1Ac∑e∈EcAc,eVe2,Kv=1Av∑e∈EvAv,eVe2,Kcv=1Ac∑e∈VcAv,cKv,Kc=αKKce+1-αKKcv Here, Ac,e and Av,e are the portions of the edge-associated area assigned to the primal cell c and dual cell v, respectively, and Vc denotes the vertices surrounding cell c. The vertex-based estimate Kcv is obtained by interpolating the vertex kinetic energy Kv to the cell center using the primal–dual overlap areas Av,c. The blending coefficient αK controls the contributions of the two estimates. In GMCORE, we take αK=0.55 to suppress the potential Hollingsworth-type computational instability (Hollingsworth et al., 1983). The kinetic contribution at edge e is obtained with the same normal-gradient operator 29∇Ke=1de∑c∈Ce-ne,cKc=Kc2-Kc1de, where Kc1 and Kc2 are the kinetic energies in the two cells adjacent to e. Because de already contains the local-radius scaling in Eq. (18), the deep-atmosphere metric enters this gradient directly.

Vertical momentum advection follows Simmons and Burridge (1981). At full level k, Me,k+12 and Me,k-12 are the dry-mass fluxes through the lower and upper interfaces, δπe,k is the layer dry-air mass at edge e, and Ve,k denotes either ue,k or ve,k. The operator is 30η˙∂Ve∂ηk=Me,k+12Ve,k+1-Ve,k+Me,k-12Ve,k-Ve,k-12δπe,k.

The interface mass fluxes Me,k+12 and Me,k-12 are diagnosed from the same layer continuity equation, in the deep-atmosphere configuration each interface flux carries its own shell-area factor.

Finally, the horizontal pressure-gradient force is written in perturbation form to reduce cancellation on the terrain-following coordinate. Here pc, ϕc, αd,c=1/ρd,c, and πc retain their definitions from Sect. 2.1; the subscript ref denotes a time-independent hydrostatically balanced reference state and a superscript prime denotes the perturbation from that state: 31pc=pref,c+pc′,ϕc=ϕref,c+ϕc′,αd,c=αd,ref,c+αd,c′,πc=πref,c+πc′.

For any cell-centered quantity qc, the overbar qe‾=qc1+qc2/2 denotes centered interpolation to edge e, while the bracketed gradients use Eq. (20). The symbol δ denotes the vertical increment across a model layer. The pressure-gradient force at an edge is 32PGFe=α‾d,e∇p′e+α‾d,e′∇prefe+∇ϕ′e+δpe′-δπe′δπe∇ϕe. The corresponding term in Eqs. (22) and (23) is -LPGFe.

The model employs a centered difference scheme for layer-mass transport, while a third-order upwind scheme is adopted for potential temperature, geopotential, and vertical velocity. Tracer advection is handled by the SWIFT scheme (Bendall and Kent, 2025), which is derived from the FFSL formulation (Lin and Rood, 1996) with specific modifications to preserve monotonicity in two-dimensional transport and relies on PPM for the spatial reconstruction of tracers.

A challenge for latitude-longitude grids is the severe convergence of meridians at the poles, which typically induces numerical noise and structural deformation in cross-polar flows. To circumvent this singularity, GMCORE redefines the relative vorticity around the poles via Stokes' theorem (Li et al., 2020). This specific physical constraint governs the evolution of potential enstrophy, effectively suppressing artificial computational noise sources precisely at the polar singularities.

After the spatial discretization described in Sect. 2.3, the governing equations can be written in the following prototype form: 33∂X∂t=-LX, where X represents any prognostic variable and LX denotes the corresponding spatially discretized tendency, including the terms described in Sect. 2.3. GMCORE advances Eq. (33) using a modified three-stage Runge–Kutta scheme (Wicker and Skamarock, 2002). Starting from the model state Xn at time level n, the three stages are 34aX*=Xn-Δt3LXn,34bX**=Xn-Δt2LX*,34cXn+1=Xn-ΔtLX**, where Δt is the dynamical time step, X* and X** denote the intermediate states of the first and second Runge–Kutta stages, respectively, and Xn+1 is the updated state. At each stage, the spatial tendency is recomputed using the most recently obtained intermediate state, whereas the stage update is based on Xn. Thus, the three evaluations of the spatial operator are successively performed at Xn, X*, and X**.

Each Runge–Kutta stage can be regarded as a forward-in-time update over the corresponding stage interval. Within such an update, most dycore tendencies are evaluated explicitly, whereas the vertically propagating fast acoustic modes require a different treatment. When using the nonhydrostatic solver, the vertical momentum and geopotential equations (in Eqs. 2 and 6) form a vertically coupled subsystem: w is coupled to the vertical pressure-gradient force, while ϕ evolves directly with w, and the pressure response is linked to them through the equation of state.

Following the vertical implicit treatment used in nonhydrostatic models GRIST (Zhang et al., 2019) and ICON (e.g., Zängl et al., 2015), the prognostic variables are computed directly as full-field quantities without relying on a stationary, global background reference profile.

In this semi-implicit formulation, horizontal advection, vertical advection and the full Coriolis force are evaluated explicitly. Conversely, the terms responsible for high-frequency vertical acoustic wave propagation – specifically the vertical pressure gradient force in the vertical momentum equation, and the associated w⋅gr term in the geopotential equation – are treated implicitly. By introducing an off-centering weight β∈0.5,1.0, the implicit equations for the vertical velocity w and the geopotential ϕ can be conceptually simplified as: 35wτ=Δτ⋅Rw*+βΔτLg0ra2δpτδπτ,36ϕτ=Δτ⋅Rϕ*+βΔτg0ar2wτ, where Rw* and Rϕ* represent the total explicit residual tendencies evaluated at time stage * and Δτ is the sub-timestep. At full levels, the temporal increment of pressure pτ-p* is diagnostically linked to vertical increment of δϕ and the thermodynamic properties through the linearized equation of state: 37pτ-p*p*≈γδπθmτδπθm*-δϕτδϕ*, where γ=cp/cv is the ratio of the specific heats at constant pressure and constant volume, Eq. (37) can be discretized in the vertical: 38δpτ=δp*+γδp*δπθmτδπθm*-γδpnδϕτδϕ*.

By substituting the implicit geopotential relation in Eq. (36) into Eq. (38), and then substituting the resulting pressure increment into the vertical momentum equation, Eq. (35), we eliminate the dependencies on other prognostic variables. This procedure reduces the coupled equations into a one-dimensional vertical equation for the vertical velocity w within each independent grid column. When discretized on the vertically staggered levels, it directly forms a standard tri-diagonal linear system: 39Ak-12wk-12τ+Bk+12wk+12τ+Ck+32wk+32τ=Rw,k+12*, where Rw,k+12* includes all the explicitly resolved forcing terms. The matrix coefficients Ak-12,Bk+12, and Ck+32 are dynamically updated at each time step based on the local atmospheric state.

This one-dimensional tri-diagonal system is solved independently for every vertical column using the highly efficient Thomas algorithm, demanding no horizontal communication. Once the vertical velocity w is obtained, the increments for the remaining implicitly treated variables (such as δϕ and δp) are sequentially back-substituted. This semi-implicit vertical nonhydrostatic solver successfully decouples the fast vertical acoustic wave speeds from the integration stability criteria. This approach allows GMCORE to avoid solving global elliptic equations during nonhydrostatic calculations and to increase the time step.

Due to the convergence of the latitude and longitude grids at the poles, the zonal grid spacing shrinks significantly in the polar regions. As a result, the local zonal CFL number increases sharply in the polar region and can impose a severe stability constraint on the explicit part of the dynamical time integration.

In GMCORE, we apply an adaptive zonal gaussian convolution filter for dynamical tendencies along each latitude circle. For an arbitrary tendency field M in u,v,θm,πs and advection tendency of w,ϕ in nonhydrostatic module, the filtered field Mfilt can be written 40Mi,j,kfilt=∑m=-HjHjGj,mMi+m,j,k, where Gj,m is the gaussian weight at offset m on the jth latitude circle, and Hj is the corresponding half-width of the stencil. Because the zonal direction is periodic, the index i+m is wrapped cyclically around the latitude circle.

The kernel width is not prescribed as a constant but diagnosed adaptively from the local zonal stability constraint. In the implementation, a dimensionless width parameter is first defined as 41Wj=αfcfΔtΔxj, where Δxj is the zonal grid spacing on latitude ϕj, Δt is the dynamical time step, and cf is a characteristic filtering wave speed now set to 300 m s⁻¹. The coefficient αf is the parameter controlling the overall filter strength. The factor cfΔtΔxj has the form of a zonal CFL number, so that the filter automatically becomes stronger where the grid is more restrictive. This latitude-dependent feature provides a smooth transition of the kernel width with latitude and helps avoid abrupt changes in filtering strength.

The diagnosed continuous width Wj is converted to the discrete stencil size by taking the smallest odd integer not smaller than Wj, with a minimum stencil size of three grid points: Nj=max⁡3,oddceilWj.The continuous kernel is represented on the discrete stencil through a gaussian profile. For a stencil of width Nj=2Hj+1, the unnormalized weights are defined by 42G̃j,m=1sj2πexp-m22sj2,m=-Hj,…,Hj, with sj=Wjσf , where σf is a prescribed shape parameter controlling the relation between the diagnosed width Wj and the standard deviation of the discrete gaussian kernel. The final gaussian filter weights are then normalized as 43Gj,m=G̃j,m/∑m=-HjHjG̃j,m,∑m=-HjHjGj,m=1.

The normalization procedure guarantees that the filter preserves constant fields. In this way, the filter automatically strengthens near the poles while remaining weak in lower latitudes. Because the method uses only finite-width local convolution, it avoids the global communication required by FFT-based polar filtering and is therefore well suited to parallel latitude–longitude grid implementations. It should be noted that the wider communication halo is required only for the variables to which the filter is applied. In the parallel implementation, processes covering higher latitudes therefore use wider halos for those variables. In practice with a horizontal resolution of 1°, this half width halo usually reaches 90 points at the two poles of north and south. Nevertheless, the additional communication cost remains manageable for the resolutions considered here.

In addition to the polar gaussian filter, GMCORE applies optional horizontal divergence damping to the momentum field (Lauritzen et al., 2011). This damping is to diffuse the divergent part of flow noise that may otherwise accumulate near steep gradients, in the upper levels, or in the polar region. The damping is formulated in terms of the horizontal divergence, let 44δh=∇h⋅V, where V=u,v is the wind vector. δh is the divergence. The second-order divergence damping is written as 45∂V∂tδ,2=νδ,2∇hδh, and a more scale-selective fourth-order option is 46∂V∂tδ,4=-νδ,4∇h∇h2δh. The damping coefficients νδ,2 and νδ,4 are scaled with the local grid area and time step and can be smoothly enhanced near the poles or upper levels. 47νδ,2∼C2AΔt,νδ,4∼C4A2Δt.

GMCORE also provides an optional upper-level Rayleigh damping of vertical velocity. This damping is not activated in the two idealized cases presented here; with their comparatively low model tops, both cases remain numerically stable without a sponge layer. For future high-top applications, the same damping option can be used as an upper-boundary treatment to attenuate upward-propagating disturbances and reduce artificial reflection, rather than as a correction to the deep-atmosphere equations themselves.

The idealized baroclinic wave test case evaluates the model's ability to simulate mid-latitude baroclinic processes. Following the model parameters proposed by Ullrich et al. (2014), we employ this test to demonstrate the characteristics of the deep-atmosphere configurations. The simulations use the 30-level vertical grid adopted from CAM, with the model top located at 2.255 hPa, corresponding to an approximate geometric height of 30 km. The horizontal resolution is 1°, corresponding to a 360 × 180 regular latitude–longitude grid. The domain is decomposed along the latitude direction for parallel computation. The simulations were performed using 2 compute nodes with 120 CPU cores in total, and the wall-clock time for the integration reported here was approximately 5 min.

The initial conditions feature a mid-latitude jet in hydrostatic and geostrophic balance (Staniforth and White, 2011), which is subsequently perturbed to trigger baroclinic instability. Before the baroclinic simulation, the barotropic steady geostrophic flow (see Williamson et al., 1992, their test Case2) is employed, for which an analytical solution is available to verify the convergence of GMCORE. Another reason for adopting this barotropic test case is to avoid the non-convergence issues that may arise from initialization in a three-dimensional baroclinic model. We then use the unperturbed standard-Earth baroclinic-wave basic state to examine the stability of GMCORE.

Steady geostrophic flow is integrated for 5 d at nominal horizontal resolutions of 2, 1, 0.5, and 0.25°, with time steps of 600, 300, 150, and 75 s, respectively. The fitted convergence orders of the relative geopotential errors are 2.25 for L2 and 2.07 for L∞, indicating near-second-order convergence (in the left panel of Fig. 2). For the three-dimensional model, stability is tested using the unperturbed standard-Earth baroclinic-wave basic state. Following Jablonowski and Williamson (2006), we monitor the weighted norms l2u‾t-u‾0. On the 1° grid, the norm remains below 0.044 m s⁻¹ throughout the 30 d integration, indicating stable preservation of the balanced basic state (in the right panel of Fig. 2).

Simulations with perturbation are conducted using both the deep-atmosphere and the shallow-atmosphere solvers. Figure 3 shows the evolution of surface pressure, 850 hPa temperature, and 850 hPa relative vorticity in the deep configuration at days 8 and 10. The developing wave exhibits the familiar life cycle of baroclinic instability: a midlatitude disturbance grows along the jet, amplifies downstream, and evolves into a mature cyclone–anticyclone structure. Overall, the simulated fields are consistent with the standard behaviour reported in previous model intercomparisons of this test case (Ullrich et al., 2014; Wood et al., 2014; Borchert et al, 2019; Skamarock et al., 2021; Kelly et al., 2026; Yatunin et al., 2026), and indicate that the GMCORE deep-atmosphere dycore is able to maintain the balanced basic state and reproduce the expected evolution of the wave under Earth-like conditions.

With a timestep of 300 s, the gravity-wave CFL number is approximately 0.27 near the equator, 0.39 at midlatitudes, and 0.64 at 60°. It exceeds unity poleward of about 70° because of the reduced zonal grid spacing, while the gaussian filter is adaptively strengthened according to Eq. (41). Vertical acoustic terms are treated semi-implicitly to relax their CFL constraint.

In a standard Earth-radius experiment (a≈6371 km), the troposphere is extremely thin relative to the planet's radius. Consequently, the deep-atmosphere geometrical scaling r/a2 and the variable gravity gr introduce very small dynamical differences compared to the shallow-atmosphere.

To amplify the dynamical differences between the deep- and shallow-atmosphere equations, we follow Ullrich et al. (2014) and employ the reduced-radius, or “small-Earth”, baroclinic-wave experiment, in which the planetary radius is decreased by a factor of 20 while the planetary angular rotation velocity is increased by the same factor. Under the standard Earth-radius experiment, the two experiments produce nearly indistinguishable baroclinic-wave evolutions, whereas in the X20 experiment the shallow-atmosphere assumption is no longer strictly valid and the two solutions separate much more clearly.

In our X20 simulations, the shallow-atmosphere solution retains a comparatively weaker and smoother baroclinic-wave structure, broadly consistent with the reference behaviour reported by Ullrich et al. (2014). In contrast, the deep-atmosphere solution exhibits a more rapid amplification of the wave and a substantially stronger cyclonic development. By day 8, the deep-atmosphere field already shows a more compact and intense low-pressure center together with a more amplified downstream wave; by day 10, the contrast becomes even more pronounced, with the deep-atmosphere solution characterized by a deeper and stronger mature cyclone, while the shallow-atmosphere counterpart remains less developed. This behaviour is consistent with Ullrich et al. (2014), who showed that in the X20 experiment the deep-atmosphere integration intensifies much more rapidly and drives the minimum surface pressure below 900 hPa by day 10, whereas the shallow-atmosphere result remains much closer to the standard baroclinic-wave evolution.

To sum up, the numerical consistency tests and the baroclinic-wave experiments serve complementary purposes. The steady geostrophic-flow test shows near-second-order convergence, while the unperturbed standard-Earth basic state remains close to its balanced state throughout the 30 d integration. For the perturbed baroclinic wave, the deep- and shallow-atmosphere solutions are nearly indistinguishable under the standard-Earth-radius setup but separate clearly in the X20 reduced-radius experiment, consistent with published deep-atmosphere benchmarks. Because part of the X20 contrast may originate from differences in the analytic initial states, we regard this experiment primarily as a benchmark-reproduction test rather than as a quantitative attribution of the deep-atmosphere terms. Together, these results provide complementary evidence for the numerical consistency and correctness of the GMCORE deep-atmosphere implementation.

Next, we try to use an idealized tropical cyclone (TC) test case, which was proposed by Reed and Jablonowski (2011). In this case, an initial warm-core vortex that satisfies gradient wind balance is inserted in an aquaplanet without topography. This eliminates the land surfaces and the real environmental flow effects, making it an ideal test-bed for isolating and evaluating the deep-atmosphere modification. More precisely, due to the low model top (about 30 km) of this case, geometric and variable-gravity effects are expected to be much smaller than the retained nontraditional Coriolis contribution. This case is therefore used mainly to examine the impact of the NCTs-related deep-atmosphere terms.

We follow the standard setup of Reed and Jablonowski (2011) and run this case at a horizontal resolution of 0.25 and 0.125° for 10 d with the same vertical grid as the baroclinic-wave. The 0.25° simulations were performed using 480 CPU cores with a walltime around 120 min, whereas the 0.125° simulations used 600 CPU cores with a walltime around 900 min. To isolate the dynamical impact of the deep-atmosphere terms, the experiment is executed using both the nonhydrostatic deep-atmosphere (NHD) and nonhydrostatic shallow-atmosphere (NHS) configurations.

During the dynamical evolution of a tropical cyclone, the full three-dimensional Coriolis force – specifically the NCTs – is conventionally neglected in traditional shallow-atmosphere models. However, in the deep tropics where cosφ approaches its maximum, and within the TC eyewall region characterized by exceptionally strong vertical updrafts (with w frequently exceeding 10 m s⁻¹), the horizontal and vertical momentum forcings generated by the NCTs become non-negligible in magnitude. Although previous idealized cloud-resolving studies (e.g., Igel and Biello, 2020) have highlighted the potential impact of NCTs on equatorial convection, the macroscopic dynamical effects of this mechanism within a global deep-atmosphere dycore during a complete TC life cycle remain largely unverified.

As shown in Fig. 5a, both NHS and NHD simulations produce a north-westward-moving tropical cyclone, but the NHD trajectory exhibits a systematic south-westward displacement relative to the NHS trajectory as the vortex develops. This result is qualitatively similar to that reported by Liang and Chan (2005), who found that inclusion of the 2Ωcosφw term had little effect on tropical-cyclone intensity but led to a westward to south-westward trajectory deviation. We also test different perturbations at different latitudes and get the same conclusion (Fig. 5b).

In our results, both simulations remain similar in terms of storm intensity and axisymmetric structure. As already indicated in the Supplementary Material, the two simulations show no essential difference in the minimum central pressure, maximum low-level wind speed, or the warm-core structure, suggesting that the leading-order thermodynamic processes controlling storm intensification are not strongly altered by the deep-atmosphere extension.

To interpret this trajectory difference, Fig. 6 shows the time-mean horizontal structure at 5 km altitude. In Fig. 6a, the mean vertical velocity field exhibits a compact core of strong ascent near the storm center. This indicates that the mature vortex maintains a well-defined eyewall-like updraft region. Closely collocated with this convective core, in Fig. 6b, we show the quantity 48Γ=2Ωcosφ∂w∂y, which represents the vorticity tendency associated with the NCTs in the x-momentum equation. Here, the dipole-like structure refers to the pair of opposite-signed Γ anomalies on the two sides of the storm center. In other words, the meridional gradient of the eyewall vertical motion produces a positive tendency on one side of the core and a negative tendency on the other, giving rise to an organized inner-core asymmetry rather than an axisymmetric distribution.

This dipole is dynamically consistent with the near-core easterly anomaly shown in Fig. 7, which in turn is consistent with the south-westward displacement of the NHD trajectory. Figure 7 shows the horizontal winds at 1.5 km. The left panel gives the circulation in the NHD run, and the right panel shows the wind difference between the deep and shallow simulations. The main feature in the difference field is an easterly anomaly near the storm center. This anomaly is important because it is consistent with the south-westward displacement of the NHD trajectory in Fig. 5. In the Liang and Chan (2005) experiments, the additional Coriolis term associated with vertical motion produced a generally easterly flow near the TC center, and this easterly flow was identified as the immediate cause of the westward or south-westward displacement. Our result is not identical to their diagnosis, but the low-level easterly anomaly in Fig. 7 points in the same direction. It suggests that the south-westward shift in the NHD trajectory is tied to a coherent change in the inner-core flow rather than to random differences accumulated during the integration.

At the same time, the present analysis remains more limited than that in Liang and Chan (2005). We have not yet constructed a full asymmetric wind composite as shown in their Fig. 7, so the phase relationship among eyewall ascent, vorticity tendency, and asymmetric gyres is not shown explicitly here. For this reason, the mechanism discussed above should be taken as an interpretation of the present results rather than as a complete dynamical proof. The result from this test is simpler: the deep-atmosphere GMCORE gives a TC evolution that is similar to the shallow-atmosphere run in intensity, but different in trajectory, and this trajectory difference is accompanied by an organized inner-core asymmetry and a low-level easterly anomaly in the deep-minus-shallow wind field.

The systematic trajectory response found here can also be viewed in the broader context of tropical-cyclone asymmetric vortex dynamics. In the present simulations, retaining the NCTs introduces an organized inner-core forcing while leaving the axisymmetric intensity comparatively weakly affected. For an approximately axisymmetric eyewall updraft w0r, the Cartesian meridional derivative can be expressed in storm-centered polar coordinates as 49∂∂y=sinθ∂∂r+cosθr∂∂θ, where θ is the azimuthal angle. Since w0 is axisymmetric, ∂w0∂θ=0, and therefore 50∂w0∂y=dw0drsinθ. Hence, combining Eqs. (48) and (50), we get Γ=2Ωcosφdw0drsinθ, which projects directly onto an azimuthal-wavenumber-one vorticity tendency. This connection is potentially important because wavenumber-one disturbances have long been associated with displacement and tilt modes of tropical-cyclone-like vortices, whose evolution can involve vortex-Rossby-wave adjustment, precession, alignment, and trochoidal vortex-center motion (Polvani, 1991; Reasor and Montgomery, 2001; Nolan et al., 2001; Schecter et al., 2002). More recent analyses of strongly tilted vortices further show that dipolar vertical-motion asymmetries can modify vortex-centerline evolution and steering depending on their structure relative to the vortex tilt (Päschke et al., 2012; Dörffel et al., 2021). The NCT-related dipole and accompanying near-core easterly anomaly identified here therefore suggest that the retained 2Ωcosφ terms preferentially modify the low-wavenumber asymmetric circulation that is dynamically relevant to vortex motion. Determining how this forcing is partitioned among vortex-Rossby-wave adjustment, trochoidal displacement, and three-dimensional tilt or centerline precession provides an interesting direction for further investigation.

In summary, the NHD and NHS simulations remain similar in storm intensity and warm-core structure but exhibit a systematic difference in trajectory, with the NHD vortex displaced south-westward across the paired experiments. The NCT-related vorticity-tendency dipole projects naturally onto a wavenumber-one asymmetry and is accompanied by a near-core low-level easterly anomaly, providing a dynamically consistent interpretation of the trajectory response. However, the present diagnostics do not establish a closed causal chain between the NCT forcing, the asymmetric circulation, and vortex translation. More complete analyses of the angular-momentum, potential-vorticity, and asymmetric-flow budgets will be required to quantify this mechanism.