| 19 Feb 2026
Development and improvement of a nonhydrostatic spectral model using non-constant coefficient semi-implicit and vertically conservative semi-Lagrangian schemes
Abstract. A two-dimensional x–z nonhydrostatic spectral model using non-constant coefficient semi-implicit and vertically conservative semi-Lagrangian schemes has been developed, which is computationally efficient by allowing for long timesteps in simulations. The model incorporates several improvements related to free-slip surface boundary conditions, prognostic variables in spectral space, and the semi-Lagrangian and semi-implicit schemes. These improvements enhance the numerical stability, especially in cases with steep orography, and improve model results. The model was tested with various test cases (e.g., mountain wave test cases), confirming that a single recalculation of the non-constant coefficient linear terms is usually sufficient to stably solve the equations associated with the non-constant coefficient semi-implicit scheme, even in the cases with a steep mountain with an average slope of 45°. The model also ran stably even in the case of an extremely steep mountain with an average slope of 63.4° by performing several iterations using the preconditioned general conjugate residual method. In all these cases, good results were obtained with long timesteps.
This paper describes several improvements to the classical
SISL method for a nonhydrostatic dycore. These include a correct
treatment of the free-slip boundary condition at the surface
and use a non-constant linear operator in the semi-implicit solver.
The results improve stability of the scheme especially in
the presence of steep topography, a known issue for SISL methods.
The improvements are well documented and demonstrated in challenging
test cases.
minor comments:
1. The results are presented in a x-z model, with Fourier series
used in the x direction. From the introduction, I concluded that
this x-z model is used for evaluation, and serves as a simplified
version of a 3D spherical harmonic based model, which is confirmed
in the paper's conclusions. This could be mentioned
explicitly in the introduction.
2. Section 2.2: Why T and Q (or ln T and P) instead of potential temperature and
PHI as prognostic variables? For a mass coordinate models,
the use of potential temperature and PHI is common in finite volume / finite
element methods, since it avoids the need compute the D3
operator. (i.e. Dubos & Tort MWR 2014, Taylor et al. JAMES 2019)
The T/Q approach used here seems common in spectral methods.
I wonder if the authors can remind the reader what these reasons are
and comment on if they are still important at nonhydrostatic resolutions.
3. Line 128: The derivation here shows that the free-slip boundary
condition gives simultaneous conditions for the surface vertical
velocity and a Neumann like pressure boundary condition. The authors
note that this formulation gives improved results over an
extrapolation (Eq. 21) used in the MRI model. It appears to this
reviewer that the MRI model is using an approximation that would
result in a boundary condition slightly inconsistent with the
free-slip condition. I think the authors should state this.
4. Line 144: The authors mention the nonconstant terms come from
Eq. 7b and Eq. 13. Given that there are choices of prognostic
variables which can eliminate Eq. 13, is it possible to also
simplify 7b through choice of prognostic variables in order to
obtain only constant coefficient linear terms?
5. Section 2.4, general comment: The authors seem to carefully
consider many choices of prognostic variables and related formulation
questions, and often state that choices are made because of observed
improvements in stability. Sometimes, a mathematical reason is given
to explain the improvements. In other cases, is it simple trial and
error (trial with test cases) or can mathematical principals behind
the choices be found in all cases?
6. Line 249: "The quadratic truncation is used..."
Is this equivalent to the fully dealiased grid used in spherical harmonic
models (i.e. quadratic terms will be computed with no aliasing error)
7. Line 250: "... diffusion is not required for numerical stability,
and is not used, except that specified diffusion is used in the
density current test case in Sect. 3.5."
Is diffusion not used because of the improved stability properties
of the formulation? Or is it that the test cases dont require
diffusion (except for the density current test case). When this formulation
is used in a 3D spherical harmonic model, will (hyper)diffusion be used?