Stability Properties of the Constant Coefficients Semi-Implicit Time Schemes Solving Nonfiltering Approximation of the Fully Compressible Equations

Petra Smolíková; Jozef Vivoda

doi:10.1175/MWR-D-22-0125.1

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Fig. 1.

Frequencies of normal modes as functions of horizontal wavenumber for (a) ν = 1, χ = α = β = δ = ϵ = 1, and several values of γ; (b) ν = 1, χ = α = γ = 1, $δ = χ / β$ , ϵ = β, and several values of β; (c) ν = 1, χ = α = δ, β = ϵ = γ = 1, and several values of χ; (d) ν = 10, χ = α = δ, β = ϵ = γ = 1, and several values of χ, used in the blended equation set. The dashed line represents the fully compressible solution, while the dotted line is the hydrostatic one.

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Fig. 2.

The amplification factor Γ for the EXTR scheme and the generalized linear model with $f_{α β ϵ} [h^{*}, 1]$ with (a) $h^{*} = 0.5$ , (b) $h^{*} = 1$ (corresponds to EE), (c) $h^{*} = 2$ , and (d) the HPE system. We set $γ^{*} = 1$ , $T^{*} = 300 K$ , ν = 1, and Δt = 10 s in all cases. No isolines are depicted for values bigger than 1.25, and the same orange color is used.

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Fig. 3.

The amplification factor Γ for a particular choice of $k = (π / 200) m^{- 1}, T^{*} = 300 K$ , ν = 1, and Δt = 10 s for the EXTR scheme with $f_{α β ϵ} [h^{*}, 1]$ . The distinct curves show different values of ϑ.

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Fig. 4.

As in Fig. 3, but for the blended equation set with f_αδ[h, h] and $k = (π / 1000) m^{- 1}$ .

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Fig. 5.

Vertical velocity at time 6000 s for the nonlinear mountain wave. The contour interval is 0.2 m s⁻¹, the time step is 2 s, and $γ^{*} = 3$ . The results are shown for (a) the reference: PC scheme, fully nonhydrostatic; (c) EXTR using the generalized linear model f_αβϵ[5, 1]; (e) the hydrostatic run with the EXTR scheme; and EXTR using the blended equation set f_αδ[h, h] with (b) h = 0.1, (d) h = 0.01, and (f) h = 0.001. For (b)–(f) the difference with the reference solution in (a) is shown in colors.

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Fig. 6.

The potential temperature field at time 300, 600, and 900 s of the Straka test, the contour interval is 1 K, the time step is 1 s, and $γ^{*} = 3$ . (a) The reference: PC scheme, fully nonhydrostatic; (b) EXTR using the generalized linear model f_αβϵ[5, 1]; (c)–(e) EXTR using the blended equation set f_αδ[h, h] with h = 0.1 in (c), h = 0.01 in (d), and h = 0.001 in (e); and (f) hydrostatic EXTR run. For (b)–(d) the difference with the reference experiment in (a) is depicted in colors.

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Fig. 7.

An illustration of results obtained for the case of lee waves generated in the flow behind the mountains of Krušné hory, at 1100 UTC 12 Feb 2019: (a) cloud cover observed by geostationary satellite Meteosat (Vis-IR channel) and (b) orography (m) of the experimental domain with the area of depicted cloudiness denoted by the blue rectangle and the cross-section line for Fig. 8 denoted by the blue line. (c)–(f) Low- and midlevel cloud cover fractions obtained by model simulations are shown with grayscale from white (overcast) to black (clear sky) by a regular step of one okta. High-level cloud cover fractions obtained by model simulations are shown by shades of blue again by a regular step of one okta. The model simulations start at 0000 UTC 12 Feb 2019, using the PC scheme, fully nonhydrostatic in (c); the EXTR scheme, the generalized linear model with f_αδ[2, 1] in (d); the EXTR scheme, the blended equation set with f_αδ[0.4, 0.4] in (e); and EXTR with the hydrostatic assumption in (f).

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Fig. 8.

Vertical velocity (m s⁻¹) in the cross section over the Krušné hory for several runs from 0000 UTC 19 Feb 2019 integrated for 11 h: (a) PC scheme, fully nonhydrostatic; (b) EXTR scheme, f_αδ[2, 1]; (c) EXTR scheme, f_αδ[0.4, 0.4]; and (d) EXTR with the hydrostatic assumption. The cross-section line is depicted in Fig. 7b with a blue line. The contour interval is 0.5 K. For (b)–(d) the difference with the reference experiment in (a) is depicted in colors.

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Fig. 9.

Stability measured as the completed integration from 0000 UTC 19 Feb 2019 for 24 h: white means stable for $γ^{*} = 5$ and $γ^{*} = 3.5$ , and yellow means stable only for $γ^{*} = 5$ . Orange means unstable (integration crashed for both values of $γ^{*}$ ). The time scheme used is EXTR with the time step of 90 s. The control parameters setting is $f_{α δ} [h^{*}, h]$ . The gray color highlights the pure EE solution that shows instability in this case.

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Fig. 10.

The amplification factor Γ for $f_{α δ} [h^{*}, h]$ , $γ^{*} = 1$ , and the particular choice of k = 0.0027 m⁻¹, ν = 1, Δt = 45 s, and ϑ = −0.3. It follows that λ = 2.325 km = Δx. No isolines are depicted for values bigger than 1.14, and the same orange color is used.

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Fig. 11.

Orography (m) of the Occitania experiment with the forecasted wind direction valid for 1200 UTC 3 Oct 2015 at 700 hPa. The blue line denotes the cross section for results presented in Fig. 12.

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Fig. 12.

Vertical velocity (m s⁻¹) in the cross section over the Massif Central for several runs from 0000 UTC 3 Oct 2015 integrated for 12 h, Δt = 10 s, and $γ^{*} = 8.5$ : (a) PC scheme, full nonhydrostatic; (b) EXTR scheme, f_αδ[0.9, 0.9]; (c) EXTR scheme, f_αδ[0.1, 0.1]; and (d) EXTR with hydrostatic assumption. The cross-section line is depicted in Fig. 11 with a blue line. The contour interval is 1 K. For (b)–(d) the difference with the reference experiment in (a) is depicted in colors.

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Editorial Type:: Article

Article Type:: Research Article

Stability Properties of the Constant Coefficients Semi-Implicit Time Schemes Solving Nonfiltering Approximation of the Fully Compressible Equations

Petra Smolíková

Petra Smolíková ^aCzech Hydrometeorological Institute, Prague, Czech Republic

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Jozef Vivoda

Jozef Vivoda ^bSlovak Hydrometeorological Institute, Bratislava, Slovakia
^cFaculty of Mathematics, Physics, and Informatics, Comenius University, Bratislava, Slovakia

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Online Publication:: 06 Jul 2023

Print Publication:: 01 Jul 2023

DOI:: https://doi.org/10.1175/MWR-D-22-0125.1

Page(s):: 1797–1819

Received:: 05 May 2022

Final Form:: 24 Mar 2023

Accepted:: 24 Mar 2023

Published Online:: 06 Jul 2023

Displayed acceptance dates for articles published prior to 2023 are approximate to within a week. If needed, exact acceptance dates can be obtained by emailing amsjol@ametsoc.org.

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Abstract

A set of control parameters is introduced in the fully elastic nonhydrostatic Euler equations formulated in the mass-based vertical coordinate of Laprise. Contrary to the classical approach, the hydrostatic limit is represented by a subspace of control parameters, instead of a single point. By finding a suitable path from the fully compressible equations to the hydrostatic subspace, we are able to construct a blended system with acoustic modes slowed down and gravity modes nearly unaffected. Numerical stability of the discretized system is thus improved, and the solution remains essentially the fully compressible one. Alternatively, control parameters can be used to redefine the linear model of the constant coefficients semi-implicit time scheme, increasing the numerical stability of the fully compressible system. With a careful choice of the control parameters in both, the linear model used in the semi-implicit temporal scheme, and in the full model, the blended system does not deteriorate the compressible solution while its semi-implicit temporal discretization is more stable. We illustrate the potential of the method in several simple examples and in real case studies using the ALADIN system.

Vivoda’s current affiliation: European Centre for Medium-Range Weather Forecasts, Bonn, Germany

© 2023 American Meteorological Society. This published article is licensed under the terms of the default AMS reuse license. For information regarding reuse of this content and general copyright information, consult the AMS Copyright Policy (www.ametsoc.org/PUBSReuseLicenses).

Corresponding author: Petra Smolíková, petra.smolikova@chmi.cz

Abstract

Vivoda’s current affiliation: European Centre for Medium-Range Weather Forecasts, Bonn, Germany

Corresponding author: Petra Smolíková, petra.smolikova@chmi.cz

Keywords: Dynamics; Nonhydrostatic models; Numerical analysis/modeling

1. Introduction

Applications of numerical weather prediction (NWP) systems vary in a wide range from global models run on planetary scales to local simulations of one convective system or one valley flow. Separate systems of equations were offered in the past for distinct purposes focusing on the given scale. Thus, the hydrostatic assumption may be applied if the horizontal to vertical scale ratio is big enough. Compressibility of the flow may be suppressed for mesoscale phenomena to eliminate fast-moving waves, as vertically propagating acoustic modes, from the solution. Separate filtered or unfiltered equation sets usually have separate numerical solutions, and the comparison of systems with different sets of basic equations may be difficult because of the impossibility of isolating the effect of the chosen system from the effect of the chosen numerical methods. Thus, for the purpose of comparative studies and for the sake of unified applications across different scales it may be beneficial to establish several equation sets in one common numerical framework. Then the theoretical analysis of the problem as well as the numerical solution may be unified. Moreover, the maintenance of the whole system is in this way significantly simplified.

Such attempts are numerous, starting from Davies et al. (2003) who introduced a switchable form of the full equation set using six control parameters. He presents a concise analysis of the hydrostatic, the pseudoincompressible (Durran 2003), the anelastic (Wilhelmson and Ogura 1972; Lipps and Hemler 1982), and the Boussinesq approximations and compares their normal modes with those of the full set of compressible Euler equations. He concludes that for the representation of Rossby modes, essential for multiscale applications, only the hydrostatic and the pseudoincompressible-filtered equation sets are suitable. Moreover, the anelastic equation sets are relevant in small-scale dynamics only.

Since then, there have been several attempts to introduce control parameters to create a framework of an all-scale blended multimodel solver. The blended soundproof to compressible system was designed through two control parameters in Benacchio et al. (2014), where one control parameter allows for a continuous inclusion of the effect of pressure perturbations on density while the second parameter allows for a thermodynamically consistent or inconsistent approach. Klein and Benacchio (2016) state that for an intermediate value of total energy between those of the fully compressible and pseudoincompressible models, energy conservation is ensured.

Similarly important is the ability of unified systems to show no degradation of solution quality when the respective regime, for example hydrostatic, is being approached. A review of such methods is given in Benacchio and Klein (2019). We may mention a doubly blended model for multiscale dynamics of Klein and Benacchio (2016) where two control parameters are introduced, enabling pseudoincompressible, hydrostatic primitive equations and unified Arakawa and Konor (2009) and fully compressible models in one unified framework. Reduced soundproof dynamics may be accessed by switching two parameters between values 0 and 1. Moreover, intermediate values of the parameters connected to compressibility of the atmosphere allow for a smooth transition from a balanced model to the fully compressible one over several model steps. Undesirable unbalanced modes may be filtered out in this way during initialization and in data assimilation. This idea is further developed in Chew et al. (2022) where the blended modeling strategy is extended with a Bayesian local ensemble data assimilation method. The pseudoincompressible regime is used solely for one filtering time step in each subsequent assimilation window. Such a solution appears sufficient to suppress imbalances coming from the initialization and data assimilation. Their model is cast in density, mass-weighted potential temperature, Exner pressure, and horizontal and vertical velocity components in the height based vertical coordinate. The unified equations of Arakawa and Konor (2009) are formulated in a mass-based sigma coordinate in Voitus et al. (2019). This equation set captures the nonhydrostatic small scale effects and retains the hydrostatic compressibility of the flow at large scales, while the vertically propagating acoustic waves are filtered out. This represents an advantage over the classical quasi-hydrostatic system where significant distortion in the dispersion of gravity modes is observed at small scales.

Here, we adopt an alternative strategy to filtering. We first formulate the nonhydrostatic fully compressible Euler equation system as a departure from the hydrostatic primitive equations (HPE) using the Laprise (1992) mass-based coordinate. Then we identify all terms that appear in the unapproximated system on top of the HPE and introduce one control parameter for each of these terms. All together, we have five control parameters in our blended system. We show that there is a constraint, called the unifying constraint, for these five parameters that allows the structure of normal modes periodic in time, without damping or amplifying its amplitude. Hence, one degree of freedom is suppressed and the full control space is restricted to four dimensions. The hydrostatic approximation may be achieved in this space through multiple choices of control parameters. Thus, there is an HPE subspace where the equation system remains hydrostatic. We establish a path through the control space from the point representing the Euler equations to the HPE subspace where the stability of the solution in the normal mode decomposition is guaranteed. We evaluate the proposed configurations of control parameters in some of the well-known test cases and in real simulations using the ALADIN system.

We show that in the case where the control parameters are nonzero and they satisfy the unifying constraint, the blended system contains all vertical normal modes present in the fully compressible nonhydrostatic solution, and no filtering is achieved. However, the frequency of the acoustic modes is modified by the factor depending on the control parameters used and on the horizontal and vertical wavenumbers, and it may be decreased with their suitable choice. On the other hand, the dispersion of gravity modes is practically unaffected for all horizontal wavenumbers with the exception of the first few vertical normal modes, again in dependence on the control parameters’ choice. Moreover, based on Davies et al. (2003), the Rossby modes are kept unchanged as well, ensuring the usefulness of the system in multiscale applications. This is the main distinction from other approximate systems proposed in literature and mentioned above. We do not filter out acoustic modes, but we slow down their propagation in order to gain numerical stability.

As another stabilizing option, we suggest using the blended system solely in the formulation of the linear model treated implicitly in the semi-implicit time-marching scheme; the set of Euler equations is kept for the nonlinear residual, treated explicitly or within an iterative method. Hence, only the semi-implicit discretization is modified, while the system being solved involves unapproximated Euler equations. This approach generalizes the method described in Bénard (2004), where the system is first linearized around an isothermal, stationary, horizontally homogeneous and hydrostatically balanced reference state, and then two values of the reference atmospheric temperature are used in different terms of the obtained linear model. This corresponds to the usage of one control parameter for modification of the relevant term in the linear model. We refer to this approach as to the generalized linear model.

It shows up that in the case of the generalized linear model one may choose the values of control parameters from a broader interval, not restricting the choice to intermediate values in [0, 1]. Large values of some control parameters may improve the stability of the time marching scheme while keeping high accuracy of the solution. The newly proposed generalized linear model is used for the separation of the linear part and the nonlinear residual as usual, and a standard Helmholtz solver is applied. The method is again tested in the standard set of test cases and in real simulations with the ALADIN system. Finally, the combination of both approaches (a blended system with a modified linear operator) is possible, yielding the best results in terms of stability and accuracy.

The paper is structured as follows. Section 2 contains a brief description of the ALADIN system and of the time schemes available within, in particular. We describe the equation system with control parameters in section 3, the time continuous linear system used for the definition of the time-marching procedure in section 4, and some details of the spatial discretization in section 5. The stability analysis is given in section 6. We confirm the results of the stability analysis on two idealized cases in section 7 and provide two real case studies in section 8. Section 9 is dedicated to final remarks and conclusions.

2. Temporal schemes

The ALADIN system (ALADIN International Team 1997; Termonia et al. 2018) is a limited-area NWP system developed by the international ACCORD consortium, which is used in many operational applications across Europe and North Africa for weather forecasting. It uses in its dynamical core either the set of HPE or the set of nonhydrostatic fully compressible Euler equations (EE). The HPE system is discretized in time using two-time-level constant coefficients semi-implicit (SI) technique with a spectral solver described in Robert et al. (1972). Further, the semi-Lagrangian advection treatment is used with the formulation of Hortal (2002). This combination of methods allows for relatively large time steps. However, the time-stepping methods designed for the HPE system are not sufficiently robust when applied to the EE system.

Cullen (2001) showed that two successive time steps (predictor and corrector) integrated using the current SI procedure could improve robustness of the time-integration scheme of the HPE system at high horizontal resolutions. Bénard (2003) proposed a generalization of this method for any number of successive iterations, called the iterative centered implicit (ICI) scheme, suitable for application on the EE system. A special case of ICI with one initial step (predictor) and one iterative step (corrector) only is referred to as the PC scheme. When the PC scheme is applied to the EE system a stable and efficient integration is usually reached for kilometric horizontal resolutions. The CPU cost is increased in this case by a factor of 1.3 (Bénard et al. 2010) with respect to the simple SI scheme. However, when going to hectometric horizontal scales the time-stepping stability of the EE system integration requires more than one iteration and the ICI scheme becomes expensive.

This motivated the work of Bénard (2004) who deduces that the linearization of the EE system around a reference state is not a unique way to define the linear system of semi-implicit schemes. When choosing the reference linear system more generally the robustness and stability properties of the time stepping method may be improved for both the simple SI procedure and the ICI scheme. Bénard introduced one control parameter into the linear system, and he rescaled in this way the term in the vertical momentum equation responsible for the vertical propagation of elastic waves. In this paper we evolve his method further by introducing a set of control parameters in the EE system. If the value 0 is set for all the control parameters, one gets the HPE system, while setting the value 1 for all the control parameters gives the full EE system. The linear system is derived from the EE system with control parameters using the traditional linearization technique as follows. For a state X of the atmosphere given as the vector of prognostic variables, the time evolution of the system

M

can be written as

\frac{\partial X}{\partial t} = M (X) .

(1)

Then the semi-implicit technique with spectral solver requires the linearization of

M

around a temporally and horizontally constant reference state

X^{*}

. The linearized system can be written as

M (X) \approx M (X^{*}) + L^{*} (X^{*}) \cdot (X - X^{*}),

(2)

where

L^{*} (X^{*})

is the linear operator [the Jacobian of

M (X)

evaluated at

X^{*}

]. We use in the following the notation

L^{*}

instead of

L^{*} (X^{*})

for simplicity. The properties of

X^{*}

yield

M (X^{*}) = L^{*} \cdot X^{*} = 0

and finally the linearization of

M

is given by

M (X) \approx L^{*} \cdot X .

(3)

The original system

M

is then divided into the linear part and the nonlinear residual:

\frac{\partial X}{\partial t} = L^{*} {\bar{[X]}}^{t} + [M (X) - L^{*} \cdot X],

(4)

where

{\bar{[]}}^{t}

is the implicit-centered temporal average operator. We denote X⁻ = X(t − Δt), X⁰ = X(t),

X^{+ 1 / 2} = X [t + (Δ t / 2)]

and X⁺ = X(t + Δt). One may time discretize (4) using two time levels through

\begin{array}{l} \frac{X^{+} - X^{0}}{Δ t} & = L^{*} (\frac{X^{+} + X^{0}}{2}) + (M - L^{*}) (X^{+ 1 / 2}) \\ = L^{*} (\frac{X^{+} - 2 X^{+ 1 / 2} + X^{0}}{2}) + M (X^{+ 1 / 2}) \end{array} .

(5)

Here

X^{+ 1 / 2}

is unavailable and it must be estimated using previous model states. We use either the first-order-in-time estimate

X^{+ 1 / 2} = X^{0}

and the resulting time discretization:

\frac{X^{+} - X^{0}}{Δ t} = L^{*} (\frac{X^{+} - X^{0}}{2}) + M (X^{0})

(6)

is referred to as the nonextrapolating scheme (NESC), or we use the second-order-in-time estimate

X^{+ 1 / 2} = (3 X^{0} - X^{-}) / 2

giving the time discretization:

\begin{array}{l} \frac{X^{+} - X^{0}}{Δ t} = L^{*} (\frac{X^{+} - 2 X^{0} + X^{-}}{2}) + M (\frac{3 X^{0} - X^{-}}{2}) \end{array},

(7)

which is referred to as the extrapolating scheme (EXTR).

In (Hortal 2002) the second-order spatiotemporal averaging along semi-Lagrangian trajectories is made, resulting in the so-called SETTLS scheme. We omit the precise position of the given state vectors on the semi-Lagrangian trajectory in all model descriptions and in the stability analysis given in the present paper. In such a context EXTR is equivalent to the SETTLS scheme and results obtained here are valid also for the SETTLS scheme. In idealized simulations and real case studies where the ALADIN system is used the SETTLS scheme (with averaging along the semi-Lagrangian trajectories) is applied instead of EXTR.

For the simple SI scheme the NESC approach is not appropriate due to its low accuracy. On the other hand, it may serve as the first guess of the ICI scheme since in the successive iterations the second-order-in-time accuracy is restored. The ICI scheme consists of the predictor step given by (6), with the resulting X⁺ denoted as X⁺⁽⁰⁾, followed by corrector steps for n = 1, 2, …, N_iter according to

\begin{array}{l} \frac{X^{+ (n)} - X^{0}}{Δ t} = L^{*} (\frac{X^{+ (n)} - X^{+ (n - 1)}}{2}) + M (\frac{X^{+ (n - 1)} + X^{0}}{2}) \end{array} .

(8)

Then the final state is taken as the last iterate

X^{+ (N_{iter})}

. Here again the scheme is called the PC scheme if N_iter = 1. In Bénard (2003), the ICI scheme with the NESC predictor computation was proven to show much better stability properties than the simple noniterative SI scheme of any kind and that is why it is successfully used in the most operational applications of the ALADIN system. As the inconvenience of this approach, one may see the overall computational cost. We aim in the present paper to study the stability properties of the SI scheme using either NESC or EXTR time discretization and to show that the careful choice of the control parameters may possibly prevent the usage of the more expensive PC scheme, or generally ICI scheme. The EXTR approach may be considered as a good candidate for a cheap, stable and accurate time scheme in real applications. Nevertheless, for high horizontal resolutions (hectometric scales) one may not prevent the usage of the ICI scheme. However, if the stability of the time procedure is enhanced, the number of needed corrector steps may be decreased leading to the CPU time saving.

We proceed in two distinct ways:

1) The control parameters are introduced only in the linear operator used in the time discretization scheme. We keep the EE system unapproximated by setting the value 1 to all the control parameters in the full model. The method may be seen as a preconditioning of the semi-implicit solver and the equation system used remains the full EE system. This approach generalizes the method of Bénard (2004).
2) We change not only the linear but also the nonlinear part of the time stepping calculation by setting values different from 1 to some of the control parameters. The equation set being subject of the integration is no longer the full EE system. Moreover, if the control parameters in the full model get values from (0, 1) then the obtained set of equations can be seen as a transition from the HPE to the EE system. We show that with a careful choice of control parameters, the significant features of the full system are preserved.

These two approaches provide two distinct and very different results. Let us remark that the second approach is possible only if the discretization methods used for the time integration of the evolution system are unified for the HPE and the EE systems. This is the case of the ALADIN code as described in section 5.

In both cases we examine the stability of the applied time discretization schemes for different values of control parameters together with the quality of the obtained solutions. We follow up Bénard (2003), where a general method to carry out space-continuous stability analysis of various time-discretization schemes was presented, based on the method described in Simmons et al. (1978).

3. Euler equations with control parameters

The vertical coordinate used in the ALADIN system is the general stretched hybrid-pressure terrain-following coordinate η of Laprise (1992). However, for the theoretical analysis presented in this paper the pure mass-based terrain-following coordinate σ was chosen for the sake of simplicity, defined through $σ = π / π_{s}$ , where π is the hydrostatic pressure and π_s its surface value. It is a particular case of η obtained through setting A(η) = 0 and B(η) = η ≡ σ in Eq. (9) of Bénard et al. (2010).

The prognostic variables used are the following: temperature T, horizontal wind component V, vertical velocity w, pressure departure

\hat{q} = \ln (p / π)

, and the surface pressure variable q_s = ln(π_s). Here, p denotes true pressure. We omit the Coriolis force here as well as other external forces and heat sources. We consider a dry, vertically unbounded atmosphere. It means that at the top the pressure vanishes with p = 0. This strategy is commonly used in mass-based atmospheric models. On the bottom boundary of the domain, the vertical movement of an air parcel is only governed by the flow along the rigid surface, the fact expressed with gw_s = V_s ⋅ ∇ϕ_s. Here g = 9.8061 m s⁻² stands for constant gravity acceleration, ϕ is geopotential, and the subscript s refers to the surface values. For horizontal components of the wind, we have free slip boundary condition given by V_s = V_L where L refers to the lowest model level. We introduce a set of control parameters α, β, γ, δ, and ϵ. Then the full system of Euler equations using T, V, w,

\hat{q}

, and q_s cast in the vertical coordinate σ is written as

\frac{d T}{d t} = κ T \frac{\dot{π}}{π} - α κ T [\frac{\dot{π}}{π} + \frac{1}{1 - κ} (D + \hat{d})],

(9a)

\frac{d V}{d t} = - R T \nabla q_{s} - \nabla ϕ - β [R T \nabla \hat{q} + (\frac{1}{π_{s}} \frac{\partial p}{\partial σ} - 1) \nabla ϕ],

(9b)

\frac{d w}{d t} = \frac{g}{π_{s}} \frac{\partial γ (p - π)}{\partial σ},

(9c)

\frac{d \hat{q}}{d t} = - δ [\frac{\dot{π}}{π} + \frac{1}{1 - κ} (D + \hat{d})],

(9d)

\frac{\partial q_{s}}{\partial t} = - \int_{0}^{1} (D + V \cdot \nabla q_{s}) d σ,

(9e)

where R = C_p − C_υ is the perfect gas constant, C_p and C_υ are the specific heats at constant pressure and constant volume, κ = R/C_p,

\hat{d}

is the modified vertical divergence,

\dot{π} = d π / d t

the mass-based vertical velocity, ∇ is the horizontal gradient on σ surfaces and D = ∇ ⋅ V the horizontal wind divergence. The system (9a)–(9e) is closed with the following diagnostic relations:

\frac{\dot{π}}{π} = V \cdot \nabla q_{s} - \frac{1}{σ} \int_{0}^{σ} (D + V \cdot \nabla q_{s}) d σ^{'},

(10a)

ϕ = ϕ_{s} + \int_{σ}^{1} \frac{R T}{σ^{'}} d σ^{'} + \int_{σ}^{1} ϵ (\frac{π}{p} - 1) \frac{R T}{σ^{'}} d σ^{'},

(10b)

\hat{d} = \frac{p}{R T π_{s}} (\nabla ϕ \frac{\partial V}{\partial σ} - g \frac{\partial w}{\partial σ}) .

(10c)

Notice that the parameter ϵ enters the horizontal wind evolution (9b) via (10b). Even if it is not required by numerical methods used to solve the system, we consider strictly the control parameters constant in space and time.

With all control parameters equal to 1, (9)–(10) represent the Euler equations. On the other hand, the hydrostatic approximation may be reached through several ways. One possibility is α = δ = 0. Then $\hat{q}$ of a parcel remains zero if it was zero initially and thus p = π at any place and time. It follows that β, γ, and ϵ are not used in this case, since they multiply terms that are identically zero. Another possibility is to make the limit γ → ∞ while keeping α = β = δ = ϵ = 1, which corresponds to the traditional approach used in Davies et al. (2003).

4. Continuous linearized system

We proceed exactly as in Bubnová et al. (1995) and Bénard et al. (2010), seeking for increased conciseness. First, we define the basic state $X^{*}$ , then we linearize our continuous set of equations around this basic state to obtain $L^{*}$ . We use asterisk (*) to denote variables, parameters, or vertical operators associated with the basic state $X^{*}$ or with the linear operator $L^{*}$ . We eliminate variables from the system and develop the structure equation and the dispersion formula for a 2D (x, σ) atmospheric slice. We refer to Bubnová et al. (1995) and Bénard et al. (2010) for all further details.

a. Basic state

The basic state $X^{*}$ is isothermal with the background temperature $T^{*}$ , resting, dry, and hydrostatically balanced. The orography is represented as a uniform background value for the ground geopotential ϕ_s. It follows that $V^{*} = (0, 0) m s^{- 1}$ , $w^{*} = 0 {m s}^{- 1}$ and $p^{*} = π^{*} = σ π_{s}^{*}, π_{s}^{*} = const$ . The basic state is thus horizontally homogeneous and fully characterized by only two values, $T^{*}$ and $π_{s}^{*}$ .

b. Linear model

For a given state X of the atmosphere, the deviation of a variable is given by the difference between the local value and that of the basic state,

X^{'} = X - X^{*}

. In the following, we omit prime which usually denotes the deviation. The linearized system is then formulated with vertical component of relative vorticity, horizontal divergence D = ∇ ⋅ V and vertical divergence

\hat{d}

defined by (10c) instead of the three wind components V, w as

\frac{\partial T}{\partial t} = - κ T^{*} S^{*} D - α \frac{κ}{1 - κ} T^{*} (S_{κ}^{*} D + \hat{d}),

(11a)

\frac{\partial D}{\partial t} = - R G^{*} Δ T - R T^{*} Δ q_{s} + R T^{*} (ϵ G^{*} - β) Δ \hat{q},

(11b)

\frac{\partial \hat{d}}{\partial t} = - \frac{g^{2}}{R T^{*}} γ L_{υ}^{*} \hat{q},

(11c)

\frac{\partial \hat{q}}{\partial t} = - \frac{δ}{1 - κ} (S_{κ}^{*} D + \hat{d}),

(11d)

\frac{\partial q_{s}}{\partial t} = - N^{*} D,

(11e)

where Δ = ∇² = ∇ ⋅ ∇ is the horizontal Laplacian operator. The evolution equation for the vertical component of relative vorticity is trivial in this case. The basic state attributes and the vertically continuous linear operators are defined according to Bubnová et al. (1995) and summarized in the appendix.

c. Structure equation

We proceed in several steps to eliminate all variables except the horizontal divergence D from the system (11a)–(11e) in the continuous context and under simplifying assumptions as it was proposed by Voitus in Degrauwe et al. (2020). It can be done through the application of

\partial / \partial t

on the horizontal and vertical divergence evolution equations with the usage of the rest of the system (11a)–(11e). To simplify the deductions, we define the following derived control parameters:

χ = \frac{ϵ δ - κ α}{(1 - κ)},

(12a)

ξ = β δ,

(12b)

ζ = γ δ .

(12c)

Notice that the derived control parameters inherit the value 1 from the primary control parameters α, β, γ, δ and ϵ, and, respectively, they inherit the value 0. Moreover, notice that χ may get negative values for small δ, ϵ and high α.

Using the vertical operators and the phase speed of sound c defined in the appendix with further manipulation to separate the terms containing D and

\hat{d}

we come to the following system of two equations:

(\frac{1}{c^{2}} \frac{\partial^{2}}{\partial t^{2}} - L_{υ}^{* ζ}) \hat{d} = L_{υ}^{* ζ} S_{κ}^{*} D,

(13a)

(\frac{1}{c^{2}} \frac{\partial^{2}}{\partial t^{2}} - B_{κ}^{* ξ, χ} Δ) D = G_{κ}^{* ξ, χ} Δ \hat{d} .

(13b)

We continue with the application of the operator

[(1 / c^{2}) (\partial^{2} / \partial t^{2}) - L_{υ}^{* ζ}]

to (13b) followed with the application of

\partial^{*}

(see the appendix for the definition) on its both sides. Using the commutativity property:

\partial^{*} L_{υ}^{* ζ} G_{κ}^{* ξ, χ} = \partial^{*} G_{κ}^{* ξ, χ} L_{υ}^{* ζ},

(14)

we get the following structure equation:

\frac{1}{c^{4}} \frac{\partial^{4}}{\partial t^{4}} \partial^{*} D - \frac{1}{c^{2}} \frac{\partial^{2}}{\partial t^{2}} \partial^{*} (L_{υ}^{* ζ} + B_{κ}^{* ξ, χ} Δ) D + \partial^{*} L_{υ}^{* ζ} B^{*} Δ D = 0 .

(15)

Now, we will examine the normal modes of (15). Let us consider a 2D (x, σ) vertical slice for simplicity. The normal modes of the 2D system have the following structure:

D (x, σ, t) = \overset{^^}{D} e^{ikx + i ω t} σ^{- \bar{n}},

(16)

with k being the horizontal wavenumber, ω being the time frequency and ν the vertical wavenumber appearing in

\bar{n} = - i ν + 1 / 2

, where i is the imaginary unit and the overbar denotes the complex conjugate, so that

\bar{n} = i ν + 1 / 2

. The real values of ω lead to a solution periodic in time, while imaginary part of ω would result in distortion, damping or amplifying the amplitude

\overset{^^}{D}

in time. Then partial derivatives satisfy the following:

\frac{\partial^{2} D}{\partial t^{2}} = - ω^{2} D,

(17a)

Δ D = - k^{2} D,

(17b)

\partial^{*} D = - \bar{n} D .

(17c)

Using the following relations:

\partial^{*} B^{*} D = \frac{κ (1 - κ)}{\bar{n} n} \partial^{*} D,

(18a)

\partial^{*} L_{υ}^{* ζ} D = - \frac{ζ}{H^{2}} \bar{n} n \partial^{*} D,

(18b)

\partial^{*} G_{κ}^{* ξ, χ} S_{κ}^{*} D = {ξ + χ \frac{{(1 - κ)}^{​ 2}}{\bar{n} n} - [(χ - ξ) n + ξ] \frac{(1 - κ)}{\bar{n} n}} \partial^{*} D,

(18c)

\partial^{*} L_{υ}^{* ζ} B^{*} D = - κ (1 - κ) \frac{ζ}{H^{2}} \partial^{*} D,

(18d)

and

κ (1 - κ) = N^{2} H^{2} / c^{2}

, and using the definition

J^{2} = \bar{n} n / H^{2}

we get the final dispersion formula:

ω^{4} - c^{2} ω^{2} {ζ J^{2} + ξ k^{2} + [(1 - χ) + (χ - ξ) \frac{\bar{n}}{κ}] k^{2} \frac{N^{2}}{c^{2} J^{2}}} + ζ c^{2} k^{2} N^{2} = 0,

(19)

which reduces to the dispersion formula of the HPE system for ζ = ξ = χ = 0:

ω^{2} - k^{2} \frac{N^{2}}{J^{2}} = 0,

(20)

and to the dispersion formula of the fully elastic EE system in case that ζ = ξ = χ = 1:

ω^{4} - c^{2} ω^{2} (J^{2} + k^{2}) + c^{2} k^{2} N^{2} = 0 .

(21)

We may notice that J² > 0. To have the blended system stable, ω must be purely real. The necessary condition for purely real solutions of the dispersion formula (19) is that all the coefficients of (19) are real. This implies

ξ = χ .

(22)

We call (22) the unifying constraint.

We choose the control parameters to satisfy the unifying constraint. Then the dispersion formula with the shape:

ω^{4} - c^{2} ω^{2} [ζ J^{2} + χ k^{2} + (1 - χ) k^{2} \frac{N^{2}}{c^{2} J^{2}}] + ζ c^{2} k^{2} N^{2} = 0,

(23)

contains not only separated terms for vertical and horizontal modes, which are moreover modified with ζ, χ, but also contains the vertical–horizontal mixed term. This term plays a role mainly for small vertical wavenumbers. The control parameters β and γ have an opposite effect (small values of β similarly as large values of γ), which affects mainly acoustic modes for long horizontal waves and gravity modes for short horizontal waves as demonstrated in Figs. 1a and 1b. Since we would like to modify the frequency of acoustic modes solely, and we are interested in short horizontal waves, we may not alter control parameters β and γ and we keep β = γ = 1. Then, we replace χ and ζ with δ in (23), choose ϵ = 1 for convenience, and get α = δ = χ from the unifying constraint. This is now the unique parameter that controls the hydrostaticity of the system. The solutions for distinct vertical modes (ν = 1 and 10) and several choices of χ are shown in Figs. 1c and 1d. We may see that the frequency of acoustic modes is modified by a factor χ, and with χ < 1 we get smaller frequencies. On the other hand, the modification of the frequency of gravity modes is negligible for higher vertical modes, which is desirable behavior. For smaller vertical wavenumbers, the vertical–horizontal mixed term is mainly responsible for the distortion of gravity modes in dependence on the value of χ.

Fig. 1.

Citation: Monthly Weather Review 151, 7; 10.1175/MWR-D-22-0125.1

To enlighten more the proposed method, we analyze vertically propagating modes of the blended system by setting k = 0 in the dispersion formula (23). After excluding the stationary solution ω = 0 we get

ω^{2} = ζ c^{2} J^{2},

(24)

which means that the speed of sound is modified by the factor

\sqrt{ζ}

. For the traditional path to HPE (γ → ∞, α = β = δ = ϵ = 1) where ζ = γ, the blended system with γ > 1 accelerates vertically propagating acoustic modes. On the other hand, for our alternate path to HPE (δ → 0, α = δ, β = γ = ϵ = 1), the blended system with δ < 1 slows down vertically propagating acoustic modes since ζ = δ in this case. Thanks to this we get the blended system that has more stable temporal discretization. We believe that the fundamental difference between the two paths to HPE is the reason why the traditional approach could not yield a numerically efficient solution.

The values of control parameters bigger than one appear as an additional unphysical moderation or acceleration of respective normal modes. It is not desirable in the full model. On the other hand, when used only in the linear model of the SI time scheme, it becomes a stabilization numerical technique that may be compared to the usage of simplifying requirements on the SI basic state.

d. Control parameters in the semi-implicit time scheme

After linearization of the full model $M$ around the basic state $X^{*}$ resulting in $L^{*}$ of (11), we introduce in $L^{*}$ generally different values of control parameters denoted $α^{*}, β^{*}, γ^{*}, δ^{*}, ϵ^{*}$ . It means that the values of the control parameters may be set independently in the full model $M$ and in its linear counterpart $L^{*}$ . Then $L^{*}$ is no longer the linearization of $M$ around any reference state and may be viewed as the linear preconditioner in the time marching process.

In case the control parameters are used solely in the linear system and the full model is not modified, even choices which modify the gravity modes may serve well since the choice of the linear model is purely algorithmic, serving for the distinction of linear and nonlinear terms treated differently in the SI time marching scheme. It is the reason why we introduce two possible settings of control parameters, both satisfying the unifying constraint and both with only one degree of freedom—the parameter $h^{*}$ —to be defined. First, $α^{*} = β^{*} = ϵ^{*} = h^{*}$ and $δ^{*} = 1$ . And second, $α^{*} = δ^{*} = h$ and $β^{*} = ϵ^{*} = 1$ . The setting of $γ^{*}$ is treated separately.

The same setting may be used in the full model as well. We denote the free parameter h in this case. Then either α = β = ϵ = h and δ = 1, or α = δ = h and β = ϵ = 1. We define two mappings assigning values from {1,

h^{*}

, h} to four control parameters in the linear model and to four control parameters in the full model by

\begin{array}{l} f_{α β ϵ} [h^{*}, h] : & (α^{*}, β^{*}, δ^{*}, ϵ^{*}) \to (h^{*}, h^{*}, 1, h^{*}) \\ (α, β, δ, ϵ) \to (h, h, 1, h) \end{array},

(25a)

\begin{array}{l} f_{α δ} [h^{*}, h] : & (α^{*}, β^{*}, δ^{*}, ϵ^{*}) \to (h^{*}, 1, h^{*}, 1) \\ (α, β, δ, ϵ) \to (h, 1, h, 1) \end{array} .

(25b)

The mapping names remind the control parameters whose value is assigned generally different from 1 (if

h^{*}

, h ≠ 1). We extensively apply f_αβϵ and f_αδ to assign values to the control parameters in the analysis and tests presented in this paper.

5. Spatial discretization

The horizontal discretization of the ALADIN system is based on the biFourier transform between the grid point representation of model variables needed for nonlinear calculation used in the advection scheme and in the physical parameterizations and the spectral representation of model variables used for the Helmholtz equation solver, horizontal diffusion and horizontal derivatives calculation. The details may be found in Bubnová et al. (1995).

The basic principles of the vertical discretization are as well described in Bubnová et al. (1995) (notice that the meaning of $α^{*}, β^{*}, δ^{*}$ is different than here) and we apply them with one major distinction being in the solution of the Helmholtz equation. In the original proposal, all the variables but the vertical divergence $\hat{d}$ were eliminated and a single space and time discretized equation for $\hat{d}$ was obtained. Then two constraints were formulated, denoted (C1) and (C2) in Bubnová et al. (1995), having consequences on the definition of discrete vertical operators. We reduce the EE system to a single space and time discretized equation for the horizontal divergence D, similarly as it is done for the HPE system. This kind of elimination, proposed by Voitus and described in Degrauwe et al. (2020), allows to formulate the Helmholtz operator ( $I - Δ t^{2} c^{2} B^{*} Δ$ ), where $B^{*}$ consists of an HPE-based part $B_{HY}^{*}$ and an additional nonhydrostatic departure matrix $B_{NH}^{*}$ . In this way, the Helmholtz problems of the HPE system and the EE system are unified allowing the extension to systems formulated with the control parameters having values distinct from 0 and 1. We describe here the modification of the process needed to propose the solution to the Helmholtz problem formulated for the system (9).

a. Space discretized equations

After time-discretization with a semi-implicit scheme with the time step Δt and the vertical discretization as described in Bubnová et al. (1995), the system (11) becomes

T + δ t κ T^{*} [S^{*} D + \frac{α^{*}}{1 - κ} (S_{κ}^{*} D + \hat{d})] = \tilde{T},

(26a)

D + δ t R T^{*} Δ [G^{*} \frac{T}{T^{*}} + q_{s} - (ϵ^{*} G^{*} - β^{*}) \hat{q}] = \tilde{D},

(26b)

\hat{d} + δ t \frac{g^{2}}{R T^{*}} γ^{*} L_{υ}^{*} \hat{q} = \tilde{d},

(26c)

\hat{q} + δ t \frac{δ^{*}}{1 - κ} (S_{κ}^{*} D + \hat{d}) = \tilde{q},

(26d)

q_{s} + δ t N^{*} D = \tilde{q_{s}},

(26e)

where

δ t = Δ t / 2, S_{κ}^{*} = I - (1 - κ) S^{*}

and

G^{*}, L_{υ}^{*}, N^{*}, S^{*}

are discrete analogs of

G^{*}, L_{υ}^{*}, N^{*}, S^{*}

defined in Bubnová et al. (1995). We do not repeat the definition of discrete vertical operators here in the attempt to keep the notation and explanation as simple as possible. All left-hand side terms are evaluated at time t + Δt. The right-hand sides with tildes contain all the explicit terms evaluated at time t or t − Δt depending on the chosen time scheme. They may be elaborated as in Degrauwe et al. (2020) and their definition is omitted here.

b. Elimination of variables

We introduce the vertical Helmholtz discrete operator:

H^{*} = I - Δ t^{2} c^{2} L_{υ}^{* ζ},

(27)

which is a constant-coefficient positive-definite matrix and we know that its inverse exists. We obtain the following discrete form of (13a)–(13b):

H^{*} \hat{d} = (I - H^{*}) S_{κ}^{*} D,

(28a)

D - Δ t^{2} c^{2} [κ G^{*} (1 - κ) S^{*} + (1 - κ) N^{*}] Δ D - Δ t^{2} c^{2} G_{κ}^{* ξ, χ} S_{κ}^{*} Δ D = Δ t^{2} c^{2} G_{κ}^{* ξ, χ} Δ \hat{d},

(28b)

where

G_{κ}^{* ξ, χ} = ξ^{*} I - χ^{*} (1 - κ) G^{*}

. After elimination of

\hat{d}

we get

(I - Δ t^{2} c^{2} B^{*} Δ) D = D^{• •},

(29)

where

D^{• •}

is developed exactly as in Degrauwe et al. (2020) using definition (27) and replacing

I - (1 - κ) G^{*}

with

G_{κ}^{* ξ, χ}

. Moreover,

B^{*} = [κ G^{*} (1 - κ) S^{*} + (1 - κ) N^{*} + G_{κ}^{* ξ, χ} H^{*}^{- 1} S_{κ}^{*}] .

(30)

Here,

B^{*}

is a sum of the hydrostatic factorization matrix

B_{HY}^{*} = κ G^{*} (1 - κ) S^{*} + (1 - κ) N^{*}

and the nonhydrostatic increment

B_{NH}^{*} = G_{κ}^{* ξ, χ} H^{*}^{- 1} S_{κ}^{*}

. Let us recall that

G_{κ}^{* ξ, χ}

reduces to

G_{κ}^{*}

for

χ^{*} = ξ^{*} = 1

and to the zero matrix if

χ^{*} = ξ^{*} = 0

. Similarly,

L_{υ}^{* ζ}

reduces to

L_{υ}^{*} / H^{2}

for

ζ^{*} = 1

and to the zero matrix for

ζ^{*} = 0

. It implies the reduction of the system for pure nonhydrostatic and pure hydrostatic systems.

Let us remark that the discretization and the solution of the Helmholtz equation differs from the one presented in Degrauwe et al. (2020) only in the usage of $L_{υ}^{* ζ}$ in the definition of $H^{*}$ , and of $G_{κ}^{* ξ, χ}$ in the definition of $B^{*}$ .

6. Stability analysis of the time marching scheme

SI time schemes combine an implicit method applied on the linear terms and an explicit method for the nonlinear terms including the possibility of the iterative treatment described by (8). The crucial step is thus the separation of the source terms of the complete system into the linear and the nonlinear part. In case of SI schemes with horizontally homogeneous coefficients constant in time this separation is usually based on the linearization of the equation system around a stationary reference basic state, as summarized in Bénard (2004) after (Simmons and Temperton 1997; Bubnová et al. 1995; Caya and Laprise 1999). The explicit treatment of the nonlinear terms may lead to poor stability of the whole time stepping method. As advocated in Bénard (2004), a more general approach with modified linear part of the equation system may lead to the improvement of stability properties.

With the presented method, after linearization of the system of equations around a SI reference state, the control parameters of the linear part of the system may get values different from the control parameters in the full system. Thus, the shape of the nonlinear residual is changed, and this may lead to enhanced stability compared to the classical method. But the resulting linear system is no longer a linearization of the original system around any SI reference state.

a. The unbounded linear system

The system (11a)–(11e) is first provided with the control parameters

α^{*}, β^{*}, γ^{*}, δ^{*}, ϵ^{*}

. Then, following exactly the method presented in Bénard (2003), it is transformed into an unbounded system in the σ coordinate by the application of vertical linear differential operators on both sides:

\begin{array}{l} (\partial^{*} + I) \frac{\partial T}{\partial t} = (α^{*} - 1) κ T^{*} D - α^{*} \frac{κ}{1 - κ} T^{*} (\partial^{*} + I) (D + \hat{d}) \end{array},

(31a)

\partial^{*} \frac{\partial D}{\partial t} = R Δ T - R T^{*} (ϵ^{*} + β^{*} \partial^{*}) Δ \hat{q},

(31b)

\frac{\partial \hat{d}}{\partial t} = - \frac{g^{2}}{R T^{*}} γ^{*} L_{υ}^{*} \hat{q},

(31c)

(\partial^{*} + I) \frac{\partial \hat{q}}{\partial t} = - \frac{δ^{*}}{1 - κ} (\partial^{*} + I) (D + \hat{d}) + δ^{*} D .

(31d)

Properties (A1) are used. In this way all vertical integrals as well as the surface values including model variable q_s are eliminated and the system no longer contains any reference to the lower and the upper boundaries. In the following, we will develop the time discretization of (31a)–(31d) for time continuous normal modes and we will examine their maximum possible growth rate to evaluate the stability of the above considered time schemes for the given structure. Following reasoning of Bénard (2003), any normal mode of the original system (11a)–(11e) is then as well the normal mode of the unbounded system (31a)–(31d) with the same frequency and the transformation does not hide any instability unless this instability is caused by the boundary conditions.

b. Time-discretized space-continuous analysis

Here, following the method elaborated in Bénard (2003), we examine the stability of the time discretization of the system (31a)–(31d) for perturbations that have the spatial structure of the time continuous normal modes:

X (x, σ) = \overset{^^}{X} e^{ikx} σ^{- \bar{n}},

(32)

where k, ν (in

\bar{n}

) are horizontal and vertical wavenumbers, and X represents the vector of prognostic variables

(T, D, \hat{d}, \hat{q})

. The stability analysis examines the response of the time discretized system to the state with spatial structure of the normal mode. The application of vertical operators on X results in

Δ X = - k^{2} X,

(33a)

\partial^{*} X = - \bar{n} X,

(33b)

(\partial^{*} + I) X = n X .

(33c)

After this application and after inversion of the left-hand scalar operators to appear on the right-hand side the original unbounded system (31a)–(31d) can be written in the following form:

\frac{\partial X}{\partial t} = L^{*} X,

(34)

where

L^{*} = {\begin{matrix} 0 & [\frac{(α^{*} - 1)}{n} - \frac{α^{*}}{(1 - κ)}] κ T^{*} & - \frac{α^{*} κ}{(1 - κ)} T^{*} & 0 \\ \frac{R k^{2}}{\bar{n}} & 0 & 0 & - (\frac{ϵ^{*} - \bar{n} β^{*}}{\bar{n}}) R T^{*} k^{2} \\ 0 & 0 & 0 & γ^{*} n \bar{n} \frac{g^{2}}{R T^{*}} \\ 0 & δ^{*} [\frac{1}{n} - \frac{1}{(1 - κ)}] & - \frac{δ^{*}}{(1 - κ)} & 0 \end{matrix}} .

(35)

With the means of the SHB-type analysis (Simmons et al. 1978) we examine the thermal stability of the system. We suppose that the full system has exactly the same shape but generally different temperature

\bar{T} > 0 K

than the reference linear system. We introduce the “nonlinearity” factor:

ϑ = \frac{\bar{T} - T^{*}}{T^{*}}

(36)

and derive

\bar{L}

from

L^{*}

through the replacement of

T^{*}

with

\bar{T} = (1 + ϑ) T^{*}

and of the control parameters

α^{*}, β^{*}, γ^{*}, δ^{*}, ϵ^{*}

with α, β, γ, δ, ϵ.

Then the unbounded system is

\frac{\partial X}{\partial t} = \bar{L} X,

(37)

which is time discretized either using the time discretization EXTR (7) or the time discretization NESC (6). Here

L^{*}

is used as the linear model and

\bar{L}

as the full model. Moreover, the iterative time step may be applied according to (8), but then the stability analysis becomes too complex and it is not presented here.

The time discretized growth of any normal mode is examined assuming

X^{+ 1 / 2} = μ (λ) X^{0},

(38a)

X^{+} = λ X^{0},

(38b)

with the unknown numerical complex growth rate λ to be quantified. If for some solution, |λ| > 1 the examined time scheme is unstable for the particular normal mode. Hence to ensure stability, we need to get Γ = max(|λ|, all possible λ) ≤ 1 and Γ is considered to be the characteristic value of the time scheme stability for the given structure. We call it the amplification factor. The function μ(λ) is introduced here to unify the treatment of both considered time schemes, EXTR and NESC. For NESC, nevertheless, we set μ(λ) = 1 since

X^{+ 1 / 2} = X^{0}

, while for EXTR, using X⁰ = λX⁻ we set

μ (λ) = 3 / 2 - 1 / (2 λ)

since

X^{+ 1 / 2} = (3 / 2) X^{0} - (1 / 2) X^{-} = [3 / 2 - 1 / (2 λ)] X^{0}

. This unification is indeed possible in all the elimination of the growth rate polynomial equation and makes it simpler.

Since up to now we keep strictly the procedure described in Bénard (2003), all conditions necessary for the stability of the state X and all conditions on the normal modes of the linear unbounded system are met.

The possible values of λ for the normal mode structure (32) are given by the roots of the polynomial equations that we derive here. We define the generalized state vector

Z = {[X^{0}, X^{+ 1 / 2}, X^{+}]}^{T}

. We now apply (5) using linearized models:

\begin{array}{l} \frac{X^{+} - X^{0}}{Δ t} = L^{*} (\frac{X^{+} - 2 X^{+ 1 / 2} + X^{0}}{2}) + \bar{L} (X^{+ 1 / 2}) \end{array} .

(39)

Then (38a), (39) and (38b) result in a set of linear equations of the following shape:

[\begin{matrix} μ (λ) I & - I & O \\ A & B & C \\ - λ I & O & I \end{matrix}] \cdot Z = M Z = 0,

(40)

where

O, I, A, B, C

are matrices of size 4 × 4,

O

with all elements equal to zero,

I

being the identity matrix, and

A = - I - \frac{Δ t}{2} L^{*},

(41a)

B = - Δ t (\bar{L} - L^{*}),

(41b)

C = I - \frac{Δ t}{2} L^{*} .

(41c)

We consider the following polynomial in λ:

\begin{matrix} Det (M) = Det [A + μ (λ) B + λ C] \\ = Det (Λ_{t} Δ t I - P Δ t L^{*} - Q Δ t \bar{L}) \end{matrix},

(42)

where the response factors are defined as

Λ_{t} = \frac{λ - 1}{Δ t},

(43a)

P = \frac{λ + 1}{2} - μ (λ),

(43b)

Q = μ (λ) .

(43c)

For NESC, the degree of the polynomial equals the number of prognostic variables, i.e., four in our case, while for EXTR it is doubled with four additional computational modes. The possible values of λ are given by the roots of the polynomial equation:

Det (M) = 0,

(44)

which will be algebraically expanded in the following section depending on the chosen values of control parameters.

c. The growth rate polynomial equation

We define the response functions

Λ^{+}, \tilde{Λ}, \hat{Λ}

with two parameters φ₁, φ₂ as

Λ^{+} (φ_{1}, φ_{2}) = φ_{1} P + φ_{2} Q,

(45a)

\tilde{Λ} (φ_{1}, φ_{2}) = φ_{1} P + φ_{2} Q (1 + ϑ),

(45b)

\hat{Λ} (φ_{1}, φ_{2}) = φ_{1} P + φ_{2} Q (\frac{1}{1 + ϑ}) .

(45c)

We shorten the notation for φ₁, φ₂ = 1 with Λ⁺(1, 1) = Λ⁺ etc. It follows that Λ⁺ = P + Q,

\tilde{Λ} = P + Q (1 + ϑ)

and

\hat{Λ} = P + Q [1 / (1 + ϑ)]

. We expand (44) into the following growth rate polynomial equation:

\begin{array}{l} Λ_{t}^{4} + Λ_{t}^{2} Λ^{+} (δ^{*}, δ) \hat{Λ} (γ^{*}, γ) c^{2} J^{2} \\ + Λ_{t}^{2} [\frac{κ (1 - κ)}{n \bar{n}} Λ^{+} \tilde{Λ} - \frac{κ}{n} (1 - \frac{κ}{\bar{n}}) Λ^{+} \tilde{Λ} (α^{*}, α) \\ + (1 - \frac{κ}{\bar{n}}) Λ^{+} (δ^{*}, δ) \tilde{Λ} (\frac{ϵ^{*} - \bar{n} β^{*}}{n}, \frac{ϵ - \bar{n} β}{n})] c^{2} k^{2} \\ + Λ^{+} Λ^{+} (δ^{*}, δ) \tilde{Λ} \hat{Λ} (γ^{*}, γ) c^{2} k^{2} N^{2} = 0 . \end{array}

(46)

We distinguish two possible approaches. First, the values of control parameters are equal to 1 in the full model

\bar{L}

while set freely in the linear model

L^{*}

and we call this approach the generalized linear model. In this case the full set of equations is being solved. Second, the control parameters have freely set values in the full model

\bar{L}

and they modify the set of equations being solved. Independently, the values of control parameters in the linear model

L^{*}

may be set. We call this approach the blended equation set.

We set always γ = 1 and β = ϵ = 1 since other choices distort the solution of gravity modes as discussed in section 4c.

1) Generalized linear model

First, we suppose that the control parameters in the full model have the value needed for the full set of Euler equations, i.e., α = β = 1 (on top of β = γ = ϵ = 1). The influence of the value of $γ^{*}$ was thoroughly discussed in Bénard (2004), and we omit these considerations here. We set $γ^{*}$ independently from the value of the other control parameters and $γ^{*} > 1$ in case it is necessary for the stabilization of the system in idealized or real cases. Let us just remark here that the limit case of the approach adopted in Bénard (2004) with $γ^{*} \to \infty$ represents SI time scheme with hydrostatic linear model but with the full set of Euler equations being solved.

The imaginary part of the left-hand side of (46) is equal to zero for all ϑ resulting in real coefficients of all terms in (46) only in case of

δ^{*} = 1

. Then with the unifying constraint

χ^{*} = ξ^{*}

we obtain the following:

\begin{array}{l} Λ_{t}^{4} + Λ_{t}^{2} Λ^{+} \hat{Λ} (γ^{*}, 1) c^{2} J^{2} \\ + Λ_{t}^{2} Λ^{+} \tilde{Λ} [β^{*} + (1 - β^{*}) \frac{κ (1 - κ)}{n \bar{n}}, 1] c^{2} k^{2} \\ + Λ^{+} Λ^{+} \tilde{Λ} \hat{Λ} (γ^{*}, 1) c^{2} k^{2} N^{2} = 0, \end{array}

(47)

restricting the coefficients of the polynomial equation to real numbers only. See the appendix for the detailed deduction. We have only two degrees of freedom, the control parameters

β^{*}, γ^{*}

in this case.

We present in Fig. 2 the visualization of the values of the amplification factor Γ obtained from (47) for the EXTR scheme with $f_{α β ϵ} [h^{*}, 1]$ and $h^{*} = 0.5$ (Fig. 2a), $h^{*} = 1$ (Fig. 2b), and $h^{*} = 2$ (Fig. 2c), for large intervals of values of the horizontal wavenumber k (on the x axis) and ϑ (on the y axis) and for $T^{*} = 300 K$ , ν = 1, Δt = 10 s. We present in Fig. 3 the amplification factor for the particular choice of $k = (π / 200) m^{- 1}$ (corresponding to the wavelength of 400 m), $T^{*} = 300 K$ , ν = 1 and Δt = 10 s for the EXTR scheme with generalized linear model using $f_{α β ϵ} [h^{*}, 1]$ for a range of values of $h^{*}$ and for four distinct values of ϑ. We set $γ^{*} = 1$ in all visualizations.

Fig. 2.

Citation: Monthly Weather Review 151, 7; 10.1175/MWR-D-22-0125.1

Fig. 3.

Citation: Monthly Weather Review 151, 7; 10.1175/MWR-D-22-0125.1

We may see that the region of stability is significantly enlarged for $h^{*} > 1$ and covers almost all the depicted domain of values of k and ϑ for h = 2; see Fig. 2c. On the other hand, the decrease of value of $h^{*}$ has a detrimental effect on stability of the scheme with the amplification factor bigger than 1 in a large region. Only for very small wavenumbers k < 0.0002 and small temperatures $\bar{T} < 180 K$ the stability is still guaranteed for $h^{*} = 0.5$ ; see Fig. 2a and compare with Fig. 2b. A similar conclusion may be drawn from Fig. 3, higher values of ϑ need higher $h^{*}$ to reach stability.

2) Blended equation set

On the other hand, if we change the values of the control parameters in the full model, the equation system being solved is modified as discussed in section 4. With

δ^{*} = δ = 0

and

α^{*} = α = 0

, we get the hydrostatic solution:

Λ_{t}^{4} + Λ_{t}^{2} \frac{κ (1 - κ)}{n \bar{n}} Λ^{+} \tilde{Λ} c^{2} k^{2} = 0 .

(48)

With all the control parameters having the value of 1, we obtain the expected nonhydrostatic fully compressible solution in the shape of

\begin{matrix} Λ_{t}^{4} + Λ_{t}^{2} Λ^{+} c^{2} (\hat{Λ} J^{2} + \tilde{Λ} k^{2}) \\ + Λ^{+} Λ^{+} \tilde{Λ} \hat{Λ} c^{2} k^{2} N^{2} = 0 \end{matrix} .

(49)

For values between 0 and 1 we may say that the system represents the transition from the HPE system to the fully compressible EE system. Here, we use strictly the same values of control parameters in the linear model and in the nonlinear residual, i.e., (

α^{*}, β^{*}, γ^{*}, δ^{*}, ϵ^{*}

) = (α, β, γ, δ, ϵ), to make the analysis tractable. Notice that then for any φ we have

Λ^{+} (φ, φ) = φ Λ^{+},

(50a)

\tilde{Λ} (φ, φ) = φ \tilde{Λ},

(50b)

\hat{Λ} (φ, φ) = φ \hat{Λ} .

(50c)

Using the unifying constraint, we get the growth rate polynomial equation in the following form:

\begin{array}{l} Λ_{t}^{4} + ζ Λ_{t}^{2} Λ^{+} \hat{Λ} c^{2} J^{2} \\ + ϱ Λ_{t}^{2} Λ^{+} \tilde{Λ} c^{2} k^{2} \\ + ζ Λ^{+} Λ^{+} \tilde{Λ} \hat{Λ} c^{2} k^{2} N^{2} = 0, \end{array}

(51)

with

ϱ = \frac{κ (1 - κ)}{n \bar{n}} - \frac{κ}{n} (1 - \frac{κ}{\bar{n}}) α + (1 - \frac{κ}{\bar{n}}) δ (\frac{ϵ - \bar{n} β}{n}),

(52)

= χ + (1 - χ) \frac{κ (1 - κ)}{n \bar{n}} .

(53)

Here,

ϱ

depends only on one parameter χ and is purely real. Since necessarily β = γ = ϵ = 1, we get α = δ from the unifyng constraint. Thus the only choice in this case is f_αδ[h, h].

The visualizations of values of the amplification factor Γ are presented in Fig. 2, as obtained for the EXTR scheme and for large intervals of values of k and ϑ. Figures 2b and 2d represent limit cases for f_αδ[h, h], h = 1 and h = 0, respectively. In Fig. 4 the amplification factor for the particular choice of $k = (π / 1000) m^{- 1}$ (corresponding to the wavelength of 2 km), $T^{*} = 300 K$ , ν = 1, and Δt = 10 s is shown for the EXTR scheme with blended equation set using f_αδ[h, h] for a range of values of h and for four distinct values of ϑ.

Fig. 4.

As in Fig. 3, but for the blended equation set with f_αδ[h, h] and $k = (π / 1000) m^{- 1}$ .

Citation: Monthly Weather Review 151, 7; 10.1175/MWR-D-22-0125.1

We may see that the region of stability is slightly enlarged to bigger wavenumbers k for positive ϑ (and high $\bar{T}$ ) values and to smaller k for negative ϑ (and small $\bar{T}$ ) values. The effect of the choice of h is much less pronounced than in case of changing $h^{*}$ in the generalized linear model. Similarly, we may see in Fig. 4 that smaller h may be beneficial for the stability of the scheme, but the effect is modest. It is worth mentioning that this weak effect of modified control parameters in the blended equation set was not to be expected and the limited character of the stability analysis method capturing only the linear part of the nonlinear residual may be blamed for. A considerably stronger impact of the choice of h in the blended equation set is observed in idealized test cases and real simulations presented in sections 7 and 8. Let us mention that similar conclusions hold for other time schemes, namely, NESC and PC. We do not illustrate this statement with figures.

7. Idealized tests

In idealized tests and real simulations, the stretched hybrid-pressure terrain-following coordinate η of Laprise (1992) is used as described in Bénard et al. (2010), generalizing the pure unstretched coordinate σ. To assess the behavior of the proposed method two idealized test cases are performed in the reduced (x, η) vertical plane version of the ALADIN system: the nonlinear nonhydrostatic flow over idealized orography according to Bubnová et al. (1995) and the density current test published in Straka et al. (1993).

a. Nonlinear nonhydrostatic flow over a hill

First, we examine a nonlinear nonhydrostatic flow over a bell-shape mountain. We use exactly the same experimental setting as in Vivoda et al. (2018) which is for the reader’s comfort recalled in the following paragraph.

We consider a flow with a constant upstream horizontal velocity U = 4 m s⁻¹ in a dry atmosphere with temperature profile determined by a constant Brunt–Vaisälä frequency N = 0.01 s⁻¹ and bottom temperature T₀ = 288 K. Let a bell-shape mountain be characterized by its height H and its half-width a with values H = a = 400 m. The surface geopotential is defined by

ϕ_{s} (x) = g H \frac{a^{2}}{a^{2} + x^{2}} .

(54)

Then NH/U = Na/U = 1 confirming the nonlinear and nonhydrostatic character of the flow. The horizontal extent of the domain is 30.64 km with horizontal grid distance of 80 m. We have 150 model levels in vertical, regularly spaced with Δz = 180 m up to 200 hPa and allowing a smooth transition up to the model top defined by π = 0 Pa. An isothermal layer starts at 220 hPa with the temperature of 216 K. Lateral boundary conditions are prescribed through a variant of the so-called Davies relaxation scheme suggested for spectral limited area model by Radnóti (1995) and a stationary coupling is used; in the boundary zones, the model is relaxed toward the initial state. No artificial dissipation algorithms like sponge, diffusion, or time decentering are applied. We integrate up to 6000 s.

The two-time-level SI time stepping EXTR is applied with additional averaging along the semi-Lagrangian trajectory according to (Hortal 2002). The results for the vertical velocity field for several choices of the time step and of the control parameters are shown in Fig. 5. Following the results of the stability analysis presented in section 6, we use in the experiments either $f_{α β ϵ} [h^{*}, 1]$ , modifying the control parameters in the linear model only, or f_αδ[h, h], using the blended equation set. We set $T^{*} = 300 K$ and $γ^{*} = 3$ . Dispersive nonhydrostatic waves should be produced that have horizontal and vertical length of similar magnitude and propagate with a pronounced downstream tilt of constant phase far away of the mountain as shown for the reference run in Fig. 5a.

Fig. 5.

Citation: Monthly Weather Review 151, 7; 10.1175/MWR-D-22-0125.1

We summarize the results obtained with the two sets of control parameters in Table 1, where “o” means that the whole integration was finished and the final results have good agreement with the reference experiment (as in Figs. 5b–d), “(o)” means that the whole integration was finished but the final results show significant departure from the reference results (as in Fig. 5f), and “x” means that the integration crashed before ending. For all the time step lengths used for the fully compressible set of Euler equations, the noniterative SI 2TL time scheme was not stable enough, while the iterative centered implicit 2TL time scheme with one iteration (PC) was stable to reach 6000 s.

Table 1.

The stability of the nonlinear nonhydrostatic flow over the Agnesi-shaped mountain for several time steps and several different settings of control parameters. We set $γ^{*} = 3$ . Here “o” means finished integration with the results in good agreement with the reference experiment, “(o)” means finished integration but the results are significantly different from the reference results, and “x” means that the integration crashed before ending.

In Table 1 we may see that there is a clearly distinguished stability region confirming the results of the stability analysis given in section 6. For the generalized linear model, values of $h^{*}$ slightly higher than one increase stability, while for short time steps the region of these values may be enlarged to relatively high values ( $h^{*} = 6$ for Δt = 2 s) keeping the generated wave pattern realistic (see Fig. 5c). On the other hand, for the blended equation set, decreasing the value of h brings more stability, and for longer time steps smaller values of control parameters are needed. For moderate values of h (h ≥ 0.1) the solution is in close agreement with the reference one and with the one published in literature. When going to very small values of control parameters in this case (h = 0.001), the solution starts to be distorted with a chessboard pattern in the lee of the mountain (see Fig. 5f). We present an unrealistic chimney-like solution given by the hydrostatic run for comparison in Fig. 5e.

b. Density current

As a second example we use the same initial configuration as in Straka et al. (1993). It simulates the evolution of a perturbation of temperature initially defined as

T^{'} (x, z) = T_{0} [\cos (π L) + 1] / 2,

(55)

with

L = \min {1, \sqrt{{[(x - x_{c}) / x_{r}]}^{2} + {[(z - z_{c}) / z_{r}]}^{2}}}

, symmetric around a central point (x_c, z_c) with a radius x_r in the horizontal direction and z_r in the vertical direction and with the minimum value of T₀ = −15 K. It represents a cold bubble in the background atmosphere at rest and neutral with a constant potential temperature θ₀ = 300 K and reference pressure p₀ = 1000 hPa. Temperature is then derived from

T (x, z) = θ_{0} {(\frac{p}{p_{0}})}^{​ κ} + T^{'} (x, z) .

(56)

The domain settings correspond exactly to those published in Vivoda et al. (2018) and are summarized here in short: equal horizontal and vertical resolution Δx = Δz = 25 m; the time step of 1 s; the horizontal domain length of 51.2 km; 160 model levels in vertical placed regularly in height followed by a transition zone of 40 model levels with smoothly increasing vertical resolution up to 25 km; the top 8 levels are set to be isothermal; x_c = 2560 m, x_r = 4000 m, z_c = 3000 m, z_r = 2000 m.

The semi-implicit two-time-level time stepping corresponding to the EXTR scheme is applied with additional averaging along the semi-Lagrangian trajectory according to Hortal (2002). We set $T^{*} = 300 K$ and $γ^{*} = 3$ . Periodic boundary conditions are imposed in the horizontal direction and second-order spectral diffusion is applied on the temperature field with the strength gradually increasing to the domain top. The results of several experiments with modified control parameters are shown in Fig. 6. The plotted field is the potential temperature at time 300, 600, and 900 s. Only a part of the right half of the domain is depicted. Several distinct choices of the values of control parameters are used. All the other settings are being kept unchanged in all the experiments. The difference from the reference experiment in Fig. 6a is shown in colors for Figs. 6b–d. The shape of the solution in Figs. 6e,f differs substantially from the reference, we omit emphasizing it with colors.

Fig. 6.

Citation: Monthly Weather Review 151, 7; 10.1175/MWR-D-22-0125.1

One can see in Fig. 6b that modifying only the linear model gives the solution with all basic features kept; compare this with the reference in Fig. 6a. On the other hand, the approximate solution of the blended equation set shows good agreement with the reference experiment for the value $h = h^{*} = 0.1$ (Fig. 6c), for smaller values the solution is unrealistically distorted (Fig. 6d, $h = h^{*} = 0.01$ ) and totally destructed for even smaller values (Fig. 6e, $h = h^{*} = 0.001$ ), evoking the result of the hydrostatic run that destructs the cold bubble very quickly (Fig. 6f). The stability of the EXTR scheme is ensured with the value of $h = h^{*} = 0.1$ (the blended equation set) for Δt = 1 s. With higher values including 1 (the pure EE system) the integration could not be finished. On the other hand, with the PC scheme stable integration may be achieved for the pure EE system using the time step up to 4 s (and the results correspond to the results of the reference run presented in Fig. 6a, not shown).

8. Real case studies

We demonstrate the stability properties and accuracy of the proposed method in real cases. First, we show an example of a lee wave being formed behind the ridge of Krušné hory. We use the ALADIN system in the current operational limited area application of the Czech Hydrometeorogical Institut (CHMI) with the horizontal resolution of 2.325 km. Then we verify the properties of the proposed methods using the ALADIN system in the configuration with a high horizontal resolution (Δx = 375 m) over the Occitania domain.

a. Lee waves in the flow

First, we show an example of cloudiness being formed realistically by a nonhydrostatic fully compressible model while being omitted by the model at the same horizontal and vertical resolution but with the hydrostatic assumption.

We use the CHMI operational domain centered above the Czech territory and covering the Alps in the 2.325-km horizontal resolution, with 87 vertical levels in vertical resolution decreasing with height, starting at the height of 10 m and with the top layer at 50 Pa. The observed formation of clouds is typical for leeward side of mountains, which represent an obstacle in the flow. Here, the strong north winds cross the mountains of Krušné hory. We start the integration at 0000 UTC 12 February 2019 and the forecast integration range is 24 h.

The semi-implicit two-time-level time stepping (Hortal 2002) is applied, with the semi-Lagrangian horizontal diffusion (Váňa et al. 2008) employed for model prognostic variables including hydrometeors and turbulence total and kinetic energies used in physics parameterizations. On top of that, supporting spectral diffusion is used close to the model top acting as a sort of sponge layer. No time decentering is applied. The background temperature of the SI reference state is 350 K and the background surface pressure is 900 hPa. The time step is 90 s.

As a parameterization package, the ALARO physics is employed as described in Termonia et al. (2018), with the 3MT scheme for moist deep convection overcoming the problem of partially resolved deep cumulus, with the radiation scheme ACRANEB, version 2, and with the turbulence model II of TOUCANS. The lateral boundary information is provided by the global system ARPEGE of Météo-France using the Davies relaxation scheme in one hour coupling frequency.

When the 2TL SI noniterative time scheme is applied on the fully compressible equation system a stable integration may not be reached. We show that with a wide range of values of the control parameters this is possible and even if the equation system is not any more fully compressible (when using the blended equation set), we obtain results of a comparable quality as with the fully compressible run.

The results for the low level cloudiness and the cross section through the vertical velocity field across the mountain ridge are shown in Figs. 7 and 8 for several choices of control parameters, as well as the orography of the domain and the position of the cross section line (see Fig. 7b).

Fig. 7.

Citation: Monthly Weather Review 151, 7; 10.1175/MWR-D-22-0125.1

Fig. 8.

Citation: Monthly Weather Review 151, 7; 10.1175/MWR-D-22-0125.1

Contrary to the results of the stability analysis presented in section 6 and corroborative 2D simulations shown in section 7, here the usage of $β^{*} \neq 1$ or $ϵ^{*} \neq 1$ was an obstacle to the stable integration. The only stabilizing setting of control parameters was $f_{α δ} [h^{*}, h]$ . Here we use $f_{α δ} [h^{*}, h]$ generally with $h^{*} \neq h$ . It follows that the condition $δ^{*} = 1$ is not satisfied here. The $γ^{*}$ value is fixed on 3.5, or 5 when indicated.

Figure 9 shows the regions of stability for varying values of $h^{*}$ and h. We may see that there is a clearly identified stability region. For generalized linear model (h = 1, the rightmost column) values of $h^{*}$ higher than 1 increase stability and in stable runs, the results (Figs. 7d and 8b) are generally comparing well with the results of the reference experiment run using the PC time scheme on the fully nonhydrostatic equations (Figs. 7c and 8a) and are dissimilar to the results of the hydrostatic run where the orographically induced wave is missing in the cloudiness field (Fig. 7f) and only weak and distorted wave is visible in the vertical velocity field (Fig. 8d). This holds for moderate values of $h^{*}$ , $1 < h^{*} \leq 3$ . For higher values, $h^{*} > 3$ , the solutions start to be distorted (not shown). For the blended equation set (h < 1), on the other hand, to decrease the value of h means to bring more stability and these values may be combined with $h^{*}$ big enough ( $h^{*} \geq h$ ); see Figs. 7e and 8c. When going to very small values of h, the solution may again be deformed.

Fig. 9.

Citation: Monthly Weather Review 151, 7; 10.1175/MWR-D-22-0125.1

For comparison, we present in Fig. 10 the results of the stability analysis for $f_{α δ} [h^{*}, h]$ with $h^{*}$ , h varying independently. The amplification factor Γ is calculated from (46) using values of $γ^{*} = 1$ , k = 0.0027 m⁻¹, ν = 1, Δt = 45 s, and ϑ = −0.3. The chosen value of the wavenumber k corresponds to the wavelength λ = 2.325 km = Δx. The value ϑ = −0.3 represents the average temperature of the atmosphere of 245 K for $T^{*} = 350 K$ . Comparing with Fig. 9 we may see that the dependence of Γ on $h^{*}$ and h is pronounced correctly with the stability analysis for values smaller than one, while for higher values of $h^{*}$ , the real experiments are stable even for settings where the stability analysis indicates the uncontrolled growth. Here we have to point out to the fact that the stability analysis is done in a simplified context with several particular choices of parameters (k, ν, Δt, ϑ) and serves only for illustration of the correspondence with the real experiments.

Fig. 10.

Citation: Monthly Weather Review 151, 7; 10.1175/MWR-D-22-0125.1

b. Occitania at high horizontal resolution

Since the stability constraints are more restraining with higher horizontal resolutions and steeper slopes of the domain orography we present here the results of model integration starting from 0000 UTC 3 October 2015 with an integration range of 24 h and domain covering Occitania with complicated topography including the western part of the Alps, the eastern part of the Pyrenees, the Massif Central, and the island of Corsica, at a horizontal resolution of 375 m with 87 vertical model levels. See Fig. 11 for more details.

Fig. 11.

Orography (m) of the Occitania experiment with the forecasted wind direction valid for 1200 UTC 3 Oct 2015 at 700 hPa. The blue line denotes the cross section for results presented in Fig. 12.

Citation: Monthly Weather Review 151, 7; 10.1175/MWR-D-22-0125.1

The lateral boundary information is taken from the Météo-France limited area operational configuration of the ALADIN system with the AROME physical package and the surface scheme SURFEX; see Seity et al. (2011) for the description of this configuration. This driving model runs at 1.3-km horizontal resolution. The lateral boundary information is coupled using the Davies relaxation scheme at 1-h coupling frequency. In the nested run, the ALARO physics parameterization package is employed similarly as in the previous subsection, with explicitly resolved moist deep convection.

If the noniterative time marching scheme SETTLS is used (corresponding to EXTR if the position on the semi-Lagrangian trajectory is omitted), the HPE system may be integrated successfully while this is not possible for the EE system except when extremely short time steps are used, as Δt = 5 s, together with high $γ^{*} = 8.5$ . In this case, it is not possible to stabilize the time scheme with the generalized linear model for any value of $h^{*}$ and any $γ^{*}$ used in case the integration was not stable for the full EE system ( $h^{*} = 1$ ). This indicates the limitations of the method when only the linear model is modified through control parameters. To the contrary, stable integration may be reached with the blended equation set. Table 2 shows the largest h for which the scheme f_αδ[h, h] is stable for several values of $γ^{*}$ and several time step lengths. We use h between 0 and 1 changing with the step of 0.1. The scheme is considered stable for the given setting if the integration reaches 24 h.

Table 2.

The value of h sufficient to achieve stability in the experiment over the Occitania domain. We use the blended equation set f_αδ[h, h] and SETTLS time scheme. Boldface indicates values higher than 0.6.

We may see that the noniterative time marching scheme may be stabilized with higher value of $γ^{*}$ and smaller value of h up to time step of 30 s. The results for such time steps and small values of h are indeed not satisfactory. On the other hand, for modest values of h and Δt = 10 s we may reach comparable results as with the full EE set and the PC scheme. We present a cross section over the Massif Central through the field of vertical velocity in Fig. 12. The example of a satisfactory pattern is shown in Fig. 12b for f_αδ[0.9, 0.9] and $γ^{*} = 8.5$ . This pattern may be compared to the reference in Fig. 12a. The hydrostatic primitive equation set does not grant to capture the wave pattern as seen in Fig. 12d and similarly, for f_αδ[0.1, 0.1] and $γ^{*} = 8.5$ in Fig. 12c the pattern is destructed.

Fig. 12.

Citation: Monthly Weather Review 151, 7; 10.1175/MWR-D-22-0125.1

Comparing the run of SETTLS using f_αδ[0.7, 0.7] and Δt = 15 s with the run of SETTLS with Δt = 5 s we use 33% of CPU time needed, while for the PC scheme used with one iteration and Δt = 15 s we need 46% of CPU time of the short time step experiment. Moreover, for some cases and domains even the PC scheme may not be capable of stabilizing the integration of the full EE system and we believe that then the blended equation set may be a good alternative with better stability properties achieved.

The largest h ensuring stable integration with the PC scheme and f_αδ[h, h] for several values of $γ^{*}$ and several time step lengths used is summarized in Table 3. The widening of the zone of stability with decreasing h is clearly visible.

Table 3.

As in Table 2, but with PC time scheme. Boldface indicates values higher than 0.6.

9. Summary and discussion

In this paper, the Euler set of equations has been formulated as the hydrostatically balanced part and the increment allowing the full elasticity of the fluid. Each incremental term in the equations may be controlled through one of the control parameters. A careful choice of control parameters then enables a smooth transition between hydrostatic and fully elastic equation sets. Together with the unified numerical methods applied on these two sets we have obtained an integration scheme with satisfactory stability properties, reasonable CPU time needed for the time integration and giving a solution which keeps the essence of the fully compressible one.

An important step in the unification of numerical methods for time integration of the HPE and EE systems in the ALADIN dynamical core is the elimination of variables up to the horizontal divergence in the Helmholtz solver of the SI time scheme described in Degrauwe et al. (2020). The discretization of the proposed set of equations in time and space has been described keeping as much as possible the original solution of the ALADIN system—the one-step SI time scheme SETTLS of Hortal (2002) or the two-step PC time scheme of Bénard (2003) and spectral method used for discretization in the horizontal direction with finite difference discretization in the vertical direction.

We have found that the dispersion formula of the unified system gives purely real frequencies in case the unifying constraint is satisfied. Necessarily, for the choice of all controls parameters in one of the utmost values, zero and one, the dispersion formula reduces to the hydrostatically balanced and the fully compressible one, respectively. Then the linear part of the nonlinear instability was studied using methods of the SHB type stability analysis described in Simmons et al. (1978). It was shown that the unifying constraint is again needed for the amplification factor having only real values. When the unifying constraint is satisfied, the transition from the HPE to the EE system may be accomplished with only four independently chosen control parameters being used in the proposed blended equation set. We have shown that the careful choice of these four parameters may lead to a stabilization of the SI time scheme without losing accuracy. The nonhydrostatic features of the flow may be preserved as shown on idealized 2D tests [the nonlinear nonhydrostatic flow over an Agnesi-shaped mountain following Bubnová et al. (1995) and the density current proposed by Straka et al. (1993)] and in real simulations realized with the ALADIN system.

The main achievement of the present paper is the possibility to control the nonhydrostatic adjustment to the hydrostatically balanced equation set in dependence on the stability attained by the integration scheme in the given context. Moreover, the control parameters can further be designed as time dependent or space dependent in the vertical. The time dependency would allow for the flow dependent stability control of the scheme throughout the integration but it would require the recalculation of the inversion of the Helmholtz matrix needed for the solution of the Helmholtz equation as a part of the SI time scheme algorithm. This would represent a substantial cost in terms of the CPU time consumption, and the practical suitability of such an approach remains to be evaluated.

The possible vertical dependency of control parameters would create a space for hydrostatic balance achieved near the domain top where the coarser vertical resolution of the mass based vertical discretization does not require to include the nonhydrostatic effects of the fluid but where the stability of the integration scheme may be endangered. On the other hand, it would allow the nonhydrostatic character of the solution to be kept elsewhere. Since the coefficients in the 3D Helmholtz equation used in the SI time scheme (see section 5) are horizontally constant, the system being solved may be decoupled in the vertical dimension and a set of N (the number of vertical levels) independent 2D Helmholtz equations can be solved. Then, the control parameters can be chosen independently for each vertical level and the corresponding 2D Helmholtz equation. These considerations are left for future work.

The linear model in the constant-coefficients SI time scheme is usually classically designed as the linearization of the full set of Euler equations around a basic reference state being resting, horizontally homogeneous and hydrostatically balanced. In this paper, the linear model is defined more generally using a different set of control parameters in the linear model originating from the linearization of the full model around a basic state and in the full model itself. In this case, the control parameters in the full model may keep the value one and hence the full set of Euler equations is being solved. We call this approach the generalized linear model. The usage of modified values of control parameters only in the linear model is purely algorithmic choice and has an influence solely on the division of the source terms between linear part and the nonlinear residuum. As the consequence of the SHB analysis, two constraints have to be applied and only three degrees of freedom remain for the control parameters used (in the linear model). The presented method extends the method designed in Bénard (2004) and one of the control parameters represents the acoustic temperature proposed by Bénard. The remaining two parameters may cause the improvement of stability of the whole integration scheme in case values bigger than one are used. This fact is demonstrated first with the means of the SHB type stability analysis and then confirmed in the idealized 2D tests. In real simulations with the ALADIN system, the performed tests verify that the accuracy of solution may not be endangered with a careful choice of control parameters in the linear model. Indeed, only setting $f_{α δ} [h^{*}, 1]$ , $h^{*} > 1$ , was able to stabilize the noniterative time scheme for the experiment in current operational resolution (2.325 km) of the limited area CHMI application, contrary to the analysis suggesting $δ^{*} = 1$ . Moreover, for a high-resolution real simulation with the ALADIN system over Occitania the proposed method does not bring enough stability to enable the usage of noniterative time scheme for any choice of the control parameters used in the linear model. This shows the limits of the stability improvements achieved with the generalized linear model approach.

On the other hand, a combination of values higher than one assigned to control parameters in the linear model ( $h^{*}$ ) and values smaller than one used for control parameters in the full model (h) with $f_{α δ} [h^{*}, h]$ may stabilize the SI time stepping as shown on the example of the lee wave created in the flow behind Krušné hory with the limited area CHMI application. This version of the blended equation set seems to be the most promising choice in terms of stability and accuracy, with the potential to preserve the nonhydrostatic features of the flow. As another promising choice appears the setting f_αδ[h, h], h = 1 − τ with moderate values of τ > 0 together with the appropriate time marching scheme for the given domain parameters and the chosen time step.

Acknowledgments.

Petra Smolíková thanks the Technology Agency of the Czech Republic for its financial support under Grant SS02030040, PERUN. Jozef Vivoda thanks the Regional Cooperation for Limited Area modeling in Central Europe (RC LACE) for funding part of the study. We thank our colleagues from the ACCORD consortia, especially Ján Mašek and Radmila Brožková for their considerate help and to Karim Yessad for the preparation of the Occitania experiment. We thank the editor Dr. Tommaso Benacchio and the three anonymous reviewers for their thoughtful comments and efforts toward improving our manuscript.

Data availability statement.

Output from idealized simulations, source data for vertical cross sections in 3D simulations in the ASCII format and Mathematica, Wolfram Research, Inc. (2021), notebooks with stability analyses are accessible via Zenodo (doi: 10.5281/zenodo.7764370). For information about other data, please contact the corresponding author.

APPENDIX

Notation and Auxiliary Calculations

a. Basic-state values

We use the basic state values of the square of the acoustic phase speed, the characteristic height of the atmosphere, and the square of the Brunt–Vaisälä frequency defined in Bubnová et al. (1995) as

c^{2} = \frac{R T^{*}}{(1 - κ)},

H = \frac{R T^{*}}{g},

N^{2} = \frac{g^{2} κ}{R T^{*}} .

b. Vertical operators

To keep the notation as formal and concise as possible, we specify the state vector

X = (T, D, \hat{d}, \hat{q}, q_{s})

. Using

I

for the identity operator, the vertically continuous linear operators are defined in the vertical coordinate σ traditionally as in Bubnová et al. (1995) by

\partial^{*} X = σ \frac{\partial X}{\partial σ},

S^{*} X = \frac{1}{σ} \int_{0}^{σ} X d σ^{'},

N^{*} X = \int_{0}^{1} X d σ,

G^{*} X = \int_{σ}^{1} \frac{1}{σ^{'}} X d σ^{'},

L_{υ}^{*} X = \partial^{*} (\partial^{*} + I) X .

We define derived vertically continuous linear operators as

S_{κ}^{*} = I - (1 - κ) S^{*},

G_{κ}^{*} = I - (1 - κ) G^{*},

G_{κ}^{* ξ, χ} = ξ I - χ (1 - κ) G^{*},

B^{*} = (1 - κ) (κ G^{*} S^{*} + N^{*}),

B_{κ}^{* ξ, χ} = B^{*} + G_{κ}^{* ξ, χ} S_{κ}^{*},

L_{υ}^{* ζ} = \frac{ζ}{H^{2}} L_{υ}^{*} .

Notice that

(\partial^{*} + I) S^{*} = I,

(A1a)

\partial^{*} G^{*} = - I .

(A1b)

c. Term Θ in the polynomial growth rate equation in case of the generalized linear model

Denote Θ as the term in the square brackets in (46). For the generalized linear model, we seek for conditions on the control parameters ensuring that Θ is purely real. With α = β = γ = δ = ϵ = 1, we get

\begin{array}{l} Θ = [\frac{κ (1 - κ)}{n \bar{n}} Λ^{+} \tilde{Λ} - \frac{κ}{n} (1 - \frac{κ}{\bar{n}}) Λ^{+} \tilde{Λ} (α^{*}, 1) \\ + (1 - \frac{κ}{\bar{n}}) Λ^{+} (δ^{*}, 1) \tilde{Λ} (\frac{ϵ^{*} - \bar{n} β^{*}}{n}, 1)] \\ = {\begin{cases} \frac{κ (1 - κ)}{n \bar{n}} Λ^{+} \tilde{Λ} - \frac{κ}{n} (1 - \frac{κ}{\bar{n}}) Λ^{+} [(α^{*} - 1) P + \tilde{Λ}] \\ + (1 - \frac{κ}{\bar{n}}) [(δ^{*} - 1) P + Λ^{+}] [(\frac{ϵ^{*} - \bar{n} β^{*}}{n} - 1) P + \tilde{Λ}] \end{cases}} \\ = [\frac{κ (1 - κ)}{n \bar{n}} - \frac{κ}{n} (1 - \frac{κ}{\bar{n}}) + (1 - \frac{κ}{\bar{n}})] Λ^{+} \tilde{Λ} \\ + (1 - \frac{κ}{\bar{n}}) [- \frac{κ}{n} (α^{*} - 1) Λ^{+} + (\frac{ϵ^{*} - \bar{n} β^{*}}{n} - 1) Λ^{+} + (δ^{*} - 1) Λ^{+} \\ + (δ^{*} - 1) (\frac{ϵ^{*} - \bar{n} β^{*}}{n} - 1) P] P + (1 - \frac{κ}{\bar{n}}) (δ^{*} - 1) ϑ P Q . \end{array}

We simplify the term using

[κ (1 - κ) / (n \bar{n})] + [1 - (κ / n)] [1 - (κ / n)] = 1

and

\tilde{Λ} = Λ^{+} + ϑ Q

. Then Θ may be real for all ϑ only in the case

δ^{*} = 1

. It follows that

\begin{array}{l} Θ = Λ^{+} \tilde{Λ} + Λ^{+} (1 - \frac{κ}{\bar{n}}) [- \frac{κ}{n} (α^{*} - 1) + \frac{ϵ^{*} - β^{*}}{n} + (β^{*} - 1)] P \end{array},

and from the unifying constraint guaranteeing in this case that

ϵ^{*} - β^{*} = (α^{*} - β^{*}) κ

we have

\begin{array}{l} Θ = Λ^{+} \tilde{Λ} + Λ^{+} (1 - \frac{κ}{\bar{n}}) (1 - \frac{κ}{n}) (β^{*} - 1) P = Λ^{+} \tilde{Λ} + Λ^{+} [1 - \frac{κ (1 - κ)}{n \bar{n}}] (β^{*} - 1) P \\ = Λ^{+} \tilde{Λ} {1 + [1 - \frac{κ (1 - κ)}{n \bar{n}}] (β^{*} - 1), 1} \\ = Λ^{+} \tilde{Λ} [β^{*} + (1 - β^{*}) \frac{κ (1 - κ)}{n \bar{n}}, 1], \end{array}

which is purely real.

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Fig. 1.

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Fig. 2.

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Fig. 3.

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Fig. 4.

As in Fig. 3, but for the blended equation set with f_αδ[h, h] and $k = (π / 1000) m^{- 1}$ .

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Fig. 5.

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Fig. 6.

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Fig. 7.

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Fig. 8.

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Fig. 9.

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Fig. 10.

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Fig. 11.

Orography (m) of the Occitania experiment with the forecasted wind direction valid for 1200 UTC 3 Oct 2015 at 700 hPa. The blue line denotes the cross section for results presented in Fig. 12.