1. Introduction
The Third Order Moments Unified Condensation and Ndependent Solver (TOUCANS) turbulence scheme was developed within the numerical weather prediction (NWP) system Aire Limitée Adaptation Dynamique Développement International (ALADIN). Because the TOUCANS scheme was designed for a spectral model with the twotime level semiimplicit semiLagrangian scheme, emphasis was put on stability and accuracy for long time steps. As one of the key components of the socalled ALARO1 canonical model configuration of ALADIN (Termonia et al. 2018), the TOUCANS scheme encompasses several scientific choices, mainly (i) no critical gradient Richardson number, (ii) two prognostic turbulence energies: turbulence kinetic energy (TKE; e_{k}) and turbulence total energy (TTE; e_{s}), and (iii) modeling of moisture influence on the turbulence mixing via the buoyancy term in the TKE equation. They are described in Bašták Ďurán et al. (2014, 2018, hereinafter BD14 and BD18, respectively). An option particularly interesting for NWP is the twoenergy scheme of BD18, employing a prognostic TKE along with a prognostic TTE. The ratio of prognostic energies e_{s}/e_{k} is used as the only stability parameter of the turbulence scheme. Even though the twoenergy scheme employs only a downgradient approach, it exhibits more continuous space–time behavior and deeper mixing than a traditional TKE scheme. Such behavior is in better agreement with the large eddy simulation results.
The numerical implementation of a downgradient turbulence scheme is prone to oscillations, the socalled fibrillations. They were described in Kalnay and Kanamitsu (1988), along with an overview of tested time schemes and with a treatment proposal. Since then, other socalled antifibrillation schemes suiting for NWP models were elaborated, e.g., the work of Girard and Delage (1990) and more recently Bénard et al. (2000). Specifically, fibrillations are bounded high frequency oscillations with a vertical wavelength equal to twice the model level spacing Δz. There are two ingredients responsible for the generation of the fibrillations: (i) the coupling between momentum and potential temperature via the turbulent exchange coefficients and (ii) the only partially implicit discretization of the vertical diffusion term, where, for feasibility reasons, the nonlinear exchange coefficient is treated explicitly in time. Fibrillations thus appear in the momentum and the thermodynamic variables under specific conditions and for model time steps exceeding ∼100 s (see section 4 of Bénard et al. 2000).
When going from the Ktype schemes (Louis 1979) to the prognostic TKE ones, the antifibrillation treatment becomes unnecessary. The reason is a calculation of the exchange coefficients from the prognostic TKE instead of its gradient estimation via wind shear. As a result of additional prognostic information about the turbulence intensity, the evolution of the exchange coefficients is smoother, transferring more information between the model time steps. This remains true also in presence of water phase changes—for example, in clouds.
Naturally, the abovementioned advantage of the TKE based computation of the exchange coefficients is preserved in TOUCANS when using the twoenergy scheme. Therefore, it was a surprise to detect noisy patterns in the turbulent fluxes, a kind of ∼2Δt oscillations, since nothing like that was expected. These oscillations tend to appear in stable stratification conditions, and, in contrast with the previous experience with fibrillations, they have a smooth vertical structure. Another important property is that their coupling with momentum and thermodynamic variables is weak. Consequently, they are harder to detect since they are not that apparent in the wind and temperature fields. The core of our work is to describe and numerically analyze this new type of oscillations, appearing in the twoenergy scheme as implemented in TOUCANS. Based on this analysis, we propose an iterative method for the time scheme, which focuses on the suppression of the oscillation generation.
The structure of the paper is as follows: section 2 briefly describes the TOUCANS turbulence scheme. Section 3 provides motivation for our work by demonstrating the problem of the ∼2Δt oscillations. Section 4 explains the mechanism by which the oscillations are generated in a simplified system. An alternative temporal discretization to avoid the ∼2Δt oscillations in a full system is also proposed. Section 5 demonstrates the performance of the proposed treatment in 1D and 3D tests. Section 6 summarizes the treatment and the method and outlines the future work on the topic.
2. The twoenergy scheme
To study the numerical aspects of the twoenergy scheme, we first highlight its essential components. We start from the continuous formulation and proceed to spatial and temporal discretization aspects. Two minor updates of the TOUCANS scheme with respect to BD18 formulation are also described: (i) separate length scales for the molecular dissipation and for the eddy diffusivity and (ii) separate turbulent exchange coefficients for the TKE and the TTE. Both updates affect our analysis of the ∼2Δt oscillations.
a. Continuous formulation
The first terms on the righthand side (RHS) of Eqs. (1)–(3) are downgradient diffusion parameterizations of the turbulence diffusion and pressure correlation terms, and they represent the turbulent transport of the prognostic energies. The turbulent transport term in Eq. (3) would be obtained from Eqs. (1) and (2) only for
All results presented in our paper were obtained with a heightdependent turbulent length scale L_{n} given by Eq. (C2) of appendix C, the vertical profile of which is determined by the height of the atmospheric boundary layer (ABL). In this formulation, there is no direct influence of the TKE on L_{n}, which simplifies the analysis of the spurious ∼2Δt oscillations.
The expressions for the TKE and the TTE exchange coefficients
b. Discretization aspects
3. Spurious ∼2Δt oscillations in the twoenergy scheme
Experimentation with the twoenergy scheme revealed that it tends to generate spurious ∼2Δt oscillations in stable atmospheric conditions. The oscillations appear in the prognostic TKE and TTE, subsequently affecting the flux Richardson number, the turbulent exchange coefficients and the turbulent fluxes, while momentum and thermodynamic variables are influenced only weakly. We demonstrate the nature of the oscillations in idealized conditions and in NWP conditions.
In all presented experiments, the TOUCANS turbulence scheme used the twoenergy option, and the socalled model II of BD14. The parameterization of the thirdorder moments was switched off for simplicity. Closure constants and other tunings of the TOUCANS scheme, including numerical thresholds, are summarized in Tables C1–C3 of appendix C.
a. Idealized simulations
Idealized simulations were performed with the single column model (SCM) setup of the Integrated Forecast System (IFS), namely, with the OpenIFS (ECMWF 2020), which is a portable version of the European Centre for MediumRange Weather Forecasts (ECMWF) model IFS. The code used in this paper is based on the IFS model cycle 40r1 (ECMWF 2014), which already contains the core TOUCANS subroutines. The interface that enables us to call the TOUCANS scheme from the IFS physics has been added for the purpose of this paper. Also, the TOUCANS implementation was updated according to the description in section 2.
The oscillatory behavior of the model was analyzed in stable conditions, which were suspected to be necessary for the generation of the oscillations. The chosen case was developed by the Global Energy and Water Exchanges project (GEWEX), and it is based on the GEWEX Atmospheric Boundary Layer Study (GABLS) project (Holtslag 2006; Beare et al. 2006; Cuxart et al. 2006). The case assumes dry atmosphere. The initial state is neutrally stratified up to the height of 100 m, with a constant potential temperature profile of 265 K. Higher up, the initial stratification is stable, with a potential temperature increase of 0.01 K m^{−1}. Boundary layer is driven by an imposed geostrophic wind of 8 m s^{−1} (the latitude is assumed to be 73°), the surface pressure is kept at constant value of 1013.2 hPa. The prescribed surface potential temperature starts from the initial value of 265 K and cools by 0.25 K h^{−1}. The roughness length for both momentum and heat is 0.1 m. The experiment design enables simulations without a radiation scheme, when the initially neutral layer stabilizes because of the prescribed surface cooling. The length of the simulations is 9 h.
We performed OpenIFS SCM simulations using hydrostatic dynamical core with the twotime level semiLagrangian scheme. The target timestep length was 90 s, corresponding to the ALARO1 configuration with a horizontal mesh size of 2.3 km. A reference solution was provided by a SCM simulation with timestep length of 1 s. All SCM simulations used 91 vertical levels with 17 levels in the lowest 2 km and 8 levels in the lowest 400 m, which was our region of interest. Given the case design, the OpenIFS SCM simulations used only the turbulence scheme; the remaining physical parameterizations were turned off.
The SCM simulations confirmed the generation of the ∼2Δt oscillations in stable stratification. The problem is illustrated in Figs. 2–4. Figure 2 shows evolution of selected quantities on the 87th full and half model levels (approximately 155 and 125 m above the ground). Configuration with a time step of 90 s (red) suffers from oscillations in the TKE, TTE, and especially in the turbulent heat flux. In contrast, the temperature evolution remains smooth. Shortening of the time step to 45 s (brown) does not remove these oscillations, it only reduces their amplitude. The reference solution with a time step of 1 s (dashed black) is oscillationfree.
Figure 3 illustrates that ∼2Δt oscillating modes have a smooth (nonoscillatory) vertical structure, which makes them different from the classical fibrillations. The differences in the vertical profiles of the turbulent heat flux between subsequent model time steps (Fig. 3a) reveal that these oscillations are most severe in the capping temperature inversion of the ABL (Fig. 3b). A more complete picture of ∼2Δt oscillations is provided by the verticaltime cross sections in Fig. 4, where the vertical coherence of the oscillations is evident.
b. NWP simulations
To demonstrate the problem also in NWP conditions, we performed a 3D test using the ALADIN model in its ALARO1 configuration. General description of the ALARO configuration can be found in Termonia et al. (2018) and Wang et al. (2018). Here we list only the key properties relevant for our work: The model is integrated using the nonhydrostatic fully elastic spectral dynamical kernel, with a vertical finitedifference discretization using the Lorenz grid. A twotime level temporal discretization with one iteration of the centeredimplicit scheme, combined with a semiLagrangian advection, allows for a time step equal to 90 s. Configuration of the TOUCANS turbulence scheme was the same as in the SCM simulations.
The 3D test was run on a domain covering Central Europe, with a horizontal mesh size of 2.3 km and 87 vertical levels, with 29 levels in the lowest 2 km. To verify the role of atmospheric stability in generating the oscillations, we have selected a 12h window starting at 0000 UTC 21 April 2020. Atmospheric conditions in the target area were characterized by few clouds, with a strongly stable ABL at night, which quickly destabilized after sunrise.
Figure 5 shows the turbulent heat flux on the 82nd model half level (∼130 m above the ground) after a 3h forecast. A large portion of the domain, for example, the whole of Poland, is covered by a shortscale noise resulting from the horizontally uncorrelated ∼2Δt oscillations. In the marked point, 12h evolutions of selected quantities are displayed in Fig. 13 by red lines (see section 5b). In this point, the 82nd model half level lies within the capping temperature inversion of the ABL. The ∼2Δt oscillations are most severe in the turbulent heat flux, but they disappear as soon as the turbulent heat flux becomes oriented upward (negative in our convention), i.e., when the local temperature stratification becomes unstable.
4. Proposed treatment of the ∼2Δt oscillations
It has been empirically shown that the ∼2Δt oscillations are generated if and only if the relaxation term
a. Simplified system displaying the 2Δt oscillations
b. Iterative treatment of the 2Δt oscillations in the simplified system
We will now examine the convergence of the solver Eqs. (40)–(41). Because of the linearity of the system, it is sufficient to check the convergence for eigenvectors of the matrix M. From Eq. (41) it follows that if Δx^{0} is directed along the eigenvector ξ_{j}, then the starting iteration Δx^{+(0)} preserves this direction. Equation (40) guarantees that all higher iterations Δx^{+(}^{n}^{)} are also directed along the eigenvector ξ_{j}. We can therefore introduce amplification factors
Properties of various temporal discretizations of the system in Eq. (30) are illustrated in Fig. 6, assuming the eigenvalue λ_{j} = 50. The desired implicit discretization (gray) is nonoscillatory, but it overestimates the exact response (black). Explicit discretization of the equilibrium term with β_{τ} = 1 (brown) yields oscillations for
Figure 8 demonstrates accuracy of the single iteration with δ = 0.25 (green) at the opposite end of the spectrum, namely, for eigenvalue λ_{j} = 2. For γ < 1 the amplification factor is remarkably close to the exact value (black). For
c. Iterative treatment of the 2Δt oscillations in the full system
In section 4b, we analyzed convergence properties of the iterative solver in Eq. (47) for a simplified linear system. We concluded that one corrective iteration of the solver with the implicitness factor δ close to 0.25 is optimal for suppressing the 2Δt oscillations. Using δ = 1 is not recommended, because it results in a narrow convergence interval, so that the desired implicit solution is not approached for the timestep lengths of interest. Using δ = 0.25 with more than one corrective iteration is also not recommended, since it makes the solution more prone to oscillations. In the following, we apply the solver procedure from section 4b on the nonlinear system in Eq. (17), omitting the advection term that is treated separately.
The update of the equilibrium energies is only partial, omitting recomputation of the turbulence length scale L_{n} in the exchange coefficients in Eq. (49) and recomputation of the wind shear S and the moist Brunt–Väisälä frequency N_{moist} in the source terms in Eq. (52). Such simplification is acceptable for the treatment of the ∼2Δt oscillations, since the oscillations are only weakly coupled to momentum and thermodynamic variables. Full update of the equilibrium energies would require a corrective iteration of a substantial part of the model time step, making the treatment rather costly.
The dissipation time scales τ_{k}_{}_{s} could be also updated by the
5. Performance in 1D and 3D models
In this section, the performance of the proposed oscillation treatment is demonstrated in both the SCM simulations and the 3D tests. The sensitivity of the treatment to changes in the timestep length and in the implicitness factor δ are particularly tested to verify the theoretical applicability limits of the method. Furthermore, the influence of the proposed treatment on the nonlocality of the turbulence scheme is analyzed.
a. 1D results
Figure 9 demonstrates that evolutions after the treatment are smooth, free from the spurious ∼2Δt oscillations, reasonably reproducing the reference solution with a time step of 1 s (dashed black). The treatment is effective even with a time step of 180 s (blue), where only weak oscillations in the TKE and in the turbulent heat flux start to be visible around hour 3. This indicates that the oscillation treatment gets close to its applicability limit. The treatment with a time step of 180 s also causes somewhat stronger drift from the reference solution than with a time step of 90 s (green). Similar observation holds for the vertical profiles shown in Fig. 10. It is important to notice that the treatment sharpens the capping temperature inversion of the ABL (Fig. 10b), and the effect is more pronounced with a longer time step. This is caused by the local downgradient estimation of the turbulent fluxes in the recomputation of the source terms at the time level t + Δt. We elaborate on this in section 5c. We also note that for a time step of 1 s the model sensitivity to δ disappears (not shown), confirming the convergence of our treatment for Δt → 0.
Overall accuracy of the proposed treatment with δ = 0.25 can be judged by comparing the verticaltime cross sections in Fig. 11 with the reference in Fig. 4. The simulations with a time step of 90 s are very close for β_{τ} = 1.5 (Figs. 11a,b) and β_{τ} = 1 (Figs. 11c,d), confirming that with the oscillation treatment, there is no need to use overimplicit discretization of the relaxation term. The target configuration with a time step of 90 s and β_{τ} = 1 (Figs. 11c,d) is oscillationfree, and it is reasonably close to the reference solution (see Figs. 4e,f). The simulation is able to capture the quick growth of the ABL between hours 1 and 2, but it does not produce its further growth to 300 m after the third hour.^{3} The simulation with a doubled timestep length (Figs. 11e,f) remains almost oscillationfree, but the ABL height prediction is less accurate, staying around 200 m most of the time. All simulations develop a deeper temperature minimum on the 87th model full level (∼155 m above the ground) than the reference solution. This is consistent with temperature profiles in Fig. 10.
Linear analysis performed in section 4b shows that the proposed oscillation treatment with δ = 0.25 extends the range of the nonoscillatory timestep lengths roughly 4 times. Bifurcation analysis performed in appendix D shows that there is an additional gain of stability in the nonlinear case, accompanied with a drift that is, however, acceptably small. The analysis explains our experimental finding that the proposed treatment is stable (nonoscillatory) up to the timestep lengths of ∼180 s.
b. 3D results
A final test of the proposed treatment of the ∼2Δt oscillations must demonstrate its proper functionality in the 3D model with full physics. It must also confirm that the treatment has affordable computational cost.
Figure 12 shows the turbulent heat flux on the 82nd model half level (∼130 m above the ground) after a 3h forecast obtained with our treatment using β_{τ} = 1 and δ = 0.25. Relative to Fig. 5 the noise is greatly reduced, remaining only over a few hardly noticeable localized areas.
Figure 13 shows the model evolution at a selected grid point on the 82nd model full and half levels (approximately 150 and 130 m above the ground). The treatment with β_{τ} = 1 and δ = 0.25 (green) lies mostly within the envelope given by the oscillating original solution (red). The 1 K drift of the temperature, which leads to an inversion sharpening, is an exception. Lowering the implicitness factor to δ = 0.2 (yellow) reduces the temperature drift. On the other hand it is not sufficient to fully suppress the 2Δt oscillations, which visibly contaminate the turbulent heat flux. These oscillations are damped, but they are also continually generated, as discussed in appendix D. They disappear for δ = 0.25, because this setting yields essentially nonnegative linear amplification factor.
Figure 14 confirms conclusions resulting from the idealized GABLS1 case: (i) the vertical structure of the ∼2Δt oscillating modes is smooth (Fig. 14a); and (ii) the treatment with δ > 0 sharpens the capping temperature inversion of the ABL (Fig. 14b). The latter effect follows from the reduced nonlocality of the twoenergy scheme, discussed in section 5c. We found this side effect to be beneficial for the ALARO1 model configuration, alleviating the problem of tooearlyeroded stable layers.
Based on these results, we propose to use a single corrective iteration of the TKE and the TTE solvers with the implicitness factors β_{τ} = 1 and δ = 0.25 to eliminate the ∼2Δt oscillations that contaminate the solution obtained with the original implementation of the twoenergy scheme. The proposed treatment increases the computational cost of the full 3D model by ∼0.15%, which is negligible.
c. Impact on the nonlocality of the turbulence scheme
Figures 10 and 14 reveal that a sideeffect of the proposed ∼2Δt oscillation treatment are sharper temperature inversions. This is a consequence of updating the turbulent fluxes in the t + Δt estimates of the source terms using the local downgradient relations in Eqs. (50) and (51). The turbulent fluxes are otherwise calculated by the vertical diffusion solver, which uses the explicit exchange coefficient and the implicit diffused variable. Therefore, the solver can propagate local forcing across several layers within a single time step, leading to nonlocality of the resulting turbulent fluxes.^{4} Updating the turbulent fluxes by Eqs. (50) and (51) does the opposite, using the implicit exchange coefficient and the explicit diffused variable. This is why the updated equilibrium energies in Eq. (53) are local, evaluated from the turbulent fluxes proportional to the local gradients. Weighting Eq. (54) then combines the nonlocal equilibrium energies
As was demonstrated in BD18, the nonlocality of the twoenergy scheme is beneficial in unstable stratification, e.g., for developing shallow convection. In stable stratification the twoenergy scheme has the tendency to overestimate the penetration of the stable layers, coming from a too intense nonlocal mixing. For this reason, decreased nonlocality due to the ∼2Δt oscillation treatment is a welcome effect, improving the performance in stable stratification without much influence on the unstable one.^{5} To illustrate the effect, we have picked a case with persisting temperature inversion, and compared the model profiles after 6h integration with an atmospheric sounding. As can be seen in Fig. 15, the measured temperature (black) increases abruptly by ∼12°C in the inversion layer. The model is unable to represent such sharp temperature gradient with or without our ∼2Δt oscillation treatment. However, the inversion with the oscillation treatment (green) is more pronounced than without it (red).
Because of the beneficial impact of the proposed ∼2Δt oscillation treatment on stable layers, there is no immediate need to decouple the treatment from the nonlocality aspect. Should such need arise, a fully nonlocal treatment of the ∼2Δt oscillations could be achieved by replacing Eqs. (50) and (51) with a vertical diffusion solver for the momentum and the moist variables, using the updated exchange coefficients given by Eq. (49). The degree of the nonlocality of the twoenergy scheme could then be controlled independently by weighting the flux Richardson number between the nonlocal value obtained from the prognostic TKE and TTE, and the local value inverted from the local gradient Richardson number. Investigation of this possibility is, however, out of scope of this paper.
6. Conclusions
The presented work is devoted to the improvement of the twoenergy configuration of the TOUCANS turbulence scheme in stable stratification. While the twoenergy scheme has better performance than the traditional TKE schemes in most situations, it tends to generate spurious oscillations in stable stratification. Evidence can be found not only in realcase 3D experiments, but also in idealized 1D simulations.
We have analyzed the origin and properties of these new type of oscillations. Because of their high frequency, the oscillations resemble the “classical” fibrillations, which are caused by the coupling between turbulent exchange coefficients and the stability parameter. However, the oscillations in the twoenergy scheme are generated for a different reason: an implicitexplicit temporal discretization of the relaxation terms. Note that the relaxation technique is used on source and dissipation terms in prognostic turbulence energy equations to ensure numerical stability for relatively long time steps.
Using a simplified system with relaxation, we have shown that the generation of the ∼2Δt oscillations is associated with a sequence of period doubling bifurcations. This process starts at particular timestep length, which depends on the local gradient Richardson number.
To suppress the spurious oscillations, we have proposed a treatment with a single corrective iteration of the TKE and the TTE solvers, making discretization of the relaxation terms more implicit. The linear stability analysis and the bifurcation analysis show that the proposed treatment is effective, increasing the timestep range for the stable (nonoscillatory) solutions more than 4 times. Experiments with idealized 1D and full 3D models confirm that for operationally used timestep lengths, the oscillations are efficiently suppressed with negligible computational overhead. The proposed treatment already entered operations at Czech Hydrometeorological Institute, where ALARO1 configuration is run with a horizontal mesh size of 2.3 km and a time step of 90 s.
There are two interesting aspects related to our treatment of the ∼2Δt oscillations in the twoenergy scheme that were not covered in this paper. First, implementation of a fully nonlocal oscillation treatment would enable to suppress the oscillations without affecting the nonlocal character of the scheme. The strength of the nonlocal mixing could be controlled independently from the oscillation treatment, via a modification of the flux Richardson number. The local value, obtained from local gradients, and the nonlocal value, obtained from the prognostic energies, would be blended together using a tunable blending ratio. Second, it is tempting to develop a fully implicit discretization of the relaxation terms, which would remove the oscillations unconditionally. Due to the nonlinearity of the problem, an iterative approach with suitable preconditioner would have to be found. These two aspects will be the subject of further research.
A particular result of Kalnay and Kanamitsu (1988) is that, for relaxation coefficient proportional to the square root of the prognostic variable (2/τ ∝
The dissipation time scales in the equilibrium terms
Here we adopt an ad hoc definition of the ABL top as the level at which the turbulent heat flux falls under 0.2 W m^{−2} (i.e. where blue and white colors meet).
This part of nonlocality in TOUCANS is a discretization aspect. It vanishes for the short Δt, when the vertical interaction distance
Weaker sensitivity in unstable stratification is given by the shape of stability functions, which saturate quickly for negative flux Richardson numbers.
Equilibrium TKE is obtained from Eq. (18), by setting
Such matching is approximate when
This follows from the fact that the ratios τ_{s}/τ_{k},
Acknowledgments.
We are grateful to Luc Gerard for reporting the problem with the twoenergy scheme, as well as for initial help with analyzing its likely causes. We thank also Filip Váňa for enlightening of the stabilization technique developed for the pseudoTKE solver, inherited by the twoenergy scheme. We also thank three anonymous reviewers for their really detailed comments and valuable suggestions, considerably improving our paper. Radmila Brožková and Ján Mašek thank the Technology Agency of the Czech Republic for its financial support under Grant SS02030040, Prediction, Evaluation and Research for Understanding National Sensitivity and Impacts of Drought and Climate Change for Czechia (PERUN). Ivan Bašták Ďurán was supported by Hans Ertel Centre for Weather Research of DWD Grant 4818DWDP4 (third phase: the Atmospheric Boundary Layer in Numerical Weather Prediction).
Data availability statement.
Output from SCM simulations in the NetCDF format is accessible via Zenodo (https://doi.org/10.5281/zenodo.5052629). For information about the data from the 3D model simulations, please contact the corresponding author.
APPENDIX A
Derivation of Eddy Diffusivity and Molecular Dissipation Length Scales
APPENDIX B
Realizability Constraints and Numerical Protections
The twoenergy scheme is a subject to three realizability constraints. They must be ensured by the numerical implementation so as to get a physically meaningful scheme.
The first realizability constraint results from the physical definition of the turbulence energies. Both e_{k} and e_{s} are by definition nonnegative (see, e.g., Zilitinkevich et al. 2013). The prognostic Eqs. (1) and (3) are consistent with this constraint. In a continuous formulation, the TKE transport processes (advection and vertical turbulent transport) cannot produce negative e_{k} when it is nonnegative in the initial and boundary conditions. Sum I + II can be negative in very stable situations when the TKE buoyant destruction −II exceeds its shear production I. However, it cannot reverse the sign of e_{k} since it vanishes for e_{k} → 0. The same is true for the TKE dissipation term.
Practical implementation of the realizability constraints in Eqs. (B1)–(B3) is as follows:

The initial prognostic energies
$({e}_{k}^{0},{e}_{s}^{0})$ are limited to positive values (≥e_{min}) to suppress eventual numerical undershoots coming from the semiLagrangian advection scheme. 
The ratio
$r={e}_{s}^{0}/{e}_{k}^{0}$ is truncated to fit interval [r_{min}, r_{max}], ensuring that the constraints (B2) and (B3) are respected. However, the prognostic energies remain untouched. In practice, it is convenient to choose the limit values${\text{Ri}}_{f}^{\text{min}}$ and${\text{Ri}}_{f}^{\text{max}}$ for the flux Richardson number within the interval$(\infty ,{\text{Ri}}_{f}^{\text{crit}})$ , and to convert them to r_{min} and r_{max} by inverting Eq. (7). Both sets of limit values, used in our experiments, are given in Table C2 of appendix C. 
Provisional value of the flux Richardson number
${\text{Ri}}_{f}^{\text{prov}}$ is calculated by Eq. (7), using the truncated r. 
To prevent “jumpiness” when turbulence has negligible intensity,
${\text{Ri}}_{f}^{\text{prov}}$ is pushed toward${\text{Ri}}_{f}^{\text{max}}$ when both e_{k} and e_{s} are small, yielding final value Ri_{f} of the flux Richardson number:${\text{Ri}}_{f}=w{\text{Ri}}_{f}^{\text{max}}+(1w){\text{Ri}}_{f}^{\text{prov}},$$w=\frac{{({e}_{\text{crit}}{e}_{\text{min}})}^{2}}{{({e}_{\text{crit}}{e}_{\text{min}})}^{2}+{({e}_{k}{e}_{\text{min}})}^{2}+{({e}_{s}{e}_{\text{min}})}^{2}},\text{\hspace{1em}and}$${e}_{\text{crit}}={10}^{7}\hspace{0.17em}\text{\hspace{0.05em}}J\hspace{0.17em}{\text{kg}}^{1}.$Protection of the flux Richardson number from the “jumpiness” prevents artificial generation of turbulence due to numerical inaccuracy. The weight w is therefore equal to 1 when e_{k} = e_{s} = e_{min}, and it is practically zero when any of the turbulence energies is at least one order of magnitude greater than the critical value e_{crit}. 
The protected flux Richardson number Ri_{f} is used in the evaluation of the stability functions χ_{3} and ϕ_{3}, as well as in Eq. (8), which delivers the TTE dissipation time scale τ_{s}.

New values of
$\left({e}_{k}^{+},{e}_{s}^{+}\right)$ are calculated by solving Eq. (19) with the limitation in (B1). Prognostic equations for both TKE and TTE are solved in the same way. 
The new values of
$\left({e}_{k}^{+},{e}_{s}^{+}\right)$ are advected by the semiLagrangian scheme, yielding initial values for the next time step.
We stress that the constraints in Eqs. (B2) and (B3) on the ratio r, limiting the deeper blue/red region in Fig. B1, do not apply to the prognostic energies (e_{k}, e_{s}). The prognostic energies can occupy the whole quadrant allowed by the realizability constraint in Eq. (B1). Enforcing the constraints in Eqs. (B2) and (B3) by modification of the prognostic energies would be prone to “jumpiness,” perturbing their smooth evolution.
APPENDIX C
Used Settings of the TOUCANS Turbulence Scheme
Numerical thresholds used in this paper. The lower limit for Ri_{f} is unnecessarily strict; experiments could have been run with
Used tunings of the Geleyn–Cedilnik turbulence length scale L_{n}. Values used in OpenIFS SCM follow from Bašták Ďurán et al. (2020).
APPENDIX D
Bifurcation Analysis of the Nonlinear System
Analysis performed in sections 4a and 4b raises an important question: What happens with the numerical solution of the nonlinear relaxation problem in Eq. (27) when the analytical fixed point x^{∞} loses linear stability in a given discretization, that is, when the absolute value of the amplification factor exceeds 1? In such case, qualitative behavior of the numerical solution is determined by the shape of
Using the relation between the gradient Richardson number and the system eigenvalues, we can set up a numerical experiment with λ_{1} = 50 as in Fig. 6. According to Fig. D1, the gradient Richardson number Ri = 1.58 must be set, corresponding to the flux Richardson number
Limit solutions of the nonlinear system in Eq. (D1), discretized in the original and the newly proposed ways, can be visualized using the socalled bifurcation diagrams presented in Fig. D2. For each nondimensional timestep length γ, the system was integrated numerically from initial conditions
According to Fig. 6, the analytical fixed point in the original discretization with β_{τ} = 1.5 and δ = 0 should remain stable until γ ≈ 0.043, where the amplification factor reaches the value of −1. This is confirmed by the left column in Fig. D2, where the period doubling bifurcation creates a stable limit cycle of length 2Δt at γ ≈ 0.042 [for an explanation of the elements of bifurcation theory, please consult Hale and Koçak (1991)]. Further increase of the timestep length leads to a sequence of the period doubling bifurcations, generating the stable limit cycles of lengths 4Δt, 8Δt, 16Δt, …, eventually resulting in a nonperiodic behavior (chaos). A common feature of the limit solutions beyond γ ≈ 0.042 is the presence of significant 2Δt component, which is the most striking characteristics of the observed oscillations. The amplitude of the oscillations remains finite, and it increases with the timestep length.
The situation for the treatment with β_{τ} = 1 and δ = 0.25 is different. According to Fig. 6, the analytical fixed point loses stability at γ ≈ 0.089, where the amplification factor reaches the value of 1. Therefore, we can expect that for the longer time steps the limit solution will not oscillate, but it will be drifted from the analytical fixed point. The right column in Fig. D2 confirms that this is the case, although for the given initial conditions the drift starts earlier at γ ≈ 0.072. It means that around this timestep length a new stable fixed point is created, drifted by few percent from the analytical fixed point. The diagrams show that this new fixed point shifts with the timestep length and it remains stable until γ ≈ 0.138, where the period doubling bifurcation creates a stable limit cycle of length 2Δt. The rest of scenario is the same as for the original discretization. In both cases, the relative drift of the oscillating limit solution is stronger for the TKE, which is mostly underestimated, than for the TTE, where the overestimation prevails (with respect to the analytical fixed point).
We complete the picture by examining the TKE and the TTE evolution during the transition toward the limit solution. They are shown in Fig. D3 for the timestep lengths indicated in the bifurcation diagrams in Fig. D2. The reference solution (dashed black) is provided by the original discretization with a short time step γ = 0.01. After initial spinup, it converges to the analytical fixed point monotonically. Using a longer time step γ = 0.1 (red) leads to the undamped ∼2Δt oscillations, with the mean value drifted from the analytical fixed point. With the same timestep length, the proposed treatment (green) yields a nonoscillatory solution, converging to the new fixed point, with a drift from the analytical fixed point comparable to the original discretization (red). With the time step shortened to γ = 0.07 (brown) the drift disappears, and the solution remains very close to the reference. On the other hand, the time step γ = 0.135 (blue), which is near the edge of drifted region in Fig. D2, generates the damped 2Δt oscillations. This is because the first period doubling bifurcation, creating undamped oscillations with the period of 2Δt, occurs when the amplification factor passes through the value of −1.
Experiments performed with the 3D model confirm that although the damped oscillations decay with time, they can visibly contaminate the solution. Possible causes are the evolving gradient Richardson number, interaction with the vertical diffusion and the advection terms, or the vertical staggering of the prognostic and equilibrium turbulence energies. All these aspects, not included in our simplified system in Eq. (D1), displace the model out of the fixed point. The relaxation term is trying to push it back, generating the damped 2Δt oscillations continually. For this reason it does not make sense to use the implicitness factor δ < 0.25, since the amplification factor in the analytical fixed point would become negative for the intermediate timestep lengths (for δ = 0.25 this happens only marginally, as can be seen in Fig. 6). Using
Bifurcation analysis performed in this appendix revealed that the numerical solution of our nonlinear relaxation problem remains bounded after the analytical fixed point x^{∞} loses linear stability. The subsequent scenario depends on the employed discretization. For the original discretization, the 2Δt oscillations appear immediately via the period doubling bifurcation. For the proposed treatment, the loss of linear stability is delayed, and there is an intermediate stage with a new stable fixed point, drifted from the analytical one. The new fixed point finally loses stability via the period doubling bifurcation. Both discretizations are thus conditionally stable (nonoscillatory), but the stability range for the proposed treatment is wider. The outlined picture, obtained for the dominant eigenvalue λ_{1} = 50, remains qualitatively valid as long as λ_{1} ≫ 1. For smaller λ_{1} the ∼2Δt oscillations are not a problem in practice, since in this case they appear for timestep lengths longer than what is allowed by the model dynamics.
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