Editorial Type:
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Article Type:
Research Article

Semi-Implicit Computation of Fast Modes in a Scheme Integrating Slow Modes by a Leapfrog Method Based on a Selective Implicit Time Filter

Mohamed Moustaoui aSchool of Mathematical and Statistical Sciences, Arizona State University, Tempe, Arizona

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Bryce M. Barclay aSchool of Mathematical and Statistical Sciences, Arizona State University, Tempe, Arizona

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Eric J. Kostelich aSchool of Mathematical and Statistical Sciences, Arizona State University, Tempe, Arizona

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Online Publication:
06 Dec 2023
Print Publication:
01 Dec 2023
DOI:
https://doi.org/10.1175/MWR-D-22-0311.1
Page(s):
3133–3149
Free access

Abstract

A scheme for integration of atmospheric equations containing terms with differing time scales is developed. The method employs a filtered leapfrog scheme utilizing a fourth-order implicit time filter with one function evaluation per time step to compute slow-propagating phenomena such as advection and rotation. The terms involving fast-propagating modes are handled implicitly with an unconditionally stable method that permits application of larger time steps and faster computations compared to fully explicit treatment. Implementation using explicit and recurrent formulation is provided. Stability analysis demonstrates that the method is conditionally stable for any combination of frequencies involved in the slow and fast terms as they approach the origin. The implicit filter used in the method damps the computational modes without noticeably sacrificing the accuracy of the physical mode. The O[(Δt4)] accuracy for amplitude errors achieved by the implicitly filtered leapfrog is preserved in applications where terms responsible for fast propagation are integrated with a semi-implicit method. Detailed formulation of the method for soundproof nonhydrostatic anelastic equations is provided. Procedures for implementation in global spectral shallow-water models are also given. Examples comparing numerical and analytical solutions for linear gravity waves demonstrate the accuracy of the scheme. The performance is also shown in more practical nonlinear applications, where numerical solutions accomplished by the method are evaluated against those computed from a scheme where the slow terms are handled by the third-order Runge–Kutta scheme. It demonstrates that the method is able to accurately resolve fine-scale dynamics of Kelvin–Helmholtz shear instabilities, the evolution of density current, and nonlinear drifts of twin tropical cyclones.

© 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: Mohamed Moustaoui, Mohamed.Moustaoui@asu.edu

Abstract

A scheme for integration of atmospheric equations containing terms with differing time scales is developed. The method employs a filtered leapfrog scheme utilizing a fourth-order implicit time filter with one function evaluation per time step to compute slow-propagating phenomena such as advection and rotation. The terms involving fast-propagating modes are handled implicitly with an unconditionally stable method that permits application of larger time steps and faster computations compared to fully explicit treatment. Implementation using explicit and recurrent formulation is provided. Stability analysis demonstrates that the method is conditionally stable for any combination of frequencies involved in the slow and fast terms as they approach the origin. The implicit filter used in the method damps the computational modes without noticeably sacrificing the accuracy of the physical mode. The O[(Δt4)] accuracy for amplitude errors achieved by the implicitly filtered leapfrog is preserved in applications where terms responsible for fast propagation are integrated with a semi-implicit method. Detailed formulation of the method for soundproof nonhydrostatic anelastic equations is provided. Procedures for implementation in global spectral shallow-water models are also given. Examples comparing numerical and analytical solutions for linear gravity waves demonstrate the accuracy of the scheme. The performance is also shown in more practical nonlinear applications, where numerical solutions accomplished by the method are evaluated against those computed from a scheme where the slow terms are handled by the third-order Runge–Kutta scheme. It demonstrates that the method is able to accurately resolve fine-scale dynamics of Kelvin–Helmholtz shear instabilities, the evolution of density current, and nonlinear drifts of twin tropical cyclones.

© 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: Mohamed Moustaoui, Mohamed.Moustaoui@asu.edu
Keywords: Anelastic models; Mesoscale models; Nonhydrostatic models; Numerical analysis/modeling

1. Introduction

The solutions to the equations governing atmospheric dynamics contain physical phenomena with differing temporal scales. Small-scale processes include acoustic modes and fast-propagating gravity waves generated when buoyancy forces tend to restore equilibrium. Larger-scale phenomena, such as those influenced by rotation and advection, have slower temporal variations. The acoustic modes can be removed in atmospheric models by employing soundproof approximations such as the weakly compressible anelastic (Ogura and Phillips 1962; Lipps and Hemler 1982) and the pseudoincompressible equations (Durran 1989, 2008). Semi-implicit soundproof solvers have been developed and utilized as a basis for simulating all-scale atmospheric dynamics (Smolarkiewicz et al. 2014; Kurowski et al. 2013; Wood et al. 2014; Chew et al. 2022; Thuburn 2017; Benacchio and Klein 2019).

Numerical integration of atmospheric equations requires utilization of small time steps to ensure stable integration of terms responsible for fast-evolving processes. To circumvent this limitation, most atmospheric solvers combine explicit and implicit schemes and treat the fast-propagating processes implicitly to allow feasible numerical integrations with practical Courant numbers. The explicit component in these solvers may be handled by Runge–Kutta schemes (e.g., Weller et al. 2013; Skamarock and Klemp 2008; Skamarock et al. 2012; Mahalov and Moustaoui 2009). Other semi-implicit schemes, based on families of Adams and backward differencing methods, have been proposed for integration of fast-wave–slow-wave problems (Durran and Blossey 2012). The ability of these schemes to achieve efficient solutions and their comparison with the leapfrog-trapezoidal implicit-explicit method were evaluated for nonlinear gravity waves simulated in a nonhydrostatic atmospheric model (Durran and Blossey 2012).

The leapfrog scheme is attractive and efficient, as it is a noniterative method that requires only one function evaluation per time step. However, it is well known that the method develops computational modes that are unconditionally unstable for diffusion equations and could interact with the physical mode in nonlinear problems (Durran 1991, 1999). Applying a second-order time filter to the leapfrog scheme can damp the computational modes (Robert 1966; Asselin 1972). On the other hand, this filter degrades the accuracy of the scheme to first order. Leapfrog schemes using higher-order time filters with less degradation of the physical mode have been recently proposed (Williams 2013; Moustaoui et al. 2014; Amezcua and Williams 2015; Yazgi et al. 2017). The schemes developed in Moustaoui et al. (2014) utilize higher-order implicit time filters that reduce computational modes without reducing the accuracy of the physical mode to first order. They are conditionally stable for any filter coefficient, employ one function evaluation per time step, have a wider region of stability compared to other filtered leapfrog schemes, and are able to control instabilities of computational modes in diffusion problems and in nonlinear atmospheric applications (Moustaoui et al. 2014).

In this paper, we propose a method that can be employed for numerical integration of atmospheric models where the equations solved can be separated into terms involving fast-propagating phenomena and terms containing contributions with slow propagation. The terms responsible for fast propagation are handled by the unconditionally stable trapezoidal implicit method, while the slow terms will be integrated by the time-filtered leapfrog scheme presented in Moustaoui et al. (2014). The method proposed here is doubly implicit because of the trapezoidal implicit scheme employed for the fast terms and the implicit filter employed in leapfrog for the slow contributions. We analyze the behavior of the method, provide its detailed formulation for soundproof nonhydrostatic anelastic equations, and give procedures for its implementation in global shallow-water spectral models. The performance of the method is shown in practical nonlinear atmospheric and fluid applications.

Section 2 presents the formulation of the method. Section 3 examines its behavior. Section 4 provides detailed numerical formulations and applications for nonhydrostatic anelastic equations and global shallow-water spectral models. The conclusions are given in section 5.

2. Formulation

Consider the differential equation ∂ψ/∂t = G(ψ) + F(ψ), where G(ψ) includes terms responsible for fast-propagating or diffusion processes while F(ψ) contains other terms involving slow propagation. Suppose that the terms in G(ψ) are compatible with implicit treatment. The formulation of the scheme proposed in this paper has the following form:
ψn+1−ψ¯n−12Δt=G(ψn+1)+G(ψ¯n−1)2+F(ψn),
ψ¯n−1=ψn−1+γ(−ψ¯n−3+4ψ¯n−2−6ψ¯n−1+4ψn−ψn+1),
where ψn is the approximation to the solution at time nΔt and ψ¯n is the solution after applying a fourth-order implicit time filter using a real constant γ. This scheme uses an implicit filter because both ψ¯n−1 and ψn+1 in Eqs. (1) and (2) are not known and depend on each other. If we assume that G is linear in ψ, in which case the trapezoidal implicit method can be applied without iteration, then [G(ψn+1)+G(ψ¯n−1)]/2 can be written in the following form:
G(ψn+1)+G(ψ¯n−1)2=G(ψn+1+ψ¯n−12).
The filtered field ψ¯n−1 can be eliminated from Eq. (1) by using the following relations:
ψn+1−ψ¯n−1=(1+7γ)ψn+1−ψ˜n−11+6γ,
ψn+1+ψ¯n−1=(1+5γ)ψn+1+ψ˜n−11+6γ,
which can be obtained from Eq. (2). In these relations, the known fields of ψ at stage nΔt are combined into ψ˜n−1, where
ψ˜n−1=ψn−1+γ(−ψ¯n−3+4ψ¯n−2+4ψn).
From Eq. (1), the equation satisfied by ψn+1 becomes
(1+7γ)ψn+1−Δt(1+5γ)G(ψn+1)=ψ˜n−1+ΔtG(ψ˜n−1)+2Δt(1+6γ)F(ψn),
where the terms on the right-hand side are already computed. This equation can be converted to a linear algebraic equation for ψn+1 depending on the form of G. It can be solved with appropriate boundary conditions. In some problems (e.g., see examples in section 4), G is in, or can be transformed through basis expansions into, the simple form G(ψ) = gψ, where g is a function of some of the problem parameters. In this case, Eq. (6) can be used to derive an explicit expression for ψn+1, and the proposed scheme can be formulated recursively as follows:
ψn+1=(1+gΔt)ψ˜n−1+2Δt(1+6γ)F(ψn)(1+7γ)−gΔt(1+5γ),
ψ¯n−1=ψ˜n−1−γψn+1(1+6γ),
ψ˜n=ψn+γ(−ψ¯n−2+4ψ¯n−1+4ψn+1).
The fields of ψ required to solve Eqs. (7)–(9) in this formulation are ψn, ψ˜n−1, and ψ¯n−2. At the end of each time step, the storage arrays occupied by these fields can be overwritten by ψn+1, ψ˜n, and ψ¯n−1. Thus, the method can be coded such that only three fields of ψ are stored per time step (here, a small temporary storage is employed).

3. Stability analysis and convergence tests

a. Stability analysis

The stability and accuracy of the method are examined by considering the oscillation equation:
dψ/dt=jωfψ+jωsψ,
where ωf and ωs are real numbers representing frequencies of fast and slow modes, respectively, and j=−1. The exact solution to the oscillation equation between the time steps tn and tn+1 = tn +Δt is ψn+1=ψnexp⁡[j(ωf+ωs)Δt]. Hence, the exact amplification factor is Ae=exp⁡[j(ωf+ωs)Δt]. If we apply the scheme [Eqs. (1) and (2)] to the oscillation equation [Eq. (10)], where the fast mode is integrated with the trapezoidal implicit method, G[(ψn+1+ψ¯n−1)/2]=jωf[(ψn+1+ψ¯n−1)/2], and the slow mode is treated with the explicit time-filtered leapfrog, F(ψn)=jωsψn, then Eq. (1) becomes
ψ¯n−1=1−jωfΔt1+jωfΔtψn+1−2jωsΔt1+jωfΔtψn.
The relations satisfied by the numerical amplification factor A, which is defined by ψn+1 = Aψn (or ψ¯n+1=Aψ¯n), can then be derived from Eqs. (2) and (11). These are as follows:
ψ¯ψ=1−jωfΔt1+jωfΔtA2−2jωsΔt1+jωfΔtA,
A2ψ¯ψ=A2+γ4(−ψ¯ψ+4Aψ¯ψ−6A2ψ¯ψ+4A3−A4).
After eliminating ψ¯/ψ from these two equations, one can derive a cubic equation satisfied by the amplification factor A:
a3A3+a2A2+a1A+a0=0,
where
a3=1−jωfΔt1+jωfΔt(6γ+1)+γ,
a2=−8γ+2jωsΔt(6γ+1)1+jωfΔt,
a1=γ1−jωfΔt+8jωsΔt1+jωfΔt−1,
a0=−2γjωsΔt1+jωfΔt.
Equation (14) has three roots corresponding to one physical mode that determines the stability of the scheme and two stable computational modes. It is numerically solved for the frequencies (ωfΔt, ωsΔt) in the two-dimensional range [−10, 10] × [−1, 1] using filter coefficients of γ = 0.03 and γ = 0.06. Figure 1 shows the field of the magnitude of the amplification factor corresponding to the physical mode. It indicates that the scheme is conditionally stable for all values of (ωfΔt, ωsΔt) as they approach (0, 0) and that the magnitude of the amplification factor is close to 1 near the origin. Figure 1 also shows that the scheme is being stabilized as the fast frequency dominates. For instance, the scheme is unstable when ωsΔt = 1 and ωfΔt ≈ 0. But as the magnitude of ωfΔt increases, the scheme becomes stable for ωsΔt = 1. The stability of the proposed scheme is restricted by its explicit component.
Fig. 1.
Fig. 1.

Magnitude of the numerical amplification factor corresponding to the physical mode from the proposed scheme as a function of the frequencies of fast ωfΔt and slow ωsΔt modes. The coefficient of the time filter is (a) γ = 0.03 and (b) γ = 0.06.

Citation: Monthly Weather Review 151, 12; 10.1175/MWR-D-22-0311.1

The behavior of the scheme can be analyzed by examining the dependence on the frequencies ωfΔt and ωsΔt as they approach (0, 0) of the magnitude and phase of the amplification factor for the physical mode and damping of the computational modes. The analysis is investigated for γ = 0.03 along lines with slopes defined by an angle Θ with respect to the horizontal axis in Fig. 1. These lines are given parametrically in the (ωfΔt, ωsΔt) plane by the equations ωfΔt = ωΔt cosΘ and ωsΔt = ωΔt sinΘ, where ω is a real parameter. Here, 0 < Θ < π/2 and π/2 < Θ < π represent the cases where the fast and slow modes are propagating in the same and opposite directions, respectively. The value of Θ = 0 corresponds to a purely trapezoidal implicit scheme (ωs = 0, no slow modes), and the value of Θ = π/2 corresponds to a purely time-filtered leapfrog scheme (ωf = 0, no fast modes).

Figure 2a shows the magnitudes of the amplification factors corresponding to the physical and computational modes for finite values of ωΔt obtained from the scheme with the value of Θ = π/8. The argument of the physical mode is shown in Fig. 2b. The magnitude of the exact amplification factor |Ae| = 1 and its argument Arg(Ae) = (ωf + ωs)Δt = (cosΘ + sinΘ)ωΔt are also superimposed in Fig. 2. For this value, both the fast and slow physical modes propagate in the same direction with ωf > ωs. The physical and computational modes are stable for all the values of ωΔt shown in Fig. 2a. One of the computational modes is strongly damped and vanishes when ωΔt = 0. The argument of the physical mode shows that the scheme produces deceleration relative to the exact argument (Fig. 2b). This behavior is also found in the case corresponding to Θ = 2π/8, where the frequency of the fast and slow modes is equal: ωf = ωs (Figs. 2c,d). In this case, the scheme produces less pronounced deceleration of the phase of the physical mode (Fig. 2d) compared to the phase found for Θ = π/8. The behavior of the scheme in the case corresponding to Θ = 3π/8 where ωf < ωs is shown in Figs. 2e and 2f. The physical and computational modes are both stable (Fig. 2e) and the scheme now produces acceleration of the physical mode (Fig. 2f). We note from Fig. 2 that both the computational modes are damped. The phase of the scheme becomes larger than the exact response as ωf decreases relative to ωs.

Fig. 2.
Fig. 2.

(a) Magnitude of the amplification factors for the physical mode (solid), the 2Δt (long dashed–dashed) computational mode, and the 4Δt (dotted) computational mode as a function of ωΔt obtained for γ = 0.03 along a line with slope Θ = π/8 in Fig. 1. (b) Argument of the physical mode (solid). The dashed lines in (a) and (b) represent the magnitude and argument of the exact amplification factor. (c),(d) As in (a) and (b), but the slope is Θ = π/4. (e),(f) As in (a) and (b), but the slope is Θ = 3π/8.

Citation: Monthly Weather Review 151, 12; 10.1175/MWR-D-22-0311.1

Figure 3 shows the magnitudes of the amplification factors corresponding to the physical and computational modes and the argument of the physical mode for finite values of ωΔt obtained for the values Θ = 5π/8, 6π/4, and 7π/8. In this case, the fast and slow physical waves propagate in opposite directions. As in Fig. 2, the computational modes are damped, the physical mode is stable, and the phase produced depends on the relative magnitudes of ωf and ωs. Here, the magnitude of the amplification factor for the physical mode is much closer to one (Figs. 3a,c,e). The response of the scheme for the physical mode equals the exact amplification factor when the fast and slow modes have equal magnitudes: Θ = 3π/4, |ωf| = |ωs| (Figs. 3c,d).

Fig. 3.
Fig. 3.

(a) Magnitude of the amplification factors for the physical mode (solid), the 2Δt (long dashed–dashed) computational mode, and the 4Δt (dotted) computational mode as a function of ωΔt obtained for γ = 0.03 along a line with slope Θ = 5π/8 in Fig. 1. (b) Argument of the physical mode (solid). The dashed lines in (a) and (b) represent the magnitude and argument of the exact amplification factor. (c),(d) As in (a) and (b), but the slope is Θ = 3π/4. (e),(f) As in (a) and (b), but the slope is Θ = 7π/8.

Citation: Monthly Weather Review 151, 12; 10.1175/MWR-D-22-0311.1

The behavior of the scheme for the value of Θ = 0, which corresponds to a purely implicit treatment with no slow modes (ωs = 0), is shown in Figs. 4a and 4b. The scheme is unconditionally stable (Fig. 4a), and the phase produced shows deceleration (Fig. 4b). Here, the scheme produces one computational mode that is damped. The presence of this mode is caused by the application of the trapezoidal implicit method between the time steps t − Δt and t + Δt. The response of the scheme for the value of Θ = π/2, for which the scheme reduces to a purely leapfrog scheme using a fourth-order implicit time filter with no fast modes (ωf = 0), is shown in Figs. 4c and 4d. The computational modes are damped, and the physical mode is stable for ωsΔt ≤ 0.953 (Fig. 4c). The scheme produces acceleration for the physical mode (Fig. 4d).

Fig. 4.
Fig. 4.

(a) Magnitude of the amplification factors for the physical mode (solid), the 2Δt (long dashed–dashed) computational mode, and the 4Δt (dotted) computational mode as a function of ωΔt obtained for γ = 0.03 along a line with slope Θ = 0 in Fig. 1. (b) Argument of the physical mode (solid). The dashed lines in (a) and (b) represent the magnitude and argument of the exact amplification factor. (c),(d) As in (a) and (b), but the slope is Θ = π/2.

Citation: Monthly Weather Review 151, 12; 10.1175/MWR-D-22-0311.1

The order and accuracy of the scheme are analyzed by expanding the amplification factor A(ωfΔt, ωsΔt) as a double Taylor series about the origin (0, 0). The constant term c0 in the expansion can be found by taking (ωfΔt, ωsΔt) = (0, 0) in Eq. (14). This yields the equation (1+7γ)c03−8γc02+(γ−1)c0=0, which has three solutions: c0 = 1, c0 = −(1 − γ)/(1 + 7γ), and c0 = 0. The first solution corresponds to the physical mode; the second and the third solutions correspond to the 2Δt and 4Δt computational modes, respectively. Thus, the 2Δt mode is damped at a rate with magnitude (1 − γ)/(1 + 7γ) and 4Δt is removed when (ωfΔt, ωsΔt) approaches (0, 0), as illustrated in Figs. 2–4. The coefficients of the higher-order terms in the Taylor expansion are derived implicitly using the equation satisfied by the amplification factor [Eq. (14)] and following a method similar to the one presented in Moustaoui et al. (2014). The resulting Taylor series expansion of the amplification factor for the physical mode has the following form:
Ap=1+j(ωf+ωs)Δt−12(ωf+ωs)2(Δt)2−12jωf(ωf+ωs)2(Δt)3−18(1+7γ)ωs−(3+5γ)ωf1+3γ×(ωf+ωs)3(Δt)4+⋯.
When ωf = 0, the amplification factor corresponding to the purely explicit leapfrog scheme based on a fourth-order implicit time filter is recovered (Moustaoui et al. 2014). From Eq. (19), the magnitude of amplification factor for the physical mode in the limit (ωfΔt, ωsΔt) → (0, 0) is given by
|Ap|=1−12γ1+3γ(ωf+ωs)4(Δt)4.
Thus, the error generated by the proposed scheme for amplitude is of O[(Δt)4]. This demonstrates that the method is third-order for amplitude. Equation (20) also shows that the scheme is conditionally stable for any signs and values of γ > 0 and for any combination of fast (ωf) and slow (ωs) frequencies of the physical modes. Therefore, the scheme can be applied for practical problems where wide ranges of frequencies propagating in different directions could be involved.
The relative phase of the physical mode produced by the scheme, which can be derived from the Taylor expansion in Eq. (19), is
Rp=1+16(ωs−2ωf)(ωf+ωs)(Δt)2.

The phase error is second-order accurate and depends on the values of ωs and ωf. For positive frequencies (ωs > 0 and ωf > 0), the slow (fast) mode tends to produce acceleration (deceleration) of the phase. The scheme is accelerating the phase when the slow mode dominates: ωs > 2ωf, while the decelerating effect of the fast mode controls the phase behavior of the scheme when ωs < 2ωf. This behavior explains the results observed in the argument of the amplification factor for the physical mode (Fig. 2). The phase error becomes fourth order when ωs = 2ωf. However, this situation is not useful in practice because the implicit component of the scheme is motivated by the need to stabilize the fast modes in cases where ωf > ωs to allow stable numerical integrations employing larger time steps. In addition, the practical atmospheric problems for which the proposed scheme is designed usually comprise a wide spectrum of slow and fast frequencies.

b. Convergence tests

The effect of the application of the implicit time filtering in the proposed scheme (hereinafter LF-MBK) on the accuracy and convergence rate of the computed solution is evaluated by conducting numerical tests where the explicit component of the scheme is applied to a scalar advection equation ∂ψ/∂t + U∂ψ/∂x = 0. The results from the scheme are compared to the accuracy and convergence rates obtained from the nonfiltered leapfrog scheme (hereinafter LF) and the leapfrog scheme using the traditional second-order Robert–Asselin time filter (hereinafter LF-RA). The numerical tests use a simple one-dimensional flow, where U = 5 m s−1 in a periodic domain centered at (0, 0) using 100 × 100 grid points with a grid spacing of Δx = 100 m. The time step is chosen such that the corresponding Courant number is μ = UΔt/Δx = 0.4. The initial condition of the scalar has the form ψo(x,y)=4exp⁡(−r2/r02), where r0 = 600 m and r=x2+y2. The spatial derivatives utilize the sixth-order centered difference approximations. The total number of time steps selected corresponds to the time taken for the initial perturbation to be transported one revolution around the domain. The proposed scheme and the LF-RA method use filter coefficients of γ = 0.03 and γ2 = 4γ/(1 + 7γ) ≈ 0.0992, respectively. This choice for γ2 ensures that the computational 2Δt modes are damped at the same rate by the two schemes as Δt approaches 0 (Moustaoui et al. 2014). The selected value for γ2 is close to the value of γ2 = 0.1 commonly used in LF-RA (Skamarock and Klemp 1992). The error in each solution is computed using the root-mean-square error, RMSE=(1/N)∑i=1N(ψi−ψio)2, where N is the number of grid points, ψi is the numerical solution, and ψio is the exact solution, which is the same as the initial condition because the scalar is transported one revolution around the domain. The convergence rates are determined by conducting a sequence of numerical simulations, where the spatial resolution is doubled while maintaining the same value for the Courant number, μ = 0.4. The convergence rate is estimated from the equation CR = log2(RMSEΔx/RMSEΔx/2) as in Wicker and Skamarock (2002). Table 1 shows the errors and convergence rates obtained from a series of experiments where the resolution is doubled four times.

Table 1.

Errors and convergence rates computed for the advection test using the fourth-order implicit time filter in the proposed method (LF-MBK), the leapfrog method (LF), and the leapfrog method using a second-order Robert–Asselin time filter (LF-RA). The finite differences use sixth-order centered approximations. The numbers in parentheses indicate the convergence rates.
Table 1.

The errors in the LF scheme decrease rapidly as the resolution increases with a convergence rate of second order. The errors in this scheme are dominated by phase errors, as the scheme is neutral for amplitude in its zone of stability. These errors are second order and consistent with the computed convergence rates. The errors and convergence rates in LF-MBK and LF-RA can be caused by both the phase errors and application of the time filters, which have the potential to degrade the accuracy of the physical modes. However, the errors and convergence rates computed for LF-MBK are very close to LF despite the application of the time filter. The results demonstrate that the fourth-order time filter in the proposed scheme is able to maintain the accuracy and the convergence rate of the physical solution while reducing the computational modes. Although LF and LF-MBK have very close accuracies and convergence rates for these linear tests, the computational modes are problematic for LF in nonlinear applications (Moustaoui et al. 2014). The errors in LF-RA are much larger than those in LF-MBK. At 1600 resolution, the LF-MBK solution is 17.34 more accurate than LF-RA even though both schemes reduce the 2Δt computational modes at the same rate as Δt → 0. The convergence rates computed for LF-RA are close to first order. This is consistent with the first-order amplitude error produced by the traditional second-order time filter used in LF-RA: |A| = 1 − γ2(ωΔt)2/(2 − 2γ2).

4. Applications

a. Mesoscale linear gravity waves

Next, analytical solutions are derived and compared with solutions using numerical integrations for linear gravity waves excited by thermal forcing in the presence of a background flow. The atmospheric equations considered are the linearized Boussinesq equations for which exact solutions can be found. These equations are as follows:
un+1−u¯n−12Δt+U∂un∂x+Pxn+1=0,
wn+1−w¯n−12Δt+U∂wn∂x+Pzn+1−bn+1+b¯n−12=0,
bn+1−b¯n−12Δt+U∂bn∂x+N2wn+1+w¯n−12=Q(x,z,t)=Q0f(x,z)g(t),
∂un+1∂x+∂wn+1∂z=0,
where u and w are the perturbations of the horizontal and vertical velocities, respectively, and the normalized perturbation of potential temperature is defined by b = g(θ/θo). The terms U and N=g/θo(∂θo/∂z) are the background velocity and the Brunt–Väisälä frequency, which are considered constant to allow analytical derivations. The term Q(x, z, t) = Q0f(x, z)g(t) is the thermal forcing given by
f(x,z)=sin⁡(πz/Lz)1+x2/Lx2  for  z≤Lz;  and f(x,z)=0  for  z>Lz,
g(t)=tTr                      for  t≤Tr;   and g(t)=1     for t>Tr.
Throughout this paper, we compare the proposed method to a scheme where the slow modes are handled by the RK3 method and the fast modes are integrated by the trapezoidal scheme. In this scheme, equations equivalent to Eqs. (22)–(25) are iterated three times every time step. For instance, the equation that corresponds to Eq. (22) in this scheme has the form
us+1−unΔτ+U∂us∂x+Pxs+1=0,
where (s+1=*, s = n, Δτ = Δt/3) for the first RK3 substep, (s+1=**, s=*, Δτ = Δt/2) for the second RK3 substep, and (s + 1 = n + 1, s=**, Δτ = Δt) for the third and final RK3 substep.
The numerical solutions are found for the following parameters: U = 20 m s−1, N2 = 4 × 10−4 s−2, Tr = 0.5 h, Lx = 30 km, Lz = 3 km, and Q0 = 4 × 10−4 m s−3. All the perturbations are equal to zero at t = 0. The integrations are conducted within a periodic domain in the horizontal using 256 grid points and a grid spacing of Δx = 4 km. In the vertical, the simulation employs 225 levels distributed uniformly between the ground and the top of the domain located at 44.8 km with a grid spacing of Δz = 200 m. The time step used, Δt = 80 s, corresponds to a Courant number of μ = UΔt/Δx = 0.4. The value of NΔt is 1.6. The vertical velocity is imposed to be w = 0 at the vertical boundaries of the domain. The simulations are conducted for a total period of T = 3 h. A semi-implicit trapezoidal scheme is employed for all contributions in Eqs. (22)–(25) except for the advection and the forcing terms. These terms are handled by the explicit time-filtered leapfrog scheme with γ = 0.03 and utilize sixth-order finite difference approximations for spatial derivatives. The numerical discretization of the scheme employed is similar to the one that will be described for the anelastic equations in the following subsection. The analytical solution to Eqs. (22)–(25) is derived by writing the equations in the streamfunction–vorticity form and expanding all the fields in the form w(x,z,t)=∑k,mw^km(t)ejkxsin⁡(mz), where k and m are the horizontal and vertical wavenumbers, respectively. The problem can be solved by applying Fourier, sine, and Laplace transforms in x, z, and t, respectively. The procedure is similar to the one presented in Moustaoui et al. (2004). The resulting spectral coefficients for vertical velocity are given by
w^km(t)=Q0f^kmωN2∫0tg(τ)sin⁡[ω(t−τ)]exp⁡[−jkU(t−τ)]dτ,
where ω=Nk/m2+k2, which is evaluated analytically using the expression of g(t) in the thermal forcing. The values of w in the physical space are computed by applying inverse Fourier and sine transforms.

Figures 5a, 5b, and 5c show the vertical velocity fields obtained at T = 3 h from the analytical solution, the numerical solution obtained from the proposed scheme, and the one computed by the scheme using the RK3 method for the slow terms, respectively. Both numerical solutions compare well with the analytical solution, indicating that the proposed method is able to accurately simulate gravity wave propagation. The errors relative to the exact solution are shown in Figs. 5d and 5e for both schemes. They are amplified by a factor of 100 in these figures. The values for RMSE in LF-MBK and RK3 are 4.77 × 10−3 and 4.70 × 10−3, respectively. The errors indicate that RK3 is slightly more accurate than LF-MBK. Although the phase accuracy of RK3 is higher than LF-MBK, its combination with the second-order trapezoidal scheme reduces the overall accuracy of the scheme and explains the close accuracy found in RK3 and LF-MBK.

Fig. 5.
Fig. 5.

Cross sections of vertical velocity after 3 h from (a) analytical calculations, (b) numerical computations with the proposed scheme using a filter coefficient of γ = 0.03, and (c) a scheme using the RK3 method for the slow modes. Negative contours are dashed, and the contour interval is 0.2 m s−1. (d) The difference between the numerical solution obtained by the proposed scheme and the analytical solution (shaded). The difference is amplified by a factor of 100. The numerical solution is also superimposed. (e) As in (d), but the scheme employs the RK3 method for the slow modes.

Citation: Monthly Weather Review 151, 12; 10.1175/MWR-D-22-0311.1

b. Nonlinear shear instability

Here, the ability of the proposed scheme to resolve detailed dynamics in high-resolution and strongly nonlinear calculations of Kelvin–Helmholtz instability is investigated. Since analytical solutions cannot be derived for this case, we compare numerical solutions computed by the proposed scheme with those employing the RK3 method, which are considered as the reference solutions. The system consists of the soundproof nonhydrostatic and nonlinear atmospheric equations employing the anelastic (Ogura and Phillips 1962; Lipps and Hemler 1982) approximation:
∂u∂t+Nu+(cpθoπ)x=Du,
∂w∂t+Nw+(cpθoπ)z−gθθo=Dw,
∂θ∂t+Nθ+θozw=Dθ,
ux+1ρo(ρow)z=0,
where u and w are the horizontal and vertical velocities. The terms Θ and π are the perturbations of the potential temperature and the Exner function. The terms Θo = θo(z) and ρo = ρo(z) are the background potential temperature and density. The terms Nu, Nw, and Nθ are the nonlinear advection terms given by Nψ = uψx + wψz, where ψ = u, w, and θ. The terms Du, Dw, and Dθ are the diffusion terms. The subscripts x and z denote the horizontal and vertical derivatives, respectively. The model equations [Eqs. (27)–(30)] are not written in the usual streamfunction–vorticity form permitted in two dimensions (Moustaoui et al. 2004) so that the extension of the formulation that follows to three dimensions is straightforward.
The temporal discretization of the anelastic system in the proposed scheme has the following form:
un+1−u¯n−12Δt+Nun+Pxn+1=Du,
wn+1−w¯n−12Δt+Nwn+Pzn+1−gθn+1+θ¯n−12θo=Dw,
θn+1−θ¯n−12Δt+Nθn+θozwn+1+w¯n−12=Dθ,
uxn+1+1ρo(ρown+1)z=0,
where the “pressure” P = cpθoπ. In this formulation, the nonlinear advection and diffusion terms are handled explicitly by the time-filtered leapfrog scheme, while all the remaining terms use semi-implicit treatments. The variables (un+1, wn+1, θn+1) and (u¯n−1,w¯n−1,θ¯n−1) are not known and depend on each other implicitly through both the trapezoidal implicit component of the scheme and the implicit time filter. However, one can use the relations given by Eqs. (4) and (5) in Eqs. (31)–(33) to derive the following equations:
(1+7γ)un+1+2Δt(1+6γ)Pxn+1=Fu,
(1+7γ)wn+1+2Δt(1+6γ)Pzn+1−gΔtθo(1+5γ)θn+1=Fw,
(1+7γ)θn+1+Δtθoz(1+5γ)wn+1=Fθ.
The terms Fu, Fw, and Fθ are all known because they involve the fields of u, w, and θ that are already computed:
Fu=u˜n−1+2Δt(1+6γ)(Du−Nun),
Fw=w˜n−1+gΔtθoθ˜n−1+2Δt(1+6γ)(Dw−Nwn),
Fθ=θ˜n−1−Δtθozw˜n−1+2Δt(1+6γ)(Dθ−Nθn).
The grid employed for the numerical integration is stretched in the vertical. The spatial discretization employs a C-grid, where P and θ are defined at the center of the grid cell (i, p); the velocities u and w are placed in the interfaces of the grid at (i − 1/2, p) and (i, p − 1/2), respectively. The nonlinear advection utilizes the sixth- and fourth-order finite difference approximations in the horizontal and the vertical, respectively. Using this grid configuration, Eqs. (35)–(37) can be combined with the continuity equation [Eq. (34)] to derive the algebraic difference equation satisfied by vertical velocity w:
aik(wi−1kn+1+wi+1kn+1)+bik(wi−1k−1n+1+wi+1 k−1n+1)+cik(wi−1 k+1n+1+wi+1 k+1n+1)+dikwikn+1+eikwik−1n+1+fikwik+1n+1=Qik,
where k = p − 1/2. The coefficients of this equation are as follows:
aik=1Δx2[1+ϵ2Δt2g(θ0z,k+1/2+θoz,k−1/2)4θok],  bik=ϵ2Δt2gθoz,k−1/24Δx2θo,k,  cik=ϵ2Δt2gθoz,k+1/24Δx2θo,k,dik=−{1ΔkΔk−1/2ρo,kρo,k−1/2+1ΔkΔk+1/2ρo,kρo,k+1/2+2Δx2[1+ϵ2Δt2g(θ0z,k+1/2+θoz,k−1/2)4θok]},eik=1ΔkΔk−1/2ρo,k−1ρo,k−1/2−ϵ2Δt2gθoz,k−1/22Δx2θo,k,   fik=1ΔkΔk+1/2ρo,k+1ρo,k+1/2−ϵ2Δt2gθoz,k+1/22Δx2θo,k,
where Δk = zk+1/2 − zk−1/2 and Δk+1/2 = zk+1 − zk are the vertical grid spacings and ϵ = (1 + 5γ)/(1 + 7γ). The forcing in the right-hand side of Eq. (40) is
Qi,k=Rikw−Riku1+7γ,
where
Rikw=Si−1kw−2Sikw+Si+1kwΔx2, Sikw=Fikw+ϵΔtg2θo,k(Fik+1/2θ+Fik−1/2θ),Riku=(Fi+1/2,k+1/2u−Fi−1/2,k+1/2u)−(Fi+1/2,k−1/2u−Fi−1/2,k−1/2u)ΔxΔk.
Equation (41) is solved for wn+1 by imposing periodic boundary conditions in the horizontal. In the vertical, a rigid lid (w = 0) is specified at the bottom and the top boundaries of the domain. θn+1 is then updated from Eq. (37). The pressure Pn+1 needed to solve for un+1 is obtained by solving a difference equation in the horizontal for Pn+1. This equation is derived by combining Eq. (35) and the continuity equation [Eq. (34)] where the updated value of wn+1 is used. The horizontal velocity un+1 is then calculated from Eq. (35). After updating the fields for the stage n + 1, Eq. (8) is used to update the filtered fields at the stage n − 1: u¯n−1, w¯n−1, and θ¯n−1.
The background potential temperature in the following example is θ0(z)=θsexp⁡(N2z/g), where N2 = 1.2 × 10−4 s−2 and θs = 300 K is the temperature at the ground. The background profile of the Exner function is calculated by integrating the hydrostatic relation: ∂π0/∂z = −g/(cpθ0). The profiles π0(z) and θ0(z) are then used in the gas law to find the background density ρ0(z) needed for the continuity equation. The initial mean flow is given by U = U0 tanh[(z − z0)/d], where U0 = 30 m s−1, z0 = 5 km, and d = 500 m. The Richardson number Ri=N2/Uz2 for these profiles is such that Ri < 0.25 in the vicinity of z0. Thus, the background profiles satisfy the necessary condition for the onset of shear instability. The diffusion terms use a diffusion coefficient υ such that Re = U0d/υ = 1500. The domain employs 1024 grid points in the horizontal with a grid spacing of Δx = 20 m. It utilizes 512 points distributed in the vertical between 0 and 10 km. The vertical grid is stretched with a grid spacing down to Δz = 5 m between 4.5 and 5.5 km, and the time step is Δt = 0.05 s. The corresponding value of NΔt is 5.48 × 10−4. The small time step utilized is imposed by the nonlinear advection, which is integrated by the explicit component of the scheme. This time step is restricted by the very small grid spacings utilized in the vicinity of the shear layer (Δx = 20 m and Δz = 5 m). The shear instability is triggered by adding a very small perturbation to potential temperature. The perturbation is centered on the level of maximum shear (z0 = 5 km) and has a magnitude of order 10−6 with the following form:
δθ={10−6[1−cos⁡(2πz−z1z2−z1)]cos⁡(4πxL), for z1≤z≤z20,otherwise,
where z1 = 3 km, z2 = 7 km, and L = 1024Δx. The other initial conditions are u = U(z) and w = 0. The reference solution employs a scheme based on the RK3 method where the fast terms are integrated by the semi-implicit trapezoidal scheme. The formulation of this scheme is similar to the one described above for the proposed method, except that the terms responsible for the slow-evolving dynamics are handled by the RK3 scheme instead of the implicit time-filtered leapfrog.
Figure 6a shows a cross section of the total potential temperature field (θT = θo + θ) simulated by the proposed method after 54 000 time steps. The strength of the implicit filter in the scheme is γ = 0.03. The primary shear instability is well developed at this stage. Strong mixing of temperature occurs within the Kelvin–Helmholtz (KH) billows. The simulated field resembles the reference solution computed from the RK3 scheme (Fig. 6b). Both fields are indistinguishable, demonstrating that the scheme accurately reproduces the nonlinear development of shear instability. A thin layer of strong shear is formed between the KH billows. As time evolves, this shear layer nonlinearly triggers a secondary instability that can be seen in Fig. 6c, which shows the simulated field after 61 000 time steps. The secondary instability can be identified by the small billows formed above x = 2 km and 13 km. The reference field computed by RK3 (Fig. 6d) shows that the secondary instability is well reproduced by the proposed method. Other structures associated with secondary instabilities with finer details develop rapidly with time. These structures are clearly seen in the fields simulated by the method after 64 000 (Fig. 6e) and 67 000 (Fig. 6g) time steps. They are indistinguishable from those found in the fields computed by RK3 (Figs. 6f,h). The computed value of RMSE relative to RK3 at time step 67 000 is 1.05 × 10−2, with a root-mean-square relative error of RMSRE = 3.27 × 10−5. The relative error is obtained using
RMSRE=1N∑i=1N(θT,iMBK−θT,iRK3θT,iRK3)2,
where N is the number of grid points. The results demonstrate that the proposed method resolves correctly fine-scale structures of KH dynamics.
Fig. 6.
Fig. 6.

Cross sections of potential temperature computed after 54 000 time steps from (a) the proposed scheme using a filter coefficient of γ = 0.03 and (b) the scheme using the RK3 method for the slow modes. (c),(d) As in (a) and (b), but the number of time steps is 61 000. The fields simulated after (e),(f) 64 000 and (g),(h) 67 000 time steps.

Citation: Monthly Weather Review 151, 12; 10.1175/MWR-D-22-0311.1

c. Density current

Next, the scheme is tested on the nonlinear density current benchmark described in Straka et al. (1993), which is widely used for evaluation of solvers in nonhydrostatic atmospheric numerical models (e.g., Wicker and Skamarock 2002; Benacchio and Klein 2019). The simulation is initialized for neutral stratification at rest by prescribing a constant background potential temperature θ0 = 300 K. A cold bubble of δθ(x, z) is placed in a region centered at 3 km in the vertical with the following form:
δθ={−152[1+cos⁡(πA)], for A≤10,otherwise,
where A={[(x−xc)/xr]2+[(z−zc)/zr]2}1/2, xc = 0, zc = 3 km, xr = 4 km, and zr = 2 km. An artificial diffusion coefficient with a value of 75 m2 s−1 is applied in both the momentum and potential temperature equations. The simulation uses 128 vertical levels distributed between the ground and the top of the domain, which is specified at ztop = 6.4 km. The horizontal width of the domain is [−25.6 km, 25.6 km] with a horizontal grid spacing of Δx = 100 m. The simulation is conducted for a total physical time of 900 s using a time step of Δt = 1 s. Figure 7 compares the potential temperature field simulated by the proposed scheme (Fig. 7a) and the RK3 method (Fig. 7b) after 900 s. The computed fields show that the proposed method is able to accurately simulate the expected distribution of the nonlinear density current. The computed value of RMSE relative to RK3 at time 900 s is 2.19 × 10−2, with a root-mean-square relative error of RMSRE = 7.40 × 10−5.
Fig. 7.
Fig. 7.

Cross sections of potential temperature computed after 900 s for (a) the proposed scheme using a filter coefficient of γ = 0.03 and (b) the scheme using the RK3 method for the slow modes.

Citation: Monthly Weather Review 151, 12; 10.1175/MWR-D-22-0311.1

d. Nonlinear global shallow-water equations

In this example, the proposed method is applied to the global spectral shallow-water equations on the sphere with rotation. We repeat the same test introduced in Moustaoui et al. (2014) where nonlinear evolution of twin tropical cyclones straddling the equator was reproduced. The method employed in Moustaoui et al. (2014) for this test was fully explicit. All terms of the equations were treated with a leapfrog scheme based on a fourth-order implicit time filter. Here, we will modify the formulation of the equations to separate terms responsible for fast-propagating surface gravity waves from those accountable for slow propagation. The hyperdiffusion and the fast terms will be handled by the trapezoidal-implicit component of the scheme.

The movement of a pair of equatorial cyclones located in the northern and southern hemispheres is strongly affected by nonlinearity, rotation, and the mutual interaction between the cyclones. The pole-westward propagation of the cyclones referred to as the β drift is caused by nonlinearities opposing the dispersion of the cyclones induced by Rossby waves. The integrity of the cyclones tends to be preserved by interactions between nonlinearity and rotation (Chan and Williams 1987; Carr and Williams 1989; Moustaoui et al. 2002). The mutual interaction between the twin cyclones tends to keep the cyclones closer to the equator. Details of the mechanisms involved are described in Moustaoui et al. (2002). The global shallow-water equations considered are written in the divergence–vorticity form (Williamson et al. 1992). They are given by
∂ζ∂t+1a(1−μ2)∂(Uη)∂λ+1a∂(Vη)∂μ=K∇6ζ,
∂δ∂t+∇2Φ−1a(1−μ2)∂(Vη)∂λ+1a∂(Uη)∂μ+∇2[U2+V22(1−μ2)]=K∇6δ,
∂Φ∂t+Φoδ+1a(1−μ2)∂(UΦ)∂λ+1a∂(VΦ)∂μ=K∇6Φ.
The radius of Earth and the longitude are denoted by a and λ. The term Φ = gh is the perturbation of the geopotential, and h is the deviation of the elevation of the fluid from the mean depth. The terms Φo and η = ζ + f are the background geopotential and absolute vorticity, respectively. The Coriolis parameter is f = 2Ωμ, μ = sinϕ, and ϕ is the latitude. The relative vorticity ζ and divergence δ are given by
ζ=1a(1−μ2)∂V∂λ−1a∂U∂μ, δ=1a(1−μ2)∂U∂λ+1a∂V∂μ,
where U = u cosϕ and V = υ cosϕ. In these relations, u and υ are the eastward and northward velocities. The right-hand side terms in Eqs. (40)–(42) are hyperdiffusion terms. The variables U and V are derived from the streamfunction ψ and the velocity potential χ as follows:
U=1a∂χ∂λ−(1−μ2)a∂ψ∂μ, V=1a∂ψ∂λ+(1−μ2)a∂χ∂μ,
where ζ =Δψ and δ =Δχ. The formulation for Eqs. (42)–(44) is
ζn+1−ζ¯n−12Δt−K∇6(ζn+1+ζ¯n−12)=Gζn,
δn+1−δ¯n−12Δt+∇2(Φn+1+Φ¯n−12)−K∇6(δn+1+δ¯n−12)=Gδn,
Φn+1−Φ¯n−12Δt+Φo(δn+1+δ¯n−12)−K∇6(Φn+1+Φ¯n−12)=GΦn,
where the terms Gζn, Gδn, and GΦn, which are evaluated at the time step nΔt and handled by the time-filtered leapfrog component of the scheme, have the following form:
Gζn=−[1a(1−μ2)∂(Uη)∂λ+1a∂(Vη)∂μ]n,
Gδn=−{−1a(1−μ2)∂(Vη)∂λ+1a∂(Uη)∂μ+∇2[U2+V22(1−μ2)]}n,
GΦn=−[1a(1−μ2)∂(UΦ)∂λ+1a∂(VΦ)∂μ]n.
The relations given by Eqs. (4) and (5) can be used in Eqs. (47)–(49) to derive the following equations:
(1+7γ)ζn+1−ΔtK(1+5γ)∇6ζn+1=Fζ,
(1+7γ)δn+1+Δt(1+5γ)∇2Φn+1−ΔtK(1+5γ)∇6δn+1=Fδ,
(1+7γ)Φn+1+Δt(1+5γ)Φoδn+1−ΔtK(1+5γ)∇6Φn+1=FΦ,
where the terms Fζ, Fδ, and FΦ involve the fields of ζ, δ, and Φ that are already computed:
Fζ=ζ˜n−1+ΔtK∇6ζ˜n−1+2Δt(1+6γ)Gζn,
Fδ=δ˜n−1−Δt∇2Φ˜n−1+ΔtK∇6δ˜n−1+2Δt(1+6γ)Gδn,
FΦ=Φ˜n−1−ΔtΦoδ˜n−1+ΔtK∇6Φ˜n−1+2Δt(1+6γ)GΦn.
Equations (53)–(55) are numerically integrated by employing the spectral transform method (Hack and Jakob 1992). The operators ∇2 and ∇6 in these equations can be converted to simple forms in the spectral space using the following relationships: ∇2Plm(μ)ejmλ=−[l(l+1)/a2] Plm(μ)ejmλ and ∇6Plm(μ)ejmλ=−[l(l+1)/a2]3 Plm(μ)ejmλ, where Plm(μ)ejmλ are the spherical harmonic functions employed in the spectral expansion. Equation (53) is then solved for ζn+1. Equations (54) and (55) can be combined to derive the values of δn+1 and Φn+1. The updated variables at the stage n + 1 are then used in Eq. (8) to compute the filtered fields at the step n − 1: ζ¯n−1, δ¯n−1, and Φ¯n−1. The model is initialized with two nondivergent cyclones placed at latitude ϕ = ±7° and longitude λ = 150°E with vorticity profiles: ζ = ±Vo/do(2 − d/do)exp(1 − d/do), where Vo = 30 ms−1, do = 200 km, and d is the distance from the center of the cyclone. The initial velocity fields U and V are calculated from Eq. (46) after using ζ =Δψ. The initial geopotential is computed from Eq. (43) by using the initial values of U, V, and η = ζ + f and assuming a nonlinear balance with ∂δ/∂t = 0 and without hyperdiffusion. The background geopotential is Φ0 = gH, where H = 234.86 m. The model is numerically integrated using the proposed scheme and a scheme using the RK3 method for the slow terms. The numerical integrations use a time step of Δt = 900 s and the triangular truncation N = 213. The strength of the implicit filter used in the scheme is γ = 0.03. Figure 8 compares the evolution of the twin cyclones obtained from the proposed method (Figs. 8a,c,d) and RK3 method (Figs. 8b,d,f) after 40, 80, and 120 h. The computed fields demonstrate that the proposed method is able to accurately simulate the nonlinear pole-westward drifts of the twin tropical cyclones. The computed value of RMSE relative to RK3 at time 120 h is 6.58 × 10−2, with a root-mean-square relative error of RMSRE = 4.60 × 10−4.
Fig. 8.
Fig. 8.

Fluid elevation and wind vector fields computed after 40 h from (a) the proposed scheme using a filter coefficient of γ = 0.03 and (b) the scheme using the RK3 method for the slow modes. The contour interval is 10 m. (c),(d) As in (a) and (b), but the time is 80 h. (e),(f) The fields simulated at 120 h.

Citation: Monthly Weather Review 151, 12; 10.1175/MWR-D-22-0311.1

e. Computational performance

The simulations conducted for this study were executed in double precision on a Linux x86_64 platform using Intel processors. The average elapsed computational times per time were calculated for all nonlinear simulations presented. The results are summarized in Table 2. The average elapsed computational times per time step obtained for the shear instability case in the proposed scheme and the scheme where the slow terms are treated by the RK3 method are 0.079 and 0.211 s, respectively. This gives a relative speedup factor of 2.671. The speedup factor calculated for the density current case is 2.6. This value is close to the one computed for the shear instability case. The simulations in the density current case are relatively less expensive computationally. The average elapsed computational times per time step for this case in the proposed scheme and the one using the RK3 method are 0.0078 and 0.0203 s, respectively. The agreement between the values of the speedup factors in the cases above can partly be explained by the fact that both cases use the same numerical discretization of the scheme, which is given by Eqs. (31)–(41). The value of the speedup factor calculated for the twin cyclones case is 2.951. This value is closer to 3. The computations for this test case have the largest average elapsed computational times per time step. These are 0.102 and 0.301 s for the proposed scheme and the method using RK3, respectively. The computation times for the twin cyclones case are dominated by the spectral transforms on the sphere, which require 3 times more executions in the RK3 case compared to those needed in the proposed scheme.

Table 2.

Elapsed computational times per time step computed for the shear instability, density current, and twin cyclones examples in the proposed method (LF-MBK) and the method using RK3. The numbers in parentheses indicate the speedup factors of the proposed method relative to RK3.
Table 2.

5. Conclusions

This paper presents a method that can be employed to compute numerical solutions for atmospheric equations. It is designed for time-dependent problems where the underlying equations can be split up into parts responsible for slow-evolving processes and terms involving faster speed of propagation. In the method, the terms accountable for slow propagation such as advection and rotation are integrated by a filtered leapfrog scheme. This scheme employs a fourth-order implicit time filter to damp the computational modes without noticeably sacrificing the accuracy of the physical modes. The parts of the equations involving fast evolution such as buoyancy and rapid internal/surface gravity waves and/or diffusion terms are treated by the trapezoidal implicit method, which is unconditionally stable. The proposed scheme is not an iterative method as it requires one function evaluation per time step only. It allows faster numerical computations as the time steps permitted are larger than those required when all terms of the equations are evaluated explicitly. The method is doubly implicit because of the implicit treatment of the fast terms and the application of the implicit time filter in the leapfrog method that handles the slow contributions. Nevertheless, we derived formulation for the explicit implementation of the method.

The behavior of the method is examined by analyzing its accuracy and stability. The magnitude of the amplification factor corresponding to the physical mode indicated that the scheme is conditionally stable for all values of the slow and fast physical frequencies as they approach the origin. These include cases where the slow and fast modes are propagating in the same direction as well as situations where these modes have opposite propagation. The dependency of the magnitude and phase of the amplification factor on the slow and fast frequencies is investigated along various slanted lines. The slopes of these lines are defined by an angle Θ in the (ωfΔt, ωsΔt) plane that characterizes the directions and the relative contributions of the slow and fast waves. In all cases, the computational modes are unconditionally stable and damped. The stability of the proposed scheme is determined by the physical mode. The stability of the physical mode increases as the contribution of the fast frequency dominates the wave field.

A double Taylor series expansion of the amplification factor for the physical mode as a function of (ωfΔt, ωsΔt) derived from the characteristic equation satisfied the scheme. It shows that the O[(Δt)4] accuracy for amplitude errors achieved by the purely and implicitly time-filtered leapfrog method is maintained when the implicit treatment of the fast mode is included. The expression of the magnitude of the amplification factor demonstrates that the scheme is conditionally stable for any (ωfΔt, ωsΔt) approaching (0, 0) and all positive coefficients employed in the implicit filter.

Comparison between numerical integrations computed by the method and corresponding analytical solutions derived for propagation of mesoscale linear gravity waves in a nonhydrostatic and incompressible example demonstrates that the method achieves accurate simulations. We provide a detailed formulation for the implementation of the method in soundproof nonhydrostatic atmospheric models based on the anelastic approximation. The formulation of the model equations avoids utilization of the streamfunction–vorticity form to allow straightforward extension to three-dimensional applications. The time-filtered leapfrog component of the scheme is utilized to evaluate the advection terms where sixth- and fourth-order finite difference approximations are used for the horizontal and the vertical, respectively. The model is then employed to compute evolution of density currents and fine-scale structures emerging in the nonlinear dynamics of Kelvin–Helmholtz shear instability. The computations were evaluated against reference solutions obtained from a scheme where the slow terms are computed with the RK3 method. The comparison demonstrates that the proposed scheme accurately reproduces structures of the primary instability and finer-scale details that develop owing to secondary shear instabilities.

Detailed formulation for global spectral models solving the shallow-water equations on the rotating sphere is also provided. Here, the terms responsible for rapid-propagating surface gravity waves and hyperdiffusion are handled by the implicit component of the scheme. The global model is then applied to simulate pole-westward drifts of twin tropical cyclones caused by nonlinear interactions between rotation and the vorticity fields induced by the cyclones. The simulated drifts resemble those found in the reference solutions computed from the RK3 scheme.

In future work, we intend to develop a three-dimensional fully compressible nonhydrostatic model based on the proposed scheme. In addition, a split-explicit version of the proposed numerical scheme is being under development. The split-explicit scheme will use small and large time steps to integrate the fast and slow terms, respectively. The method will have the potential to speed up computations as the slow terms are more computationally intensive in real three-dimensional fully compressible atmospheric applications.

Acknowledgments.

This material is based upon work supported by the U.S. Department of Energy, Office of Science, Office of Biological and Environmental Research’s Urban Integrated Field Laboratories research activity, under Award DE-SC0023520.

Data availability statement.

The data utilized to test the performance of the proposed scheme in this study are based entirely on idealized simulations. The study does not use any external or real dataset. The ingredients required to replicate these simulations are described in this manuscript. The program code and data generated by these idealized simulations are available from Mohamed Moustaoui (Mohamed.Moustaoui@asu.edu) at Arizona State University.

REFERENCES

  • Amezcua, J., and P. D. Williams, 2015: The composite-tendency Robert-Asselin-Williams (RAW) filter in semi-implicit integrations. Quart. J. Roy. Meteor. Soc., 141, 764–773, https://doi.org/10.1002/qj.2391.

    • Search Google Scholar
    • Export Citation

  • Asselin, R., 1972: Frequency filter for time integrations. Mon. Wea. Rev., 100, 487–490, https://doi.org/10.1175/1520-0493(1972)100<0487:FFFTI>2.3.CO;2.

    • Search Google Scholar
    • Export Citation

  • Benacchio, T., and R. Klein, 2019: A semi-implicit compressible model for atmospheric flows with seamless access to soundproof and hydrostatic dynamics. Mon. Wea. Rev., 147, 4221–4240, https://doi.org/10.1175/MWR-D-19-0073.1.

    • Search Google Scholar
    • Export Citation

  • Carr, L. E., III, and R. T. Williams, 1989: Barotropic vortex stability to perturbations from axisymmetry. J. Atmos. Sci., 46, 3177–3191, https://doi.org/10.1175/1520-0469(1989)046<3177:BVSTPF>2.0.CO;2.

    • Search Google Scholar
    • Export Citation

  • Chan, J. C. L., and R. T. Williams, 1987: Analytical and numerical studies of the beta-effect in tropical cyclone motion. Part I: Zero mean flow. J. Atmos. Sci., 44, 1257–1265, https://doi.org/10.1175/1520-0469(1987)044<1257:AANSOT>2.0.CO;2.

    • Search Google Scholar
    • Export Citation

  • Chew, R., T. Benacchio, G. Hastermann, and R. Klein, 2022: A one-step blended soundproof-compressible model with balanced data assimilation: Theory and idealized tests. Mon. Wea. Rev., 150, 2231–2254, https://doi.org/10.1175/MWR-D-21-0175.1.

    • Search Google Scholar
    • Export Citation

  • Durran, D. R., 1989: Improving the anelastic approximation. J. Atmos. Sci., 46, 1453–1461, https://doi.org/10.1175/1520-0469(1989)046<1453:ITAA>2.0.CO;2.

    • Search Google Scholar
    • Export Citation

  • Durran, D. R., 1991: The third-order Adams-Bashforth method: An attractive alternative to leapfrog time differencing. Mon. Wea. Rev., 119, 702–720, https://doi.org/10.1175/1520-0493(1991)119<0702:TTOABM>2.0.CO;2.

    • Search Google Scholar
    • Export Citation

  • Durran, D. R., 1999: Numerical Methods for Wave Equations in Geophysical Fluid Dynamics. Springer-Verlag, 465 pp.

  • Durran, D. R., 2008: A physically motivated approach for filtering acoustic waves from the equations governing compressible stratified flow. J. Fluid Mech., 601, 365–379, https://doi.org/10.1017/S0022112008000608.

    • Search Google Scholar
    • Export Citation

  • Durran, D. R., and P. N. Blossey, 2012: Implicit-explicit multistep methods for fast-wave-slow-wave problems. Mon. Wea. Rev., 140, 1307–1325, https://doi.org/10.1175/MWR-D-11-00088.1.

    • Search Google Scholar
    • Export Citation

  • Hack, J. J., and R. Jakob, 1992: Description of a global shallow water model based on the spectral transform method. NCAR Tech. Note NCAR/TN-343+STR, 41 pp., https://opensky.ucar.edu/islandora/object/technotes%3A112/datastream/PDF/download/Description_of_a_Global_Shallow_Water_Model_Based_on_the_Spectral_Transform_Method.citation.

  • Kurowski, M. J., W. W. Grabowski, and P. K. Smolarkiewicz, 2013: Towards multiscale simulation of moist flows with soundproof equations. J. Atmos. Sci., 70, 3995–4011, https://doi.org/10.1175/JAS-D-13-024.1.

    • Search Google Scholar
    • Export Citation

  • Lipps, F. B., and R. S. Hemler, 1982: A scale analysis of deep moist convection and some related numerical calculations. J. Atmos. Sci., 39, 2192–2210, https://doi.org/10.1175/1520-0469(1982)039<2192:ASAODM>2.0.CO;2.

    • Search Google Scholar
    • Export Citation

  • Mahalov, A., and M. Moustaoui, 2009: Vertically nested nonhydrostatic model for multi-scale resolution of flows in the upper troposphere and lower stratosphere. J. Comput. Phys., 228, 1294–1311, https://doi.org/10.1016/j.jcp.2008.10.030.

    • Search Google Scholar
    • Export Citation

  • Moustaoui, M., H. Teitelbaum, C. Basdevant, and Y. Boughaleb, 2002: Linked behavior of twin tropical cyclones. J. Geophys. Res., 107, 4378, https://doi.org/10.1029/2000JD000066.

    • Search Google Scholar
    • Export Citation

  • Moustaoui, M., B. Joseph, and H. Teitelbaum, 2004: Mixing layer formation near the tropopause due to gravity wave critical level interactions in a cloud-resolving model. J. Atmos. Sci., 61, 3112–3124, https://doi.org/10.1175/JAS-3289.1.

    • Search Google Scholar
    • Export Citation

  • Moustaoui, M., A. Mahalov, and E. J. Kostelich, 2014: A numerical method based on leapfrog and a fourth-order implicit time filter. Mon. Wea. Rev., 142, 2545–2560, https://doi.org/10.1175/MWR-D-13-00073.1.

    • Search Google Scholar
    • Export Citation

  • Ogura, Y., and N. A. Phillips, 1962: Scale analysis of deep and shallow convection in the atmosphere. J. Atmos. Sci., 19, 173–179, https://doi.org/10.1175/1520-0469(1962)019<0173:SAODAS>2.0.CO;2.

    • Search Google Scholar
    • Export Citation

  • Robert, A. J., 1966: The integration of a low order spectral form of the primitive meteorological equations. J. Meteor. Soc. Japan, 44, 237–245, https://doi.org/10.2151/jmsj1965.44.5_237.

    • Search Google Scholar
    • Export Citation

  • Skamarock, W. C., and J. B. Klemp, 1992: The stability of time-split numerical methods for the hydrostatic and the nonhydrostatic elastic equations. Mon. Wea. Rev., 120, 2109–2127, https://doi.org/10.1175/1520-0493(1992)120<2109:TSOTSN>2.0.CO;2.

    • Search Google Scholar
    • Export Citation

  • Skamarock, W. C., and J. B. Klemp, 2008: A time-split nonhydrostatic atmospheric model for research and NWP applications. J. Comp. Phys., 227, 3465–3485, https://doi.org/10.1016/j.jcp.2007.01.037.

    • Search Google Scholar
    • Export Citation

  • Skamarock, W. C., J. B. Klemp, M. G. Duda, L. D. Fowler, S.-H. Park, and T. D. Ringler, 2012: A multiscale nonhydrostatic atmospheric model using centroidal Voronoi tesselations and C-Grid staggering. Mon. Wea. Rev., 140, 3090–3105, https://doi.org/10.1175/MWR-D-11-00215.1.

    • Search Google Scholar
    • Export Citation

  • Smolarkiewicz, P. K., C. Kuhnlein, and N. P. Wedi, 2014: A consistent framework for discrete integrations of soundproof and compressible PDEs of atmospheric dynamic. J. Comput. Phys., 263, 185–205, https://doi.org/10.1016/j.jcp.2014.01.031.

    • Search Google Scholar
    • Export Citation

  • Straka, J. M., R. B. Wilhelmson, L. J. Wicker, J. R. Anderson, and K. K. Droegemeier, 1993: Numerical solutions of a non-linear density current: A benchmark solution and comparisons. Int. J. Numer. Methods Fluids, 17, 1–22, https://doi.org/10.1002/fld.1650170103.

    • Search Google Scholar
    • Export Citation

  • Thuburn, J., 2017: Use of the Gibbs thermodynamic potential to express the equation of state in atmospheric models. Quart. J. Roy. Meteor. Soc., 143, 1185–1196, https://doi.org/10.1002/qj.3020.

    • Search Google Scholar
    • Export Citation

  • Weller, H., S.-J. Lock, and N. Wood, 2013: Runge-Kutta IMEX schemes for the horizontally explicit/vertically implicit (HEVI) solution of wave equations. J. Comput. Phys., 252, 365–381, https://doi.org/10.1016/j.jcp.2013.06.025.

    • Search Google Scholar
    • Export Citation

  • Wicker, L. J., and W. C. Skamarock, 2002: Time-splitting methods for elastic models using forward time schemes. Mon. Wea. Rev., 130, 2088–2097, https://doi.org/10.1175/1520-0493(2002)130<2088:TSMFEM>2.0.CO;2.

    • Search Google Scholar
    • Export Citation

  • Williams, P. D., 2013: Achieving seventh-order amplitude accuracy in leapfrog integrations. Mon. Wea. Rev., 141, 3037–3051, https://doi.org/10.1175/MWR-D-12-00303.1.

    • Search Google Scholar
    • Export Citation

  • Williamson, D. L., J. B. Drake, J. J. Hack, R. Jakob, and P. N. Swarztrauber, 1992: A standard test set for numerical approximations to the shallow water equations in spherical geometry. J. Comput. Phys., 102, 211–224, https://doi.org/10.1016/S0021-9991(05)80016-6.

    • Search Google Scholar
    • Export Citation

  • Wood, N., and Coauthors, 2014: An inherently mass-conserving semi-implicit semi-Lagrangian discretization of the deep-atmosphere global non-hydrostatic equations. Quart. J. Roy. Meteor. Soc., 140, 1505–1520, https://doi.org/10.1002/qj.2235.

    • Search Google Scholar
    • Export Citation

  • Yazgi, D., A. R. Mohebalhojeh, and S. Ghader, 2017: Using polynomial regression in designing the time filters for the leapfrog time-stepping scheme. Mon. Wea. Rev., 147, 1779–1795, https://doi.org/10.1175/MWR-D-16-0380.1.

    • Search Google Scholar
    • Export Citation
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Monthly Weather Review

  • Amezcua, J., and P. D. Williams, 2015: The composite-tendency Robert-Asselin-Williams (RAW) filter in semi-implicit integrations. Quart. J. Roy. Meteor. Soc., 141, 764–773, https://doi.org/10.1002/qj.2391.

    • Search Google Scholar
    • Export Citation

  • Asselin, R., 1972: Frequency filter for time integrations. Mon. Wea. Rev., 100, 487–490, https://doi.org/10.1175/1520-0493(1972)100<0487:FFFTI>2.3.CO;2.

    • Search Google Scholar
    • Export Citation

  • Benacchio, T., and R. Klein, 2019: A semi-implicit compressible model for atmospheric flows with seamless access to soundproof and hydrostatic dynamics. Mon. Wea. Rev., 147, 4221–4240, https://doi.org/10.1175/MWR-D-19-0073.1.

    • Search Google Scholar
    • Export Citation

  • Carr, L. E., III, and R. T. Williams, 1989: Barotropic vortex stability to perturbations from axisymmetry. J. Atmos. Sci., 46, 3177–3191, https://doi.org/10.1175/1520-0469(1989)046<3177:BVSTPF>2.0.CO;2.

    • Search Google Scholar
    • Export Citation

  • Chan, J. C. L., and R. T. Williams, 1987: Analytical and numerical studies of the beta-effect in tropical cyclone motion. Part I: Zero mean flow. J. Atmos. Sci., 44, 1257–1265, https://doi.org/10.1175/1520-0469(1987)044<1257:AANSOT>2.0.CO;2.

    • Search Google Scholar
    • Export Citation

  • Chew, R., T. Benacchio, G. Hastermann, and R. Klein, 2022: A one-step blended soundproof-compressible model with balanced data assimilation: Theory and idealized tests. Mon. Wea. Rev., 150, 2231–2254, https://doi.org/10.1175/MWR-D-21-0175.1.

    • Search Google Scholar
    • Export Citation

  • Durran, D. R., 1989: Improving the anelastic approximation. J. Atmos. Sci., 46, 1453–1461, https://doi.org/10.1175/1520-0469(1989)046<1453:ITAA>2.0.CO;2.

    • Search Google Scholar
    • Export Citation

  • Durran, D. R., 1991: The third-order Adams-Bashforth method: An attractive alternative to leapfrog time differencing. Mon. Wea. Rev., 119, 702–720, https://doi.org/10.1175/1520-0493(1991)119<0702:TTOABM>2.0.CO;2.

    • Search Google Scholar
    • Export Citation

  • Durran, D. R., 1999: Numerical Methods for Wave Equations in Geophysical Fluid Dynamics. Springer-Verlag, 465 pp.

  • Durran, D. R., 2008: A physically motivated approach for filtering acoustic waves from the equations governing compressible stratified flow. J. Fluid Mech., 601, 365–379, https://doi.org/10.1017/S0022112008000608.

    • Search Google Scholar
    • Export Citation

  • Durran, D. R., and P. N. Blossey, 2012: Implicit-explicit multistep methods for fast-wave-slow-wave problems. Mon. Wea. Rev., 140, 1307–1325, https://doi.org/10.1175/MWR-D-11-00088.1.

    • Search Google Scholar
    • Export Citation

  • Hack, J. J., and R. Jakob, 1992: Description of a global shallow water model based on the spectral transform method. NCAR Tech. Note NCAR/TN-343+STR, 41 pp., https://opensky.ucar.edu/islandora/object/technotes%3A112/datastream/PDF/download/Description_of_a_Global_Shallow_Water_Model_Based_on_the_Spectral_Transform_Method.citation.

  • Kurowski, M. J., W. W. Grabowski, and P. K. Smolarkiewicz, 2013: Towards multiscale simulation of moist flows with soundproof equations. J. Atmos. Sci., 70, 3995–4011, https://doi.org/10.1175/JAS-D-13-024.1.

    • Search Google Scholar
    • Export Citation

  • Lipps, F. B., and R. S. Hemler, 1982: A scale analysis of deep moist convection and some related numerical calculations. J. Atmos. Sci., 39, 2192–2210, https://doi.org/10.1175/1520-0469(1982)039<2192:ASAODM>2.0.CO;2.

    • Search Google Scholar
    • Export Citation

  • Mahalov, A., and M. Moustaoui, 2009: Vertically nested nonhydrostatic model for multi-scale resolution of flows in the upper troposphere and lower stratosphere. J. Comput. Phys., 228, 1294–1311, https://doi.org/10.1016/j.jcp.2008.10.030.

    • Search Google Scholar
    • Export Citation

  • Moustaoui, M., H. Teitelbaum, C. Basdevant, and Y. Boughaleb, 2002: Linked behavior of twin tropical cyclones. J. Geophys. Res., 107, 4378, https://doi.org/10.1029/2000JD000066.

    • Search Google Scholar
    • Export Citation

  • Moustaoui, M., B. Joseph, and H. Teitelbaum, 2004: Mixing layer formation near the tropopause due to gravity wave critical level interactions in a cloud-resolving model. J. Atmos. Sci., 61, 3112–3124, https://doi.org/10.1175/JAS-3289.1.

    • Search Google Scholar
    • Export Citation

  • Moustaoui, M., A. Mahalov, and E. J. Kostelich, 2014: A numerical method based on leapfrog and a fourth-order implicit time filter. Mon. Wea. Rev., 142, 2545–2560, https://doi.org/10.1175/MWR-D-13-00073.1.

    • Search Google Scholar
    • Export Citation

  • Ogura, Y., and N. A. Phillips, 1962: Scale analysis of deep and shallow convection in the atmosphere. J. Atmos. Sci., 19, 173–179, https://doi.org/10.1175/1520-0469(1962)019<0173:SAODAS>2.0.CO;2.

    • Search Google Scholar
    • Export Citation

  • Robert, A. J., 1966: The integration of a low order spectral form of the primitive meteorological equations. J. Meteor. Soc. Japan, 44, 237–245, https://doi.org/10.2151/jmsj1965.44.5_237.

    • Search Google Scholar
    • Export Citation

  • Skamarock, W. C., and J. B. Klemp, 1992: The stability of time-split numerical methods for the hydrostatic and the nonhydrostatic elastic equations. Mon. Wea. Rev., 120, 2109–2127, https://doi.org/10.1175/1520-0493(1992)120<2109:TSOTSN>2.0.CO;2.

    • Search Google Scholar
    • Export Citation

  • Skamarock, W. C., and J. B. Klemp, 2008: A time-split nonhydrostatic atmospheric model for research and NWP applications. J. Comp. Phys., 227, 3465–3485, https://doi.org/10.1016/j.jcp.2007.01.037.

    • Search Google Scholar
    • Export Citation

  • Skamarock, W. C., J. B. Klemp, M. G. Duda, L. D. Fowler, S.-H. Park, and T. D. Ringler, 2012: A multiscale nonhydrostatic atmospheric model using centroidal Voronoi tesselations and C-Grid staggering. Mon. Wea. Rev., 140, 3090–3105, https://doi.org/10.1175/MWR-D-11-00215.1.

    • Search Google Scholar
    • Export Citation

  • Smolarkiewicz, P. K., C. Kuhnlein, and N. P. Wedi, 2014: A consistent framework for discrete integrations of soundproof and compressible PDEs of atmospheric dynamic. J. Comput. Phys., 263, 185–205, https://doi.org/10.1016/j.jcp.2014.01.031.

    • Search Google Scholar
    • Export Citation

  • Straka, J. M., R. B. Wilhelmson, L. J. Wicker, J. R. Anderson, and K. K. Droegemeier, 1993: Numerical solutions of a non-linear density current: A benchmark solution and comparisons. Int. J. Numer. Methods Fluids, 17, 1–22, https://doi.org/10.1002/fld.1650170103.

    • Search Google Scholar
    • Export Citation

  • Thuburn, J., 2017: Use of the Gibbs thermodynamic potential to express the equation of state in atmospheric models. Quart. J. Roy. Meteor. Soc., 143, 1185–1196, https://doi.org/10.1002/qj.3020.

    • Search Google Scholar
    • Export Citation

  • Weller, H., S.-J. Lock, and N. Wood, 2013: Runge-Kutta IMEX schemes for the horizontally explicit/vertically implicit (HEVI) solution of wave equations. J. Comput. Phys., 252, 365–381, https://doi.org/10.1016/j.jcp.2013.06.025.

    • Search Google Scholar
    • Export Citation

  • Wicker, L. J., and W. C. Skamarock, 2002: Time-splitting methods for elastic models using forward time schemes. Mon. Wea. Rev., 130, 2088–2097, https://doi.org/10.1175/1520-0493(2002)130<2088:TSMFEM>2.0.CO;2.

    • Search Google Scholar
    • Export Citation

  • Williams, P. D., 2013: Achieving seventh-order amplitude accuracy in leapfrog integrations. Mon. Wea. Rev., 141, 3037–3051, https://doi.org/10.1175/MWR-D-12-00303.1.

    • Search Google Scholar
    • Export Citation

  • Williamson, D. L., J. B. Drake, J. J. Hack, R. Jakob, and P. N. Swarztrauber, 1992: A standard test set for numerical approximations to the shallow water equations in spherical geometry. J. Comput. Phys., 102, 211–224, https://doi.org/10.1016/S0021-9991(05)80016-6.

    • Search Google Scholar
    • Export Citation

  • Wood, N., and Coauthors, 2014: An inherently mass-conserving semi-implicit semi-Lagrangian discretization of the deep-atmosphere global non-hydrostatic equations. Quart. J. Roy. Meteor. Soc., 140, 1505–1520, https://doi.org/10.1002/qj.2235.

    • Search Google Scholar
    • Export Citation

  • Yazgi, D., A. R. Mohebalhojeh, and S. Ghader, 2017: Using polynomial regression in designing the time filters for the leapfrog time-stepping scheme. Mon. Wea. Rev., 147, 1779–1795, https://doi.org/10.1175/MWR-D-16-0380.1.

    • Search Google Scholar
    • Export Citation
  • <sc>Fig</sc>. 1.
    View in gallery
    Fig. 1.

    Magnitude of the numerical amplification factor corresponding to the physical mode from the proposed scheme as a function of the frequencies of fast ωfΔt and slow ωsΔt modes. The coefficient of the time filter is (a) γ = 0.03 and (b) γ = 0.06.

  • <sc>Fig</sc>. 2.
    View in gallery
    Fig. 2.

    (a) Magnitude of the amplification factors for the physical mode (solid), the 2Δt (long dashed–dashed) computational mode, and the 4Δt (dotted) computational mode as a function of ωΔt obtained for γ = 0.03 along a line with slope Θ = π/8 in Fig. 1. (b) Argument of the physical mode (solid). The dashed lines in (a) and (b) represent the magnitude and argument of the exact amplification factor. (c),(d) As in (a) and (b), but the slope is Θ = π/4. (e),(f) As in (a) and (b), but the slope is Θ = 3π/8.

  • <sc>Fig</sc>. 3.
    View in gallery
    Fig. 3.

    (a) Magnitude of the amplification factors for the physical mode (solid), the 2Δt (long dashed–dashed) computational mode, and the 4Δt (dotted) computational mode as a function of ωΔt obtained for γ = 0.03 along a line with slope Θ = 5π/8 in Fig. 1. (b) Argument of the physical mode (solid). The dashed lines in (a) and (b) represent the magnitude and argument of the exact amplification factor. (c),(d) As in (a) and (b), but the slope is Θ = 3π/4. (e),(f) As in (a) and (b), but the slope is Θ = 7π/8.

  • <sc>Fig</sc>. 4.
    View in gallery
    Fig. 4.

    (a) Magnitude of the amplification factors for the physical mode (solid), the 2Δt (long dashed–dashed) computational mode, and the 4Δt (dotted) computational mode as a function of ωΔt obtained for γ = 0.03 along a line with slope Θ = 0 in Fig. 1. (b) Argument of the physical mode (solid). The dashed lines in (a) and (b) represent the magnitude and argument of the exact amplification factor. (c),(d) As in (a) and (b), but the slope is Θ = π/2.

  • <sc>Fig</sc>. 5.
    View in gallery
    Fig. 5.

    Cross sections of vertical velocity after 3 h from (a) analytical calculations, (b) numerical computations with the proposed scheme using a filter coefficient of γ = 0.03, and (c) a scheme using the RK3 method for the slow modes. Negative contours are dashed, and the contour interval is 0.2 m s−1. (d) The difference between the numerical solution obtained by the proposed scheme and the analytical solution (shaded). The difference is amplified by a factor of 100. The numerical solution is also superimposed. (e) As in (d), but the scheme employs the RK3 method for the slow modes.

  • <sc>Fig</sc>. 6.
    View in gallery
    Fig. 6.

    Cross sections of potential temperature computed after 54 000 time steps from (a) the proposed scheme using a filter coefficient of γ = 0.03 and (b) the scheme using the RK3 method for the slow modes. (c),(d) As in (a) and (b), but the number of time steps is 61 000. The fields simulated after (e),(f) 64 000 and (g),(h) 67 000 time steps.

  • <sc>Fig</sc>. 7.
    View in gallery
    Fig. 7.

    Cross sections of potential temperature computed after 900 s for (a) the proposed scheme using a filter coefficient of γ = 0.03 and (b) the scheme using the RK3 method for the slow modes.

  • <sc>Fig</sc>. 8.
    View in gallery
    Fig. 8.

    Fluid elevation and wind vector fields computed after 40 h from (a) the proposed scheme using a filter coefficient of γ = 0.03 and (b) the scheme using the RK3 method for the slow modes. The contour interval is 10 m. (c),(d) As in (a) and (b), but the time is 80 h. (e),(f) The fields simulated at 120 h.

Fig. 1.

Magnitude of the numerical amplification factor corresponding to the physical mode from the proposed scheme as a function of the frequencies of fast ωfΔt and slow ωsΔt modes. The coefficient of the time filter is (a) γ = 0.03 and (b) γ = 0.06.
Fig. 2.

(a) Magnitude of the amplification factors for the physical mode (solid), the 2Δt (long dashed–dashed) computational mode, and the 4Δt (dotted) computational mode as a function of ωΔt obtained for γ = 0.03 along a line with slope Θ = π/8 in Fig. 1. (b) Argument of the physical mode (solid). The dashed lines in (a) and (b) represent the magnitude and argument of the exact amplification factor. (c),(d) As in (a) and (b), but the slope is Θ = π/4. (e),(f) As in (a) and (b), but the slope is Θ = 3π/8.
Fig. 3.

(a) Magnitude of the amplification factors for the physical mode (solid), the 2Δt (long dashed–dashed) computational mode, and the 4Δt (dotted) computational mode as a function of ωΔt obtained for γ = 0.03 along a line with slope Θ = 5π/8 in Fig. 1. (b) Argument of the physical mode (solid). The dashed lines in (a) and (b) represent the magnitude and argument of the exact amplification factor. (c),(d) As in (a) and (b), but the slope is Θ = 3π/4. (e),(f) As in (a) and (b), but the slope is Θ = 7π/8.
Fig. 4.

(a) Magnitude of the amplification factors for the physical mode (solid), the 2Δt (long dashed–dashed) computational mode, and the 4Δt (dotted) computational mode as a function of ωΔt obtained for γ = 0.03 along a line with slope Θ = 0 in Fig. 1. (b) Argument of the physical mode (solid). The dashed lines in (a) and (b) represent the magnitude and argument of the exact amplification factor. (c),(d) As in (a) and (b), but the slope is Θ = π/2.
Fig. 5.

Cross sections of vertical velocity after 3 h from (a) analytical calculations, (b) numerical computations with the proposed scheme using a filter coefficient of γ = 0.03, and (c) a scheme using the RK3 method for the slow modes. Negative contours are dashed, and the contour interval is 0.2 m s−1. (d) The difference between the numerical solution obtained by the proposed scheme and the analytical solution (shaded). The difference is amplified by a factor of 100. The numerical solution is also superimposed. (e) As in (d), but the scheme employs the RK3 method for the slow modes.
Fig. 6.

Cross sections of potential temperature computed after 54 000 time steps from (a) the proposed scheme using a filter coefficient of γ = 0.03 and (b) the scheme using the RK3 method for the slow modes. (c),(d) As in (a) and (b), but the number of time steps is 61 000. The fields simulated after (e),(f) 64 000 and (g),(h) 67 000 time steps.
Fig. 7.

Cross sections of potential temperature computed after 900 s for (a) the proposed scheme using a filter coefficient of γ = 0.03 and (b) the scheme using the RK3 method for the slow modes.
Fig. 8.

Fluid elevation and wind vector fields computed after 40 h from (a) the proposed scheme using a filter coefficient of γ = 0.03 and (b) the scheme using the RK3 method for the slow modes. The contour interval is 10 m. (c),(d) As in (a) and (b), but the time is 80 h. (e),(f) The fields simulated at 120 h.
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