## 1. Introduction

^{1}proposed a consistency relationship for nonequatorial quasigeostrophic flowswhere Δ

*x*is the horizontal grid interval, Δ

*z*is the vertical grid interval (in pseudo height coordinates),

*N*is the buoyancy frequency, and

*f*is the Coriolis parameter. Pecnick and Keyser (1989) and Persson and Warner (1991, hereafter PW91) focused on fronts and derived a relationship,which precludes numerical modes near fronts. In the relationship,

*s*is the slope of the front. PW91 show that both (1) and (2) give criteria similar to Δ

*z*/Δ

*x*∼ 0.005–0.02 for typical midlatitude values.

Spurious waves due to inconsistency of horizontal and vertical resolutions have been indicated in the past studies related to front (e.g., Gall et al. 1988; Snyder et al. 1993; Bush et al. 1995; Jorgensen et al. 2003; Lean and Clark 2003). PW91 proposed a generation mechanism for spurious waves and it has been referred to as an explanation of the spurious waves by following works. They use a two-dimensional *σ*-coordinate hydrostatic model. Temperature fields in the frontal plane are represented as a staircase because of insufficient vertical resolution (see Fig. 6 in PW91). The geopotential Φ(*z*) above the frontal plane is also modified because it represents the vertical integral of a term including temperatures with intervals from the surface to level *z*, and the interval includes the altitude of the front. This induces spurious waves.

However, the following two points are not considered in their interpretation. First, their theory predicts gravity waves extending upward from the front, but their simulation shows waves that were confined to the thin region of warm air. Second, their formation theory uses hydrostatic equations and it may not fully explain spurious waves that develop in nonhydrostatic models. To explain the spurious waves obtained in our simulation with the nonhydrostatic system, we propose an another formation mechanism of spurious waves that is based on a mountain-wave theory. It includes staircase temperatures as in PW91, but not perturbations in the geopotential. One notable difference from PW91 is that, in our theory, basic wind plays an important role in determining the strength and propagation of spurious waves. It is shown in the discussion that wind distribution could explain the confinement of waves in PW91.

For practical calculation, conditions (2) and (1) are too severe to be satisfied in models, especially with small horizontal grid interval Δ*x* of a few kilometers where clouds are resolved. If *s* = 0.005, then (2) requires Δ*z* < 50 m for Δ*x* = 10 km and Δ*z* < 5 m for Δ*x* = 1 km. If *f*/*N* = 0.008, as is typical for midlatitudes, (1) requires Δ*z* = 80 m for Δ*x* = 10 km and Δ*z* = 8 m for Δ*x* = 1 km. In general, cloud-resolving models do not use vertical grid intervals as fine as Δ*z* ∼ 10 m. This study will investigate the existence of spurious waves in high-resolution models with nonhydrostatic effects, evaluate the intensity of the spurious waves, and describe their generation mechanism. Finally, an effective method to eliminate numerical modes is proposed.

The model used in this study is a new global nonhydrostatic model [Nonhydrostatic Icosahedral Atmospheric Model (NICAM)] developed at the Frontier Research Center for Global Change (FRCGC). Recent cloud-resolving experiments were performed for an aqua planet with horizontal grid intervals down to 3.5 km (Tomita et al. 2005). Prior to the aqua planet experiments, a series of life cycle experiments for extratropical cyclones were performed using the dynamical core of NICAM to study numerical convergence in the model. Results from these experiments will be shown in section 3. These experiments revealed a consistency relationship between horizontal and vertical grid intervals that is one focus of this study.

Section 2 describes the experimental design of NICAM. Numerical convergence of the solution is investigated in section 3 using the dynamical core experiment proposed by Polvani et al. (2004). Numerical viscosity is fixed so the synoptic structure remains the same as the increases in resolution. Resolution-dependent numerical viscosity is used in section 4 to study the resolution dependency of the synoptic structure using the same dynamical core experiment. Spurious waves are generated as numerical modes if the resolution consistency is not satisfied. The numerical modes are quantitatively analyzed using a linear model in section 5. Numerical modes can be eliminated if a vertically stretched grid is used, as noted in section 6. Section 7 includes the discussion and conclusions.

## 2. Description of the numerical model

The dry dynamical core of NICAM (Tomita and Satoh 2004) is used for calculations in this study. NICAM is a nonhydrostatic global model with icosahedral grids. The icosahedral grids are quasi-uniformly distributed on the sphere. Therefore, NICAM does not suffer from polar problems that arise in conventional latitude–longitude grids models. As the horizontal grid spacing decreases, computer performance in NICAM becomes more effective than in spectral models with Legendre transformations (Satoh et al. 2005).

*n*, where

*n*is the number of grid divisions. The equivalent grid interval is conveniently defined aswhere 10 × 4

^{glevel}is the half number of triangles, and

*a*is the radius of the earth. NICAM has the same amount of grid per area to a model with square grid Δ

*x*on a side.

Table 1 shows the relationship between glevel and Δ*x*. Variables are all at the vertex of each triangle as in an Arakawa-A-type coallocation grid. A Lorenz grid (Lorenz 1960) is adopted for vertical discretization. Vertical velocity is at the midpoint of other variables.

NICAM uses fully compressible nonhydrostatic equations (Tomita and Satoh 2004). The time-splitting method (Klemp and Wilhemson 1978) efficiently calculates fast propagating waves such as acoustic waves. Integration is achieved using a second-order Runge–Kutta scheme for large time steps, and a forward–backward scheme based on horizontally explicit and vertically implicit schemes, together with a flux division method for small time steps.

*ν*

_{2n}∇

^{2n}

_{H}(

**u**,

*T*), where

*ν*

_{2}

*is the viscosity coefficient,*

_{n}**u**is the velocity,

*T*is the temperature, and ∇

*is the horizontal component of ∇;*

_{H}*n*= 1 is chosen in section 3 and

*n*= 2 in section 4, and their coefficients are shown later. Divergence damping is required for the time-splitting method to reduce acoustic waves. Its original form in Skamarock and Klemp (1992) is

*κ*

_{2}∇

*(∇ ·*

_{H}**u**), where

*κ*is the damping coefficient. In NICAM, instead, the fourth-order form

_{n}*κ*

_{4}∇

*[∇*

_{H}*· ∇*

_{H}*(∇ ·*

_{H}**u**)] is adopted to damp short-scale modes selectively. The coefficient

*κ*

_{4}, shown in Tables 2 and 3, is chosen so that the damping time for the grid scale

*dx*,is halved as Δ

*x*is halved. The present study includes no vertical diffusion.

## 3. Numerical convergence of the solutions

*z̃*≡ −

*H*log(

*p*/

*p*

_{0}) and

*f*= 2Ωsin

*ϕ*. Values of

*u*

_{0},

*z̃*

_{0}, Δ

*z̃*

_{0},

*z̃*

_{1},

*H*,

*p*

_{0},

*R*,

*a*, and Ω are taken from Table 1 in the test case paper of Polvani et al. (2004). Here

*T*

_{0}(

*z̃*) is chosen so that, at each pressure

*p*, the global average of

*T*is identical to the

*U.S. Standard Atmospheric, 1976*temperature value. Variables are hydrostatically and geostrophically balanced. In addition to the basic field, temperature perturbation,is defined where

*λ*is the longitude and

*T̂*= 1 K for all cases, except for glevel = 10 and 11 where

*T*′(

*λ*,

*ϕ*) = 0 for

*λ*< 0.

^{2}The vertical domain extends from 0–30 km with no surface friction.

*ν*

_{2}= 7.0 × 10

^{5}m

^{2}s

^{−1}. For the smallest scale Δ

*x*, the corresponding diffusion timeis about 19.9–0.31 h for glevels 5–8, respectively. These times are much smaller than the damping time

*τ*

_{κ4}, which is 69.1–8.64 h for glevels 5–8 (Table 3), and therefore divergence damping effects are negligible.

Figure 1 shows vorticity at *σ* = 0.975 and *t* = 12 days for four horizontal resolutions with glevel = 5–8. Vorticity values at *σ* = 0.975 are derived from the velocity at *z* = 300 and 900 m so that results could be compared to those of Polvani et al. (2004). Solutions converge as horizontal resolution increases, and the shape, amplitude, and phase of the converged solution are very close to those of Fig. 4 of Polvani et al. (2004). However, the amplitude of the synoptic cyclone is slightly weaker to the east (developing-cyclone region) and slightly larger to the west (decaying-cyclone region). This discrepancy may arise from differences in the diffusion operator; numerical diffusion in NICAM is calculated on *z* planes, whereas numerical diffusion in Polvani et al. (2004) is calculated on *σ* planes.

*ζ*is the vorticity and

*ζ*is the vorticity at glevel 8;

_{T}*l*

_{2}on

*σ*= 0.975 is shown in Fig. 2;

*l*

_{2}at

*t*= 12 is 0.44, 0.12, and 0.02 for glevels 5, 6, and 7, respectively, which shows that

*l*

_{2}decays roughly on the order of Δ

*x*

^{−2}. This decay is reasonable because NICAM uses second-order horizontal discretizations. Dependency on the time step interval was also examined, but it does not affect the solution if the time step satisfies the Courant–Friedrichs–Lewy (CFL) condition.

## 4. Dependency of frontal structure on resolutions

In the experiments of previous section, horizontal diffusion was fixed for all resolutions to support an examination of numerical convergence. However, smaller values of numerical diffusion are normally used as resolution increases to resolve finer structures. This section describes LCE with ultrahigh horizontal resolution up to glevel = 11, that is, Δ*x* = 3.5 km. Diffusion coefficients that depend on resolution are used.

*ν*

_{4}is defined byThe diffusion coefficient

*υ*

_{4}was chosen so that the diffusion time

*τ*

_{ν}_{4}is halved as Δ

*x*is halved. Horizontal diffusion terms in the experiments in this section are added not only to the equations of (

*u, υ, w*) and

*T*, but also

*ρ*to efficiently dampen high wavenumber computational modes. Table 4 shows horizontal and vertical resolutions.

### a. Cases with sufficient vertical resolution

Results are presented from runs with a vertical resolution of Δ*z* = 150 m for horizontal resolution glevel = 5, 7, and 9. In these cases, vertical resolution is sufficient in the sense that spurious waves are not generated. Figure 3 shows temperature fields at *z* = 75 m at 9 days in the simulation. Four baroclinic cyclones are moving eastward. They are, from west to east, A: decayed cyclone, B: decaying cyclone, C: developed cyclone, and D: developing cyclone. Cyclone B develops from the initial disturbance. Cyclones A, C, and D are secondary and are generated as Rossby waves propagate from the initial disturbance. Synoptic behaviors and propagating speeds are roughly independent of horizontal resolutions at least for glevel ≥ 5. This is consistent with results from Methven and Hoskins (1998), who used a spectral model to show that the overall behaviors in cyclone life cycles did not change as resolution increased from T95 (Δ*x* ∼140 km) to T341 (Δ*x* ∼39 km).

Structural differences between various resolutions include sharper fronts in cyclones as glevel increases. Furthermore, cyclone B has longer swirl in the central part of the vortex. Spiral structures in the cyclone are also apparent in PV contours (not shown), as discussed at length in Methven and Hoskins (1998). Two regions of strong upward motion are present in cyclone C along the northern occluded front and the southeastern cold front (Fig. 4). Figure 5 is a cross section (40°N, 28°–18°W) denoted by A–B in Fig. 4.^{3} The regions of upward motion are generated at the leading edge of the fronts (Figs. 5a, 5b and 5g), and they narrow and intensify as glevel increases. For glevel = 9, the maximum vertical velocity is about 0.06 m s^{−1}. Vertical resolution dependence is also investigated for Δ*z* = 150 and 300 m with glevel = 8 in Fig. 5c. Vertical resolution does not affect frontal structure as long as Δ*z*/Δ*x* < *s _{c}* holds, where

*s*is defined in the next subsection.

_{c}*u*,

_{f}*υ*) is moving velocity of the front and

_{f}*γ*is angle between the front and meridian. This is the horizontal velocity component normal to the front in a Doppler-shifted frame that moves with the front. Here, we roughly estimate the front tilt as

*γ*= 28° and the frontal velocity as (

*u*,

_{f}*υ*) = (12.6, 0) using the figures at

_{f}*t*= 9 and 8.75 days. Figure 6 shows

*u*. Positive (negative)

_{d}*u*indicates flow from the warm (cold) air, directed from right to left (left to right) in the figure. Upward motion originates in regions of warm southerly flow near the ground. Horizontal flows in these experiments strengthen near the ground because of the frictionless lower boundary condition. In a more realistic situation with friction, boundary flow and upwelling will be weaker.

_{d}Gravity wave generation from fronts was first suggested in Gall et al. (1988). In our case of glevel ≥ 8, the upward motion from the edge generates a secondary gravity wave that propagates up and to the west (Figs. 5a,b). Such kinds of waves generated from the edge of fronts are well discussed in Snyder et al. (1993). In our case, the amplitude of the gravity wave decreases with height, and the wave eventually dissipates at a level that is near the critical level, which is indicated by the *u _{d}* = 0 contour. The critical level marks the upper limit of propagation of front-generated gravity waves. It will also play an important role in the dynamics of spurious waves, as shown next.

### b. Cases with insufficient vertical resolution

Figures 5d–f,h show cases where vertical resolutions in these cases are insufficient. Results in these experiments all are degraded by spurious gravity waves that extend up from the cold front. Gravity waves are prominent when the cold front is sharp, and they vanish as the cold front becomes weak. The shape of these waves resembles that of Gall et al. (1988). Snyder et al. (1993) pointed out that the waves in Gall’s results are spurious due to insufficient vertical resolution, indicating that the waves in Figs. 5d, 5e, 5f and 5h are also spurious. Table 5 shows that maximum vertical velocity in these waves increases as the horizontal resolution increases and the vertical resolution decreases. The horizontal scale of the waves is proportional to the horizontal grid interval. The propagating direction becomes upward as the horizontal resolution increases. Spurious waves accompany cyclones B, C, and D. Spurious waves do not occur in cyclone A, where frontogenesis is incomplete at *t* = 10 days.

*z*/Δ

*x*exceeds a critical value

*s*of 0.0107–0.0214, which is 2–4 times larger than the value derived from the frontal slope

_{c}*s*= 0.00523 and

*f*/

*N*= 0.00763 at a latitude of 40°. The critical value

*s*obtained by experiments and from the frontal slope

_{c}*s*introduced in (2) may differ because the horizontal viscosity reduces the effective horizontal resolution of the NICAM. Discretization errors in the horizontal difference scheme may also reduce the effective horizontal resolution. If the effective horizontal grid intervalreplaces Δ

*x*in (2), condition (2) correctly explains the appearance of the spurious waves In next section Δ

*x*is used to analyze the mechanisms behind spurious waves.

_{e}## 5. Linear model for spurious waves

Spurious waves are characterized by features that resemble flow over steep orography (Gallus and Klemp 2000), especially for glevel = 11. The following generation mechanism is therefore proposed for spurious waves. Temperature fields in the frontal plane have a staircase structure because of insufficient vertical resolution (as in PW91). The staircase frontal plane behaves as if it is real terrain. Then, flow parallel to the frontal plane collides with the staircase, generating gravity waves from the staircase edges. The gravity waves resemble mountain waves. Figure 8 schematically outlines the mechanism. The Doppler-shifted wind speed parallel to the frontal plane controls the strength and propagation of the waves.

*x*–

*z*plane) linear system that uses the Boussinesq approximation:where (

*u, υ, w*) are the velocity components in (

*x, y, z*) directions,

*p*is the pressure,

*f*is the Coriolis parameter,

*θ*is the potential temperature, and

*α*is a damping coefficient that is nonzero only at upper levels. Basic field variables are denoted by an overbar, and anomalies relative to the basic field are denoted by primes. Consider stationary mountain waves, where ∂/∂

*t*is neglected, for which the lower boundary is a staircase front plane. For simplicity, the focus is on a single staircase step, defined bywhere

*z*is the height of the lower boundary, and

_{b}*β*is a smoothing parameter.

^{4}Consider a domain in the range −0.5

*R*≤

_{f}*x*≤ 0.5

*R*. The parameter

_{f}*β*represents the difference between Δ

*x*and Δ

*x*in (12);

_{e}*β*= 1 corresponds to the case with Δ

*x*= Δ

_{e}*x*. Generally,

*β*> 1, and from (12), a value

*β*= 3 is chosen. The second term on the right-hand side is added to use the Fourier transformation such that

*z*has no jump at

_{b}*x*= ±0.5

*R*when horizontally periodic boundary conditions are adopted;

_{f}*R*is sufficiently large that the second term does not affect the region near

_{f}*x*= 0, where topography abruptly changes. The solution for a multistep staircase arises from a superposition of one-step solutions. If

*w*′ = 0 at the upper boundary and damping is added to upper levels, then waves that do not originate from mountains are eliminated. Waves with downward group velocity or waves that increase exponentially with

*z*do not appear in the linear model. The following function

*(*u

_{d}*z*) is used for the basic wind

*. It mimics the profile obtained in results shown in Fig. 6:A value of*u

*N*

^{2}= 0.00015 s

^{−2}is used above the frontal plane below the tropopause. The linear system (13)–(17) can be solved with the above boundary conditions and basic field in (19).

Figure 9 compares spurious waves obtained by NICAM at *t* = 9 days to mountain waves obtained from the linear analysis. For glevel = 10 and 11, the linear solutions match the spurious waves obtained by NICAM both in shape and amplitude. However, the spurious waves that develop in NICAM fracture and spread laterally above 3000 m, eventually disappearing. Spurious waves become obscure above 3000 m because the finite horizontal and vertical resolutions in NICAM cannot resolve fine structure. For glevel = 9, the simulated wave train is directed horizontally, in contrast to the linear theory wherein the wave train is directed vertically. In this case, the distance from the next staircase is relatively short, and we should probably consider the superposition of waves from each staircase.

The linear model confines the gravity waves to levels below the critical level, where * u _{d}* = 0. Gravity waves at the three resolutions shown in Fig. 9 are indeed confined, until

*t*= 10 days when the front dissipates and weak secondary gravity waves develop above the critical level. Overall, the linear model explains the generation of the spurious waves, except when the front decays.

## 6. Frontal structure with a vertically stretched grid

The linear analysis in the previous section guides an effective way to choose a vertical grid that eliminates spurious waves: A smaller grid interval Δ*z* is required in a layer with larger *u _{d}*. If

*u*is small or the front is obscure, the large grid interval Δ

_{d}*z*is also permissible. In the present case, smaller grid intervals are required in layers below

*z*< 4000 m. Atmospheric general circulation models commonly incorporate vertically stretched grids such that grid points are densely packed in the boundary layer and more widely distributed in the free atmosphere. Hence, a stretched grid, which is similar to the grid used in a cloud-resolving model to investigate squall lines (Redelsperger et al. 2000), is used to investigate the behavior of spurious waves. The vertical grid interval is 35.5 m at the surface and slowly increases with height until

*z*= 13 km, above which height the grid interval is 700 m.

Figure 10 shows results using the stretched vertical grid for glevel = 9 and 10. Spurious waves are completely eliminated for glevel = 9. For glevel = 10, spurious waves are eliminated below *z* = 2500 m, but remain near *z* = 2500 m. Maximum vertical velocity is 0.06 m s^{−1}. These results indicate that the vertical grid spacing is too broad near the wave source level *z* = 2500 m for glevel = 10. At *z* = 2500 m, Δ*z* is about 450 m. To eliminate these spurious waves, Δ*z* must be less than half that at glevel = 10.

## 7. Discussion and conclusions

This study includes life cycle experiments for baroclinic waves in a model with varying horizontal and vertical resolutions. When the ratio of vertical and horizontal grid intervals Δ*z*/Δ*x* is sufficiently small compared to the frontal slope *s*, results are not sensitive to resolution. However, as horizontal resolution increases, fronts become sharper, vertical motions generated at the leading edge of the fronts strengthen, and the spiral structure in the center of decaying cyclones lengthens. Synoptic features, such as the location and growth rates of baroclinic cyclones, are similar when glevel ≥ 5 (corresponding horizontal resolution Δ*x* ∼223 km), which is consistent with results of past studies. Vertical resolution does not affect model results unless the aspect ratio of the vertical and horizontal grid intervals is smaller than the slope of the front.

When Δ*z*/Δ*x* exceeds a critical value between 2*s* and 4*s*, spurious gravity waves are generated at cold fronts. The generation mechanism of these spurious waves can be quantitatively explained using a linear model based on mountain-wave theory. In the linear model, cold fronts acquire a staircase shape because of insufficient vertical resolution, and the corner of the staircase cold front generates gravity waves as if it was a step mountain. For the cases Δ*z*/Δ*x* < *s* in our simulation, the linear model does not explain the absence of spurious waves. In these cases, horizontal numerical viscosity determined by Δ*x* is relatively large and it probably suppresses the spurious waves.

PW91 used as an initial condition a thin wedge of slantwise distributed warm air in a two-dimensional model, producing conditional symmetric instability (CSI). They showed spurious waves that were confined to the region of warm air. However, such confinement is inconsistent with their theory that predicts gravity waves extending beyond the formation region. A mechanism in the linear model discussed in this study may explain the wave confinement of these waves. In PW91, warm air is rising, and is sandwiched between ambient air with weak winds (for simplicity, assume zero). At both boundaries of the warm air, gravity waves with phase velocity zero are generated by mountain wave mechanism and propagate through the warm air. (Note that in the ambient air, waves are not promoted because velocity of both wind and boundary are zero). Then they are absorbed at the other boundary, which is a critical level for the waves with phase velocity zero. Thus, the waves are confined to the region of warm air.

One might suspect that the obtained spurious waves are property inherent in the Lorenz grid that NICAM adopts. It is well known that the Lorenz grid supports computational modes with a zigzag vertical structure, and it also deforms physical modes causing spurious growth of short baroclinic waves (Arakawa and Moorthi 1988). However, at least for glevel = 8 or 9, dz = 300 or 600 m, vertical wavelength is longer than the scale expected by the effect of the Lorenz grid. Obtained spurious waves are mostly consistent to our theory for all resolution. Therefore, it seems that the adoption of the Lorenz grid does not largely affect our conclusion. One may also think that the choice of vertical scheme affects our results. Although some scheme might smooth the staircase leading to weaker spurious waves, our theory is essentially independent from vertical schemes.

Strong spurious waves developed in the simulation with the highest horizontal resolution. The maximum vertical velocity of the waves, ∼0.4 m s^{−1} for Δ*x* = 3.5 km and Δ*z* = 600 m, is sufficient to generate clouds if moist effects are included. But the impact of the spurious waves on the basic field is temporary and limited to the area near the front. During the simulation, cyclones develop in a similar way for all resolutions, and it seems that the spurious waves merely add noise, at least in our resolutions.

Some adjustments are required to apply our results to reality, because our simulations are on a dry system without any surface friction. Many of the real cold fronts have slopes steeper than ours, alleviating the condition (2). Multiple slantwise circulation by CSI (e.g., Persson and Warner 1993; Lean and Clark 2003) may form critical levels in basic wind near the front, confining spurious waves near the front. Realistic surface friction may weaken basic wind near the surface reducing spurious waves.

Results from a linear model suggest an effective way to eliminate these waves. A smaller Δ*z* must be chosen for levels with larger * u _{d}*. If spurious waves are generated only at cold fronts, then small Δ

*z*is required in the lower troposphere near the ground, and larger Δ

*z*is allowed at upper levels. Spurious waves were eliminated in model simulations that used the stretched grid, although weak spurious waves still remain for glevel = 10 (Δ

*x*= 7 km).

We are grateful to Dr. Tetsuya Takemi and our coworkers in Next-Generation Model Development Group/FRCGC for their helpful discussion. We are also grateful to anonymous reviewers for their helpful comments. This research was supported by CREST, JST. The Earth Simulator in Japan Agency for Marine-Earth Science and Technology is used for the simulations by NICAM.

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The glevel and the equivalent grid interval.

Time step and divergence damping coefficients used for resolution convergence experiments. Damping time is defined as *τ _{κ}*

_{4}≡ Δ

*x*

^{4}/

*κ*

_{4}.

Time step, horizontal diffusion, and divergence damping rate used for the experiments with resolution-dependent diffusion. Diffusion time is *τ _{ν}*

_{4}≡ Δ

*x*

^{4}/

*ν*

_{4}and damping time is

*τ*

_{κ}_{4}≡ Δ

*x*

^{4}/

*κ*

_{4}.

Horizontal and vertical resolutions used for the experiments in section 4.

Maximum vertical velocity (m s^{−1}) of spurious waves in Fig. 5.

^{1}

They also proposed another relationship for flows that contain gravity waves near a critical layer.

^{2}

The exception is because of our mis-setting. Developments of cyclones with this exceptional initial condition are not so different from those with the correct one, although the phase differs by about 10°.

^{4}

When Δ*z*/Δ*x* is close to *s*, each step is close relative to the horizontal wave scale, and spurious waves are superposition of waves from each staircase. When Δ*z*/Δ*x* = *s*, each wave interferes with each other by the superposition, and if *z _{b}* is replaced by max[min(

*x*/Δ

*x*, 0.5), 0.5] Δ

*z*, spurious waves vanish in theory.