1. Introduction
Developers of atmospheric dynamical cores regularly struggle against time step restrictions. The full Euler system of equations, which describe the terrestrial atmosphere below the rarefied exosphere (Izakov 1971), admits motions due to advection, gravity waves, and sound waves. Dynamical cores with local, explicit timestepping find their maximum stable time steps limited by a Courant number restriction, whereby the fastestpropagating waves must travel no more than O(1) grid lengths per time step. For nearly five decades (Robert et al. 1972), researchers have sought to extend the usable range of time steps, both through approximations to the Euler equation [Robert et al. (1972) using a hydrostatic approximation, others such as Arakawa and Konor (2009) and Dubos and Voitus (2014) removing sound waves through alternative formulations of the Euler equations including anelastic and pseudoincompressible forms] and through implicit treatment of some or all terms of the equation.
The semiLagrangian method is a popular treatment for the advective terms in the Euler equations. This method treats the material derivative nonlocally by integrating it in a Lagrangian frame, deriving “departurepoint” values for fluid parcels that arrive on the designated computational grid at the end of each time step. First applied to weather modeling by Robert (1981) [with a historical review given by Staniforth and Côté (1991)], semiLagrangian advection further extended the admissible, stable time step size over the nextbest algorithms by (often) allowing simulations with advective Courant numbers much larger than one. The approach quickly became popular, and it remains in use today at the heart of many operational atmospheric forecasting models (Girard et al. 2014; Wood et al. 2014; ECMWF 2020) in use today.
Despite the method’s popularity, it can also be the source of computational “noise” in long simulations. The most wellknown source of such noise is spurious orographic resonance, first described by Rivest et al. (1994). Here, in the case of subcritical flow over topography, the advection terms are treated accurately via a semiLagrangian approach, but a timecentered implicit treatment of the gravity wave terms slows down their phase speed, resulting in c → u_{0} in the hypothetical limit of an arbitrarily large time step. As a result, gravity waves that would ordinarily propagate upstream (u_{0} − c < 0) can become trapped over topography, leading to resonance and inaccuracies. McDonald (1998) discusses additional sources of noise, including noise from some formulations of trajectory calculations, modeltop boundary conditions, and physical parameterizations.
Despite these descriptions, there is no exhaustive list of “noise” sources in implicit, semiLagrangian simulations. Husain et al. (2020) has recently reported anomalous “valley warm pools” arising from flow over topography, and the anomaly is not adequately explained by the above phenomena. Topographymediated effects such as this may be more common in highresolution simulations over mountainous regions, where the topographic signal can have both a large amplitude and short wavelength.
Although some sources of noise have been addressed individually, the operational GEM model (Girard et al. 2014; Husain et al. 2019) still sees a stubborn need for relatively large damping (Husain and Girard 2017) to address forecast noise. This is particularly concerning for emerging higherorder model discretizations, where overly diffusive or dissipative “fixes” can eat away at any performance advantage from an improved numerical foundation.
In this work, we further expand on the literature of potential instabilities and “noise” in semiLagrangian, timecentered simulations of the Euler equations by focusing on a type of secondary instability in the context of the shallowwater equations. In this family of instabilities, flow over topography first induces a steadystate response, but linearizing the discretized equations of motion about this background state reveal a rich field of instabilities. The largest of these instabilities can extract energy from the background flow into spurious modes over a wide range of topography scales and Courant numbers.
Overview
In section 2, we first discuss the shallowwater formulation used in this work, beginning with the continuous equations and ending with the linearized, discretized equations used to explore the system’s instabilities. Section 3 discusses the instabilities in more detail, deriving necessary conditions based on the dispersion relation of the linearized shallowwater equations. Section 4 discusses potential mitigation strategies, and we conclude in section 5 with a discussion of the implications for future analysis and model development.
The numerical development in this work is intended to closely follow the discretizations and approximations made in the GEM model.
2. Shallowwater equations
a. Timecontinuous equations
While (1) is relatively simple, it captures the interaction between advection, topographic forcing, and gravity waves in a manner similar to atmospheric flow over mountainous regions. As presented, these equations neglect the effects of planetary rotation in order to simplify the system as much as possible, and the subsequent numerical experiments will focus on regimes that would have a Rossby number greater than one if rotation terms were present.
1) Steadystate solutions
In this work, we investigate instabilities in discretized versions of (1) under steady flow over oscillatory topography, which we use as a stylized reduction of flow over mountainous regions. The topography height H is small relative to the characteristic layer thickness ϕ_{0}, and the background flow consists of a mean flow u_{0} with small variations, where u_{0} is significantly smaller than the characteristic phase speed
When (3) is used as an approximate solution to (2), the residual unsteady oscillation remains small and bounded.
2) Linearization
For the steadystate background flows considered in this paper, (5) is linearly stable, without exponentially growing solutions.^{1}
In this work, we investigate instabilities in the discretized version of (1), rather than continuous instabilities that would reflect physical phenomena such as extraction of energy from a background current. The consequential approach is to discretize (1) in its full nonlinear form, then to look at the linear response of an infinitesimal perturbation about a steady background state.
b. Time discretization
In the Lagrangian interpretation, Eq. (6c) expresses the simple idea that fluid parcels are advected by the local velocity field, and in other notational conventions this may be written as
Equation (7c) acts to define fluid parcel trajectories over the finite interval. Rearranged, it gives SL(x) = x − (Δt/2) [u^{+} + SL(u^{−})], which is a restatement of the implicit trapezoidal rule for calculating trajectories.
Steadystate solutions and linearization
Unlike the fully continuous equations, (8) is not guaranteed to have wellbehaved solutions. For
For realistic scenarios where H is small compared to the mean fluid depth, however, these restrictions are not very binding. In the smalltopography limit,
As a result, steadystate solutions to (8) are generally available. As with section 2a, we are interested in solutions that are close to a “topographyfree” state of (ϕ_{0}, u_{0}). The steadystate
It is important to stress here that the spatial derivative ∂_{x} is that induced by an infinitesimal increment to the fluid parcel trajectory. In the case of the fully discrete system, this implied derivative operator is not necessarily the same as the derivative operator used for the gravity wave terms of (7a) and (7b).
The timediscretized system (7), with a steadystate background flow given by (8), is unstable to infinitesimal perturbations if any of the eigenvalues in the linearized system (13) have a magnitude greater than one. Since (13) is not analytically tractable, we instead proceed by discretizing in space and solving the discrete eigenvalue problem.
c. Spatial discretization and numerical implementation
To investigate the stability of (13), we implement a spatial discretization in MATLAB, alongside an equivalent discretization of (7) (to allow for time integration) and (8) (to compute steadystate solutions). Overall, this discretization follows the basic framework of GEM, using an Arakawa Cgrid. In one dimension, this becomes a simple grid staggering, where u and ϕ points are offset by (1/2)Δx, and for the sake of this analysis we take the underlying grid to be periodic in x with an overall domain size of L_{x}. The topographic height H is also assigned to the ϕ grid.
The spatial derivatives used in the gravity wave terms [(⋅)_{x}] are approximated as divided differences, where the approximate derivative “lives” on the other grid.
For the initial analysis, the semiLagrangian operator will be implemented through a Fourier expansion, expanding the interpolated function in its Fourier series and evaluating the terms at the departure point (Boyd 2001). This preserves the amplitude of wavelike functions, and it will allow us to more directly examine the stability implications of the temporal discretization. The use of polynomial interpolation, which is more conventional and used in the operational GEM model, is discussed in section 4. The ∂_{x} terms which appear in (13) are evaluated in the semiLagrangian context. For the Fourier interpolation this is computed by evaluating the derivative of the Fourier series offgrid, and for polynomial interpolation this is computed by evaluating the derivative of the respective Lagrange interpolating polynomials.
For timevarying and steadystate computations, the semiLagrangian trajectories (those implied in the SL and
The linear system corresponding to (13) is full because of the Fourier interpolation in the semiLagrangian terms, but computation of its eigenvalues and eigenvectors is still quite tractable because the onedimensional system is relatively small. To explore this system, we consider a parameter space that is stylistically representative of flow past topography in highresolution, localarea simulations of the atmosphere:

Δx = 1 km, with N_{x} = 513 equally spaced points^{2} on both the ϕ and u grids giving an overall domain size (L_{x}) of 513 km

ϕ_{0} = 6 km, approximately the atmospheric scale height, and u_{0} = 20 m s^{−1}, borrowed from Williamson et al. (1992)’s first test case.

H(x) = H_{0} sin(k_{hill}x) to present topography with a single wavelength. H_{0} = 100 m, and k_{hill} will vary below.
This specification ensures that the magnitude of the topographic perturbation (H_{0}/ϕ_{0}) is small (about 1.7 × 10^{−2}), so the steadystate solution to (8) is a small perturbation to the uniform state (ϕ_{0}, u_{0}). The characteristic Froude number
A typical example of this parameter regime is shown in Fig. 1, which depicts the layer height (ϕ + H) after 4 h of simulated time under a reference solution (a Fouriertransform spectral method for the spatial discretization and Matlab’s ode45 routine for time integration) and this discretized system at Courant numbers (C = Δtu_{0}/Δx) of 0.5 and 1.5, with k_{hill} ≈ 2π/(5Δx), and initial conditions given by the approximation (3).^{3} While the reference solution and smalltimestep semiLagrangian solution are well behaved, generating a small gravity wave field of amplitude comparable to the initial perturbation, the largetimestep semiLagrangian solution develops an extremely large gravity wave, with an amplitude several times larger than the topography’s sinusoidal amplitude..
This is one example of the instabilities present in the timediscretized system (7), and these instabilities are present over a large fraction of the reasonable parameter space. Varying Δt and k_{hill} changes the effective Courant number and topographic forcing wavenumber. Varying C between 0.05 and 4 (Δt between 2.5 and 200 s) over 200 equally spaced steps and k_{hill} between 2π/2.5L_{x} and 2π/3Δx over 167 equally spaced steps and finding the most unstable eigenmode of the linearized system (13) gives the graph of Fig. 2.
Normalized maximum growth rates for the system of (13) under the conditions described in section 2c, showing the growth rate per hour of the most unstable mode for each (C, k_{hill}) pairing. Four different unstable branches appear: (a) is spurious orographic resonance, (b) is advective aliasing, (c) is wave spatial aliasing, and (d) is wave temporal aliasing. The instability growth rate is shown via the colored regions, with a lower bound of 1.5 (50% amplification per hour) chosen to highlight the (b) and (c) branches.
Citation: Monthly Weather Review 150, 3; 10.1175/MWRD210054.1
Normalized maximum growth rates for the system of (13) under the conditions described in section 2c, showing the growth rate per hour of the most unstable mode for each (C, k_{hill}) pairing. Four different unstable branches appear: (a) is spurious orographic resonance, (b) is advective aliasing, (c) is wave spatial aliasing, and (d) is wave temporal aliasing. The instability growth rate is shown via the colored regions, with a lower bound of 1.5 (50% amplification per hour) chosen to highlight the (b) and (c) branches.
Citation: Monthly Weather Review 150, 3; 10.1175/MWRD210054.1
Normalized maximum growth rates for the system of (13) under the conditions described in section 2c, showing the growth rate per hour of the most unstable mode for each (C, k_{hill}) pairing. Four different unstable branches appear: (a) is spurious orographic resonance, (b) is advective aliasing, (c) is wave spatial aliasing, and (d) is wave temporal aliasing. The instability growth rate is shown via the colored regions, with a lower bound of 1.5 (50% amplification per hour) chosen to highlight the (b) and (c) branches.
Citation: Monthly Weather Review 150, 3; 10.1175/MWRD210054.1
Throughout this parameter space, the Lipschitz trajectory crossing criterion (Δ_{t}u_{x} < 1) is well respected, and the obstacle Courant number (Δtu divided by the topographic wavelength) remains below one everywhere except the upperright corner of Fig. 2.
3. Analysis
Most of the instabilities depicted in Fig. 2 take on a generic form: a wavelike eigenmode that oscillates at a single, dominant wavenumber k, with smallerscale oscillations at shorter wavelengths, with corresponding wavenumbers k ± jk_{hill} for integer j. A typical example is shown in the bottom panel of Fig. 1.
The exact mechanics of the coupling change depending on the branch of the instability marked in Fig. 2.
a. Spurious orographic resonance
The most striking feature of Fig. 2 is the large central region excluded from the contour plot. Here, the steadystate solution process described in section 2b fails through the wellknown “spurious orographic resonance” process, with the excluded region matching the first unstable region in Fig. 1 of Rivest et al. (1994). In the region of Fig. 2 with orographic resonance, ω(k_{hill}) ≈ 0, and perturbation of the initially undisturbed system by topographic forcing generates standing waves with zero phase speed. In the language of this section, the k = 0 “wave”—the steadystate solution—shares ω = 0 with gravity waves that would have k = ±k_{hill}, and the topography at k = k_{hill} provides the coupling mechanism.
In a timedependent simulation initialized from a uniform state, the topographic forcing accumulates with no linear decay or other damping, and this leads to no physically relevant steadystate solution.
In practice, weather models such as GEM use offcentering to partially mitigate this issue, which will be discussed further in section 4.
b. Advective aliasing
Figure 3 depicts the situation with k_{hill} ≈ 0.633 km^{−1} (wavelength = 9.92Δx) and Δt = 55.5 s, corresponding to point (b) in Fig. 2, giving an advective Courant number of 1.11—comfortably within the expected range of validity for a semiLagrangian, implicit method. In this figure, the eigenmodes of the advective operator
(top) Eigenvalue magnitudes and (bottom) dispersion relation for the (left) advective operator and (right) shallowwater operators (open circles in blue for downstreampropagating waves and solid circles in red for upstreampropagating waves) at the point marked (b) in Fig. 2. The dashed black vertical lines mark ±k_{hill}. In the bottom panels, symbols show the calculated dispersion relation while the line shows the linear estimate. To improve readability, only a selection of modes with nearunitamplitude eigenvalues are shown.
Citation: Monthly Weather Review 150, 3; 10.1175/MWRD210054.1
(top) Eigenvalue magnitudes and (bottom) dispersion relation for the (left) advective operator and (right) shallowwater operators (open circles in blue for downstreampropagating waves and solid circles in red for upstreampropagating waves) at the point marked (b) in Fig. 2. The dashed black vertical lines mark ±k_{hill}. In the bottom panels, symbols show the calculated dispersion relation while the line shows the linear estimate. To improve readability, only a selection of modes with nearunitamplitude eigenvalues are shown.
Citation: Monthly Weather Review 150, 3; 10.1175/MWRD210054.1
(top) Eigenvalue magnitudes and (bottom) dispersion relation for the (left) advective operator and (right) shallowwater operators (open circles in blue for downstreampropagating waves and solid circles in red for upstreampropagating waves) at the point marked (b) in Fig. 2. The dashed black vertical lines mark ±k_{hill}. In the bottom panels, symbols show the calculated dispersion relation while the line shows the linear estimate. To improve readability, only a selection of modes with nearunitamplitude eigenvalues are shown.
Citation: Monthly Weather Review 150, 3; 10.1175/MWRD210054.1
This sinusoidal perturbation is replicated in
This gives rise to aliasing when k_{0} is near the Nyquist limit k_{nyq} = π/Δx. For such a positive k_{0}, k_{0} + k_{hill} > k_{nyq}, and on the discrete grid this highwavenumber wave appears identical to a wave with k_{1} = k_{0} + k_{hill} − 2k_{nyq}. This aliasing in turn leads to an instability when the corresponding temporal frequencies are aliased, with exp(−ik_{0}Δtu_{0}) ≈ exp(−ik_{1}Δtu_{0}). When this condition is satisfied, waves with wavenumbers k_{0} and k_{1} oscillate together in time, and when the topographic forcing produces waves of mode k_{1} out of the k_{0} mode (or vice versa) this energy accumulates.
In brief, topographic forcing via the advection operator causes a propagating, singlewavenumber mode to produce daughter waves separated in wavenumber by k_{hill}, which may spatially alias across the Nyquist frequency. When the leading mode and its daughter waves share a timealiased temporal frequency, this wave production remains coherent, leading to a growing response with amplitude proportional to H_{0}/ϕ_{0}.
This instability of this section arises from the semiLagrangian operator itself, and it can potentially act as a noisegenerating process for passively advected tracers such as moisture or chemical constituents. This instability is inherent to a finitetime semiLagrangian advection, and it does not depend on any particular trajectory calculation methodology: a function that is barely resolved on a discrete grid becomes unresolved, with aliasing error, after its locally compressed by semiLagrangian trajectories that converge.
c. Wave spatial aliasing
An analogous effect can also occur for gravity wave modes, even if the instability of section 3b is not present. Figure 4 depicts the situation with k_{hill} ≈ 0.953 km^{−1} (wavelength = 6.59Δx) and Δt = 12.43 s, giving an advective Courant number of approximately 0.25. This set of parameters is marked as (c) in Fig. 2, along the thin line of instability at low Courant numbers.
(top) Eigenvalue magnitudes and (bottom) dispersion relation for the shallowwater operator at the point marked (c) in Fig. 2. The dashed vertical lines denote ±k_{hill}. As in Fig. 3, the blue open circles correspond to downstreampropagating modes, while the red filled circles correspond to upstreampropagating modes. For clarity, a reduced set of nearunitamplitude modes are shown, and in the bottom panel the solid lines show the dispersion relation calculated from (15).
Citation: Monthly Weather Review 150, 3; 10.1175/MWRD210054.1
(top) Eigenvalue magnitudes and (bottom) dispersion relation for the shallowwater operator at the point marked (c) in Fig. 2. The dashed vertical lines denote ±k_{hill}. As in Fig. 3, the blue open circles correspond to downstreampropagating modes, while the red filled circles correspond to upstreampropagating modes. For clarity, a reduced set of nearunitamplitude modes are shown, and in the bottom panel the solid lines show the dispersion relation calculated from (15).
Citation: Monthly Weather Review 150, 3; 10.1175/MWRD210054.1
(top) Eigenvalue magnitudes and (bottom) dispersion relation for the shallowwater operator at the point marked (c) in Fig. 2. The dashed vertical lines denote ±k_{hill}. As in Fig. 3, the blue open circles correspond to downstreampropagating modes, while the red filled circles correspond to upstreampropagating modes. For clarity, a reduced set of nearunitamplitude modes are shown, and in the bottom panel the solid lines show the dispersion relation calculated from (15).
Citation: Monthly Weather Review 150, 3; 10.1175/MWRD210054.1
Finding any instability at such a low Courant number is surprising, but downstreampropagating gravity waves suffer from the same sort of aliasing as in Fig. 3. In this case, the shallowwater operators themselves contribute the necessary secondary modes to a perturbation that initially oscillates with a single wavenumber, but the full dispersion relation possesses the same sort of temporal aliasing.
Unlike the conditions for advective aliasing in (19), the more complicated functional form of (20) results in at most a few nearresonant spatial frequencies for any given configuration.
d. Wave temporal aliasing
With larger Courant numbers, the temporal aliasing can occur independently of spatial aliasing. Figure 5 depicts the eigenspectrum with k_{hill} ≈ 1.22 km^{−1} (wavelength = 5.14Δx) and Δt = 83.60 s, giving an advective Courant number of approximately 1.672. This set of parameters is marked as (d) in Fig. 2, and this sort of behavior (for various unstable perturbation wavenumbers) is typical in the large region of instability near the limit of spurious orographic resonance.
(top) Eigenvalue magnitudes and (bottom) dispersion relation for the shallowwater operator at the point marked (d) in Fig. 2. The dashed vertical lines mark ±k_{hill}. The symbols and solid lines have the same meaning as in Fig. 4.
Citation: Monthly Weather Review 150, 3; 10.1175/MWRD210054.1
(top) Eigenvalue magnitudes and (bottom) dispersion relation for the shallowwater operator at the point marked (d) in Fig. 2. The dashed vertical lines mark ±k_{hill}. The symbols and solid lines have the same meaning as in Fig. 4.
Citation: Monthly Weather Review 150, 3; 10.1175/MWRD210054.1
(top) Eigenvalue magnitudes and (bottom) dispersion relation for the shallowwater operator at the point marked (d) in Fig. 2. The dashed vertical lines mark ±k_{hill}. The symbols and solid lines have the same meaning as in Fig. 4.
Citation: Monthly Weather Review 150, 3; 10.1175/MWRD210054.1
The form of the most unstable mode is shown in Fig. 6. For this set of parameters, the unstable mode takes the form of a long wave (with wavelength of about 85.7 km, or k_{0} ≈ 7.33 × 10^{−2} km^{−1}), which has a shorter wave superimposed on it at lower amplitude. That smaller wave has a wavelength comparable to the topography, with k_{1} ≈ 1.15 km^{−1}. Both waves are well resolved on the grid.
(top) Perturbation in the ϕ field corresponding to the most unstable mode in Fig. 5. (bottom) The unstable mode consists of a large, longwavelength oscillation modified by an oscillation with a wavelength similar to that of the underlying topography.
Citation: Monthly Weather Review 150, 3; 10.1175/MWRD210054.1
(top) Perturbation in the ϕ field corresponding to the most unstable mode in Fig. 5. (bottom) The unstable mode consists of a large, longwavelength oscillation modified by an oscillation with a wavelength similar to that of the underlying topography.
Citation: Monthly Weather Review 150, 3; 10.1175/MWRD210054.1
(top) Perturbation in the ϕ field corresponding to the most unstable mode in Fig. 5. (bottom) The unstable mode consists of a large, longwavelength oscillation modified by an oscillation with a wavelength similar to that of the underlying topography.
Citation: Monthly Weather Review 150, 3; 10.1175/MWRD210054.1
This instability requires a higher Courant number than that of section 3c, since the frequency wraparound necessary for (15) to alias must occur over two waves separated by k_{hill} rather than the much larger k_{nyq}. This occurs more readily for downstreampropagating waves, where the background velocity and gravity wave propagation effects combine to give a much higher effective gravity wave Courant number.
Overall, the semiLagrangian treatment of advection is “doing its job” quite well: from one time step to the next it advects the ϕ and u fields with the background velocity. However, the net phase speed of a linear mode is given not only by the background velocity, but also by its propagation as a gravity wave. When the conditions of (21) are met, a pair of modes separated by k_{hill} oscillate at the same frequency, while topographic forcing causes each mode of the pair to feed energy into its partner. This instability is a “moving spurious orographic resonance,” and it does not depend on the precise details of semiLagrangian trajectory calculations.
4. Mitigations
Section 3 demonstrates that the shallowwater system, adapted to steady flow over topography, is unstable to certain infinitesimal perturbations over a wide parameter range. However, operational dynamical cores already contain diffusive and dissipative elements that are not present in the idealized, timecentered discretization discussed here, and some of these stabilizing elements may also affect these instabilities. This section discusses the extent to which realistic choices made in dynamical cores might mitigate the presence or strength of the instabilities.
For this purpose, the instability described in section 3d will be used as a prototypical example, with modifications as noted.
a. Need for mitigations
The first approach to mitigate these instabilities may be to do nothing at all. Over most of the parameter space in Fig. 2, these instabilities grow relatively slowly (by a factor of 3–10 per hour). Additionally, a typical forecast domain does not look like a sinusoidal mountain range, so an instability will generally have only a finite “contact time” to extract energy from the background flow. If left totally unmitigated, how badly can these instabilities corrupt an otherwise smooth, steady solution?
To observe this, we take the configuration of section 3d, perturb ϕ pointwise by white noise with a standard deviation of 1 m, or 10^{−2} times the amplitude of the hill, and simulate the resulting system via (7) with a Newton solver used for the nonlinear implicit terms. The simulation is run for a period of 8 h, and typical results are shown in Fig. 7. Despite the random perturbation, the system effectively selects the most quickly growing modes, and those begin to dominate over the 2–4h interval.
(top) Layer height (ϕ + H, solid) corresponding to the case of section 3d after 8 h of simulated time, beginning with a random ϕ perturbation from the steady state (dashed). (bottom) Rootmeansquare amplitude of the perturbation
Citation: Monthly Weather Review 150, 3; 10.1175/MWRD210054.1
(top) Layer height (ϕ + H, solid) corresponding to the case of section 3d after 8 h of simulated time, beginning with a random ϕ perturbation from the steady state (dashed). (bottom) Rootmeansquare amplitude of the perturbation
Citation: Monthly Weather Review 150, 3; 10.1175/MWRD210054.1
(top) Layer height (ϕ + H, solid) corresponding to the case of section 3d after 8 h of simulated time, beginning with a random ϕ perturbation from the steady state (dashed). (bottom) Rootmeansquare amplitude of the perturbation
Citation: Monthly Weather Review 150, 3; 10.1175/MWRD210054.1
After reaching an initial rootmeansquare amplitude of 100 m, the growth rate of the perturbation begins to slow down. By the end of the simulation, the perturbation has not become large enough to cause the model to “crash,” but the perturbation amplitude has reached about 600 m, several times the amplitude of the underlying topography. While the perturbation growth has effectively saturated because of nonlinear effects, it has effectively converted all of the energy from the background flow variation into the unstable mode. The overall flow structure no longer reflects reality.
b. Adjustments to the nonlinear solver
Operational weather models typically do not solve the nonlinear implicit system of (7) to convergence. As an example, in most operational configurations the GEM dynamical core truncates the equivalent solution process (for the Euler or hydrostatic equations) at four iterations, consisting of two outer trajectory iterations and two inner, nonlinear iterations (with frozen semiLagrangian trajectories). The inner solution step involves a linear solve using the common quasilinear approximation, where the full Jacobian of the Euler equation system is replaced with a linearization about a prescribed atmospheric state.
Under a wide range of circumstances, iterating (22) converges to the solution of (7), giving stability results identical to those of section 4a. Using even two iterations leads to instability growth at a rate very close to that predicted for the full nonlinear system; this is shown in the top panel of Fig. 8.
Rootmeansquare amplitude (solid) of the random perturbation whose evolution was shown in Fig. 7 over an 8h simulated period, with an iterative solver truncated after (top) two and (bottom) one iterations, using a basic state of ϕ_{ref} = 1.1ϕ_{0}. The bottom panel corresponds to semiimplicit timestepping, and in each panel the growth rate curve of Fig. 7 is shown (dashed) for reference.
Citation: Monthly Weather Review 150, 3; 10.1175/MWRD210054.1
Rootmeansquare amplitude (solid) of the random perturbation whose evolution was shown in Fig. 7 over an 8h simulated period, with an iterative solver truncated after (top) two and (bottom) one iterations, using a basic state of ϕ_{ref} = 1.1ϕ_{0}. The bottom panel corresponds to semiimplicit timestepping, and in each panel the growth rate curve of Fig. 7 is shown (dashed) for reference.
Citation: Monthly Weather Review 150, 3; 10.1175/MWRD210054.1
Rootmeansquare amplitude (solid) of the random perturbation whose evolution was shown in Fig. 7 over an 8h simulated period, with an iterative solver truncated after (top) two and (bottom) one iterations, using a basic state of ϕ_{ref} = 1.1ϕ_{0}. The bottom panel corresponds to semiimplicit timestepping, and in each panel the growth rate curve of Fig. 7 is shown (dashed) for reference.
Citation: Monthly Weather Review 150, 3; 10.1175/MWRD210054.1
However, taking just one iteration is special. With the initial guess ϕ* = ϕ^{−} and u* = u^{−}, this constitutes a semiimplicit timestepping algorithm^{4} equivalent to that described for the Euler equations by Simmons and Temperton (1997), where the time centering of the gravity wave terms is applied only to the reference state. This algorithm is generally stable when ϕ_{ref} > max(ϕ), and here we take ϕ_{ref} = 1.1ϕ_{0}. Under these circumstances, the random perturbation decays with time, leading to the amplitude graph shown in the bottom panel of Fig. 8.
Unfortunately, semiimplicit time stepping is also in general less accurate than the fully implicit trapezoidal rule. Because the timeaveraging is not applied to ϕ − ϕ_{ref} or u in the gravity wave terms, only small perturbations from the reference state are propagated with second order in time. Large “nonlinear” motions are evaluated on a forwardintime basis at first order, and the ϕ_{ref} > max(ϕ) requirement also contributes to the loss of accuracy by pushing the reference state further from a true linearization of the problem.
The SETTLS approach (Hortal 2002), incorporated into the IFS model (ECMWF 2020), regains a secondorder discretization by extrapolating the nonlinear terms along the semiLagrangian trajectory, but a straightforward formulation has a reduced stability region compared to the fully implicit formulation (Durran and Reinecke 2004), such that dissipation included in the linear term may still be required for overall, unconditional stability. It is not clear whether the secondary instability described in this work has an equivalent with SETTLS extrapolation.
c. SemiLagrangian dissipation
Another potential mitigation for these instabilities is the dissipation inherent in practical semiLagrangian interpolation. Thus far, the semiLagrangian interpolation used has been a Fourier interpolation that exactly reproduces single frequencies up to the grid resolution (Nyquist) limit. This requires O(N^{2}^{d}) operations to interpolate N^{d} points over d dimensions, so it is impractical in any realistic threedimensional model. Instead, these models generally rely on local polynomial interpolation, and cubic interpolation is a popular choice (e.g., ECMWF 2020; Girard et al. 2014).
The effect of both linear and cubic interpolation is to selectively damp higher frequencies, and in the context of the shallowwater equations this damping becomes preferentially stronger when the advective Courant number is close to n + 1/2 for integer n. For linear interpolation, this effect is quite severe, and in the lowk limit it acts as a strong diffusive term. For cubic interpolation, the effect is one of a higher order (in k) hyperdiffusion, and as a result cubic interpolation is preferred for horizontal interpolation in modern dynamical cores.
Either form of polynomial interpolation eliminates the highwavenumber instabilities shown in section 3b and 3c because they both strongly dissipate waves near the Nyquist limit. However, neither the strong diffusion of linear interpolation nor the weaker hyperdiffusion of cubic interpolation eliminates the instabilities described in section 3d. Figures 9 and 10 show growth rates over the (C, k_{hill}) parameter space analogously to Fig. 2 for linear and cubic semiLagrangian interpolation respectively, and the same basic pattern is evident.
As in Fig. 2, but computed using linear interpolation for the semiLagrangian operators in the computation of the steadystate and linearized operators. A similar pattern of instability exists, but it is somewhat diminished in amplitude because of the dissipation of the semiLagrangian operator. Thin bands of instability remain at integer Courant numbers.
Citation: Monthly Weather Review 150, 3; 10.1175/MWRD210054.1
As in Fig. 2, but computed using linear interpolation for the semiLagrangian operators in the computation of the steadystate and linearized operators. A similar pattern of instability exists, but it is somewhat diminished in amplitude because of the dissipation of the semiLagrangian operator. Thin bands of instability remain at integer Courant numbers.
Citation: Monthly Weather Review 150, 3; 10.1175/MWRD210054.1
As in Fig. 2, but computed using linear interpolation for the semiLagrangian operators in the computation of the steadystate and linearized operators. A similar pattern of instability exists, but it is somewhat diminished in amplitude because of the dissipation of the semiLagrangian operator. Thin bands of instability remain at integer Courant numbers.
Citation: Monthly Weather Review 150, 3; 10.1175/MWRD210054.1
As in Figs. 2 and 9, but computed using cubic interpolation for the semiLagrangian operators in the computation of the steadystate and linearized operators. The resulting region of instability is larger and more intense than that arising from linear interpolation, and the instability bands at integral Courant numbers are less apparent.
Citation: Monthly Weather Review 150, 3; 10.1175/MWRD210054.1
As in Figs. 2 and 9, but computed using cubic interpolation for the semiLagrangian operators in the computation of the steadystate and linearized operators. The resulting region of instability is larger and more intense than that arising from linear interpolation, and the instability bands at integral Courant numbers are less apparent.
Citation: Monthly Weather Review 150, 3; 10.1175/MWRD210054.1
As in Figs. 2 and 9, but computed using cubic interpolation for the semiLagrangian operators in the computation of the steadystate and linearized operators. The resulting region of instability is larger and more intense than that arising from linear interpolation, and the instability bands at integral Courant numbers are less apparent.
Citation: Monthly Weather Review 150, 3; 10.1175/MWRD210054.1
The growth rates are overall lower for linear interpolation, but this comes at the cost of unacceptable levels of diffusion—especially for tracerlike constituents such as atmospheric moisture content. Additionally, the case of linear semiLagrangian interpolation remains unstable at nearinteger Courant numbers, with this residual instability caused by abrupt changes of the interpolating stencil when departure points of adjacent cells lie alternately just to the left and just to the right of grid points. However, these bands of instability are very thin, and even if they were observed in a realistic simulation it would take only a small adjustment to the time step to stabilize the simulation.
Cubic interpolation gives instability growth rates that are intermediate between the linear and Fourier interpolation, as would be expected from its higher order.
d. Offcentering
Dynamical cores commonly implement an offcentering parameter to Eqs. (7) for stability purposes. Offcentering adds dissipation to the gravity wave modes of the system, and this dissipation acts to limit the effects of spurious orographic resonance. However, the impact on the resonant growth discussed in this work is less clear.
Although not directly applicable to the more complicated basic state of flow past topography, this functional form gives us reason to worry: dissipation is relatively weak for longer waves, and the instabilities generated by the mechanism of section 3d have a longwave component.
Testing the interaction of offcentering with the case shown in Fig. 5 initially shows promise—an offcentering parameter as small as α = 0.525 effectively eliminates the instability at the cost of dissipating higherfrequency gravity waves, giving the gravity wave spectrum shown in Fig. 11. The maximum eigenvalue in this case remains slightly greater than one (about 1.003), but the resulting growth rate is about 15% h^{−1}, small enough that it would likely be unnoticed in a realistic simulation.
As in Fig. 5, with an offcentering parameter of α = 0.525. The maximum eigenvalue magnitude is now 1.000 394, leading to a negligible growth rate for the unstable mode, but higher frequency gravity waves have an eigenvalue magnitude of about 0.9, representing a 10% dissipation per time step.
Citation: Monthly Weather Review 150, 3; 10.1175/MWRD210054.1
As in Fig. 5, with an offcentering parameter of α = 0.525. The maximum eigenvalue magnitude is now 1.000 394, leading to a negligible growth rate for the unstable mode, but higher frequency gravity waves have an eigenvalue magnitude of about 0.9, representing a 10% dissipation per time step.
Citation: Monthly Weather Review 150, 3; 10.1175/MWRD210054.1
As in Fig. 5, with an offcentering parameter of α = 0.525. The maximum eigenvalue magnitude is now 1.000 394, leading to a negligible growth rate for the unstable mode, but higher frequency gravity waves have an eigenvalue magnitude of about 0.9, representing a 10% dissipation per time step.
Citation: Monthly Weather Review 150, 3; 10.1175/MWRD210054.1
However, one other degree of freedom remains: the topography amplitude.
From the analysis in section 3d, the overall magnitude of the instability is closely related to the ratio of A_{hill} to ϕ_{0}. Thus far we have worked with A_{hill} = 100 m in dimensional units to ensure that the background flow is still a small perturbation from uniform motion, but mountains tend to be significantly taller than 100 m. Figure 12 depicts the growth rate when A_{hill} and α are modified simultaneously, and the results are not encouraging—a hill amplitude of 398 m recovers the instability with α as large as 0.6.
Growth rates for the most unstable mode of the case discussed in section 3d, for hill heights 10 ≤ A_{hill} ≤ 398 m, spaced evenly on a logarithmic scale, and offcentering parameter 0.5 ≤ α ≤ 0.6. The A_{hill} = 100 m line corresponds exactly to that shown in section 3d, and its growth rate is trivial for α ≥ 0.52. Larger hill heights quickly overcome the damping effect of offcentering.
Citation: Monthly Weather Review 150, 3; 10.1175/MWRD210054.1
Growth rates for the most unstable mode of the case discussed in section 3d, for hill heights 10 ≤ A_{hill} ≤ 398 m, spaced evenly on a logarithmic scale, and offcentering parameter 0.5 ≤ α ≤ 0.6. The A_{hill} = 100 m line corresponds exactly to that shown in section 3d, and its growth rate is trivial for α ≥ 0.52. Larger hill heights quickly overcome the damping effect of offcentering.
Citation: Monthly Weather Review 150, 3; 10.1175/MWRD210054.1
Growth rates for the most unstable mode of the case discussed in section 3d, for hill heights 10 ≤ A_{hill} ≤ 398 m, spaced evenly on a logarithmic scale, and offcentering parameter 0.5 ≤ α ≤ 0.6. The A_{hill} = 100 m line corresponds exactly to that shown in section 3d, and its growth rate is trivial for α ≥ 0.52. Larger hill heights quickly overcome the damping effect of offcentering.
Citation: Monthly Weather Review 150, 3; 10.1175/MWRD210054.1
5. Discussion and conclusions
An instability, driven by the interaction between the steady response of flow over topography and shortwave perturbations, is present in the semiLagrangian, timecentered implicit discretization of the shallowwater equations over a significant portion of the Courant number and topographic wavelength parameter space. The most prevalent instabilities are similar to the wellknown phenomenon of spurious orographic resonance, but they occur under more general circumstances. These instabilities take their energy from the variations in the background state, transferring energy into their respective unstable modes. The instability strength is proportional to the amplitude of the topography relative to the background layer height ϕ_{0}.
In timedependent simulations, the growth of the instability tends to saturate at a level proportional to the difference between the steadystate background flow and a hypothetical notopography, constant background flow. In a largerscale model, this sort of saturation would be more likely to appear as “noise” or implausible results rather than model crashes or infinite results, but interactions with other aspects of the model such as physical parameterizations may cause further corruption of the simulation. Without additional feedback, there does not appear to be a direct mechanism to fuel unlimited, exponential growth.
The more specialized instabilities of advective resonance (section 3b) and spatial aliasing (section 3c) involve modes with high spatial frequencies that are not well resolved on the grid. These instabilities are effectively suppressed by modest damping from offcentering or through the use of polynomial interpolation for the semiLagrangian advection. In contrast, the temporal resonance described in section 3d consists of an instability with a longwavelength component and a shortwavelength component comparable to the length scale of the topography itself, and it is much less affected by these sorts of damping mechanisms. Since its oscillations are at least as well resolved as the topography itself, it is also unlikely to benefit from changes exclusively in the semiLagrangian operator, such as positivitypreserving interpolation.
The instabilities shown here do not appear to depend strongly on the exact form of the semiLagrangian operator, and in particular there seems to be no strong dependency on how the trajectories are computed. The numerical examples presented here all used iteratively computed trajectory calculations, and the form of the instability remains the same with even small numbers of iterations that do not result in fully converged trajectories. Even more extreme adjustments such as timeexplicit trajectory computations (basing the trajectories only on u^{−}, not shown for brevity) fail to dislodge the instability.
a. Appearance in realistic simulations
If this orographic instability is present in the shallowwater system, then why do operational systems based on this algorithm nonetheless produce successful forecasts and analyses?
The answer may lie in the slow growth rates of the instability and the nature of Earth itself. The problem domain established in section 2c is specifically designed to highlight the instability: it consists of periodic topography with a single wavelength over a relatively large domain. The calculated growth rates over much of the parameter space in Fig. 2 are of the order of a factor of 10 per hour, but the phase speed of gravity waves in this system is about 250 m s^{−1}. In 1 h, a gravity wave will have propagated about 900 km.
As a general rule, Earth’s surface does not look like a perfectly periodic, sinusoidal undulation of a single wavelength. A mode that is unstable in one region will likely find itself in a different regime 900 km downstream, having had only a small “contact time” to grow from the instability. Since the unstable modes (e.g., Fig. 6) depend on k_{hill}, a mode that grows in one region is not likely to grow quickly in another.
Higherresolution, local area simulations, however, are more likely to have topographies that resemble those studied here. The relevant time scale for the instability is proportional to the Courant number, so higher resolution simulations that use a shorter time step would see this instability grow proportionally more quickly in time. Very highresolution simulations with resolutions finer than 1 km over mountainous regions (with correspondingly large “hill amplitudes”) may be particularly susceptible to this instability.
To a researcher or meteorologist looking at the outputs from such a simulation, this instability would likely appear as a stubborn but small amplification of “noise” otherwise present in any output from an operational system. This “noise” would be generated from the dynamical core itself, absent physical parameterizations of moist processes, but since it would occur over mountainous regions the vertical coordinate may receive undue blame.
b. Extensions to the Euler equations
Further tests involving extensions of this analysis to the threedimensional Euler system will help characterize to what extent current and nearfuture forecasting models are impacted by this instability.
It seems very likely that these instabilities are still present in the more complicated system. The root cause of the instability is the interaction between gravity waves, which are slowed by the timecentered discretization, advection by the variable background flow, which is treated quite accurately via the semiLagrangian approach, and topographic forcing, which produces sum and difference modes. This same set of effects should be present in the threedimensional system regardless of its choice of vertical coordinate, but adding the extra dimension will also extend the gravity wave spectrum in ways that may allow for more coupling and resonance.
c. Future work
In the shortest term, the approach to controlling these instabilities must be to control the “operating envelope” of the dynamical core by restricting the topographic scales presented to the model. Filtering topography with a characteristic wavelength shorter than 5–6Δx would eliminate the most unstable regions show in Fig. 2, particularly in combination with cubic interpolation in the semiLagrangian operator (Fig. 10). This restriction may be unsatisfying as a physical representation, however, since in the presence of a radiative forcing model topography is also responsible for driving anabatic and katabatic winds, which are not fundamentally propagating gravity waves. If instead the time step must be reduced for stability, the criticism of Bartello and Thomas (1996) to the claimed performance advantages of semiLagrangian advection will become even more applicable.
In the longer term, the best approach will likely be to revisit the timestepping algorithm. The semiimplicit approach computes the nonlinear, timeimplicit terms through a single linear iteration based on a reference background state, and it appears to be immune to the temporal aliasing instability. However, this comes at the cost of reducing the time accuracy of the system or introducing time extrapolation, which itself comes with stability constraints. There may be room for compromise through a split approach, where a full nonlinear iteration applies to the coarsest features of the solution go through several iterations of a nonlinear solution process and the finest scales are handled semiimplicitly.
Another promising approach may ultimately be to return to an Eulerian formulation of advection, avoiding the instability by treating advection consistently with the discretization of the gravity wave restoring forces. This approach is under study with spectral element codes (Thomas and Loft 2005). Alternatively, the consistency can be enforced at the timediscretization level through exponential propagation as described in Gaudreault and Pudykiewicz (2016). Peixoto and Schreiber (2019) also demonstrates the combination of a semiLagrangian treatment of advection with exponential propagation of the other terms in the shallowwater equations.
Ultimately, the potential for this sort of instability must be considered by those designing dynamical cores. These instabilities affect a wide section of the parameter space spanned by Courant numbers and the topography wavelength, but they are invisible to an analysis linearized about an atmosphere at rest.
This can also be shown from an energy perspective, following Eldred and Randall (2017). The total energy
Using an odd number of points eliminates the exact, 2Δx “sawtooth” mode on the grid, which slightly simplifies the subsequent numerical analysis by allowing every remaining mode to be identified as upstream or downstream propagating.
This example does not use the steady state computed after the time discretization because (8) depends on the time step and time discretization. Since each panel of the figure would then have slightly different initial conditions, it would not be a likeforlike comparison.
The nomenclature in the literature is not quite consistent here: Girard et al. (2014) and Wood et al. (2014) use similar timestepping algorithms that treat the nonlinear terms implicitly with a restricted number of iterations, but the former describes the method as “implicit” and the latter “semiimplicit,” while Benacchio and Wood (2016) suggests the phrase “iterative implicit” for a nonlinear, implicit system solved with a fixed number of iterations. Here, we use the phrase “semiimplicit” to mean that the time averaging is not applied to some portion of (1).
Linearizing about a constant u_{0} ≠ 0 gives a dispersion relation of essentially the same functional form, just shifted by exp(iku_{0}Δt), because the semiLagrangian advection of u and ϕ fully acts before the offcentering modification.
Acknowledgments.
The author would like to thank Drs. Syed Husain and Janusz Pudykiewicz for helpful comments on an early draft of this paper. Additionally, the author would like to thank Dr. Nigel Wood and two other anonymous reviewers for their helpful comments during peer review.
Data availability statement.
The Matlab script files used to generate the figures in this paper are available on Github at https://github.com/csubich/subich_mwr_instabilities_2021, under an LGPL license.
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