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Toward Transient Subgrid-Scale Gravity Wave Representation in Atmospheric Models. Part I: Propagation Model Including Nondissipative Wave–Mean-Flow Interactions

Gergely Bölöni Institut für Atmosphäre und Umwelt, Goethe-Universität Frankfurt am Main, Frankfurt, Germany

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Young-Ha Kim Institut für Atmosphäre und Umwelt, Goethe-Universität Frankfurt am Main, Frankfurt, Germany

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Sebastian Borchert Deutscher Wetterdienst, Offenbach, Germany

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Ulrich Achatz Institut für Atmosphäre und Umwelt, Goethe-Universität Frankfurt am Main, Frankfurt, Germany

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Online Publication:
07 Apr 2021
Print Publication:
01 Apr 2021
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Multi-Scale Dynamics of Gravity Waves (MS-GWaves)
DOI:
https://doi.org/10.1175/JAS-D-20-0065.1
Page(s):
1317–1338
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Abstract

Current gravity wave (GW) parameterization (GWP) schemes are using the steady-state assumption, in which an instantaneous balance between GWs and mean flow is postulated, thereby neglecting transient, nondissipative interactions between the GW field and the resolved flow. These schemes rely exclusively on wave dissipation, by GW breaking or near critical layers, as a mechanism leading to forcing of the mean flow. In a transient GWP, without the steady-state assumption, nondissipative wave–mean-flow interactions are enabled as an additional mechanism. Idealized studies have shown that this is potentially important, and therefore the transient GWP Multiscale Gravity Wave Model (MS-GWaM) has been implemented into a state-of-the-art weather and climate model. In this implementation, MS-GWaM leads to a zonal-mean circulation that agrees well with observations and increases GW momentum-flux intermittency as compared with steady-state GWPs, bringing it into better agreement with superpressure balloon observations. Transient effects taken into account by MS-GWaM are shown to make a difference even on monthly time scales: in comparison with steady-state GWPs momentum fluxes in the lower stratosphere are increased and the amount of missing drag at Southern Hemispheric high latitudes is decreased to a modest but nonnegligible extent. An analysis of the contribution of different wavelengths to the GW signal in MS-GWaM suggests that small-scale GWs play an important role down to horizontal and vertical wavelengths of 50 km (or even smaller) and 200 m, respectively.

Bölöni’s current affiliation: Deutscher Wetterdienst, Offenbach am Main, Germany.

© 2021 American Meteorological Society. For information regarding reuse of this content and general copyright information, consult the AMS Copyright Policy (www.ametsoc.org/PUBSReuseLicenses).

This article is included in the Multi-Scale Dynamics of Gravity Waves (MS-GWaves) Special Collection.

This article has a companion article which can be found at http://journals.ametsoc.org/doi/abs/10.1175/JAS-D-20-0066.1.

Corresponding author: Gergely Bölöni, gergely.boeloeni@dwd.de

Abstract

Current gravity wave (GW) parameterization (GWP) schemes are using the steady-state assumption, in which an instantaneous balance between GWs and mean flow is postulated, thereby neglecting transient, nondissipative interactions between the GW field and the resolved flow. These schemes rely exclusively on wave dissipation, by GW breaking or near critical layers, as a mechanism leading to forcing of the mean flow. In a transient GWP, without the steady-state assumption, nondissipative wave–mean-flow interactions are enabled as an additional mechanism. Idealized studies have shown that this is potentially important, and therefore the transient GWP Multiscale Gravity Wave Model (MS-GWaM) has been implemented into a state-of-the-art weather and climate model. In this implementation, MS-GWaM leads to a zonal-mean circulation that agrees well with observations and increases GW momentum-flux intermittency as compared with steady-state GWPs, bringing it into better agreement with superpressure balloon observations. Transient effects taken into account by MS-GWaM are shown to make a difference even on monthly time scales: in comparison with steady-state GWPs momentum fluxes in the lower stratosphere are increased and the amount of missing drag at Southern Hemispheric high latitudes is decreased to a modest but nonnegligible extent. An analysis of the contribution of different wavelengths to the GW signal in MS-GWaM suggests that small-scale GWs play an important role down to horizontal and vertical wavelengths of 50 km (or even smaller) and 200 m, respectively.

Bölöni’s current affiliation: Deutscher Wetterdienst, Offenbach am Main, Germany.

© 2021 American Meteorological Society. For information regarding reuse of this content and general copyright information, consult the AMS Copyright Policy (www.ametsoc.org/PUBSReuseLicenses).

This article is included in the Multi-Scale Dynamics of Gravity Waves (MS-GWaves) Special Collection.

This article has a companion article which can be found at http://journals.ametsoc.org/doi/abs/10.1175/JAS-D-20-0066.1.

Corresponding author: Gergely Bölöni, gergely.boeloeni@dwd.de
Keywords: Atmosphere; Gravity waves; Parameterization; Subgrid-scale processes

1. Introduction

Gravity waves (GW) play an important role in atmospheric dynamics. They are excited mostly in the troposphere—for example, by flow over orography, convection, and jets and front systems. In the course of their propagation, they affect the momentum and energy balance in the atmosphere everywhere up to the thermosphere (see, e.g., Kim et al. 2003). The direct impact of GWs on the large-scale circulation is largest in the middle atmosphere; however, they also affect tropospheric weather and climate significantly (e.g., Scaife et al. 2005, 2012).

In general circulation models (GCMs) and numerical weather prediction (NWP) models, effects of GWs must be parameterized, given the wide spatial and temporal spectrum they act on, part of it being far below the effective resolution of global model applications. Wentzel–Kramer–Brillouin (WKB) theory (Bretherton 1966; Grimshaw 1975; Achatz et al. 2017) is the basis of most GW parameterizations (GWP) in climate simulations and weather predictions (Lindzen 1981; Medvedev and Klaassen 1995; Warner and McIntyre 1996; Hines 1997a,b; Lott and Miller 1997; Alexander and Dunkerton 1999; Scinocca 2003; Orr et al. 2010; Lott and Guez 2013). There is, however, an increasing appreciation that the present handling of this technique needs improvements: a simplification typically used is the neglect of 1) horizontal GW propagation (single-column approximation) and 2) transient effects such as nondissipative GW–mean-flow interactions (steady-state approximation). The former has been shown to be an important weakness of state-of-the-art parameterizations, by, for example, Sato et al. (2009), Ribstein et al. (2015), Ribstein and Achatz (2016), and Ehard et al. (2017); Bölöni et al. (2016), Muraschko et al. (2015), and Wilhelm et al. (2018) propose improvements with regard to the latter aspect. Another drawback of GWPs in current climate and weather codes is that their applicability outside of the tropics (where Coriolis effects are nonnegligible) relies on the assumption of balanced (hydrostatic, geostrophic) resolved flows, which might not be valid with the increasing spatial resolutions applied nowadays. If, however, the resolved flow is not balanced, additional forcing terms due to the GW dynamics appear both in the momentum and the entropy equation representing, for example, elastic effects (Achatz et al. 2017; Wei et al. 2019). Potential triad wave–wave interactions in the atmosphere are also not taken into account in current GWPs, although their neglect has never been justified explicitly, to the best of our knowledge. In addition to the propagation issues listed above, faithful representation of GW sources is a key to success, and is another area where one finds room for improvement: theory and applications for orographic (Palmer et al. 1986; Bacmeister et al. 1994; Lott and Miller 1997) and convective GW sources (Beres et al. 2005; Song and Chun 2005) are relatively well developed, but the representation of GW emissions by jets and fronts—despite the efforts of Charron and Manzini (2002), Richter et al. (2010), and de la Cámara and Lott (2015)—remains difficult.

This paper focuses on the issues of GW propagation. In a novel framework, transient effects are incorporated by removing the steady-state approximation. This work is an extension of the study by Bölöni et al. (2016), in which effects of the transient, nondissipative GW–mean-flow interactions have been assessed in an idealized set-up, whereas here the same is done in a more complex framework in which the proposed transient GWP has been implemented into a state-of-the-art GCM/NWP model. The single-column approximation has been kept for the sake of simplicity, with the intention to give it up in a later step of our developments.

Section 2 motivates the implementation of a transient GWP to a state-of-the-art GCM and recalls the necessary theoretical background for the rest of the paper. This is followed by the actual implementation details in section 3 and by the presentation of the GCM-simulation results in section 4. A summary of the most important findings is given in section 5.

2. Theory

In the following we outline the theoretical basis of the proposed transient GWP called the Multiscale Gravity Wave Model (MS-GWaM). In section 2a we do so for locally monochromatic GWs together with the simplifying assumptions applied and a comparison to standard parameterization approaches. In section 2b the monochromatic perspective is generalized to full GW spectra.

a. Locally monochromatic GW fields

In this section we first sketch the general WKB theory on which MS-GWaM is built [section 2a(1)] and then describe the simplifying pseudomomentum-flux approach and single-column approximation that are used in the current study [sections 2a(2) and 2a(3)]. Then, our transient formulation is compared with the one with the steady-state approximation on which present-day GW parameterizations are based [section 2a(4)].

1) General WKB

Following WKB theory as applied, for example, by Grimshaw (1975) and Achatz et al. (2017), the spatiotemporal structure of a locally monochromatic small-scale GW field in a larger-scale flow is characterized by a local wavenumber k(x, t) = exk + eyl + ezm and local frequency ω(x, t), while its amplitude can be deduced from its wave-action density A⁡(x,t), all depending on position x = exx + eyy + ezz and time. Frequency and wavenumber are connected by the dispersion relation
ω^=ω−kh⋅U=±N2Kh2+f2m2K2,
where U(x, t) is the local horizontal wind of the large-scale flow, N2(z) is the squared Brunt–Väisälä frequency, f is the Coriolis frequency, and K = (k2 + l2 + m2)1/2 and Kh = (k2 + l2)1/2 are the magnitudes of the total and horizontal wavenumbers, respectively. Wave-action density A=Egw/ω^ is the ratio between the GW energy density Egw and the intrinsic frequency ω^. Because all fields are real valued, amplitudes corresponding to the negative branch in Eq. (1) can be determined directly from those corresponding to the positive branch, so that henceforth only the latter will be considered. The local group velocity then is
cg⁡(x,t)=∇kΩwithΩ⁡(x,k,t)=U⁡(x,t)⋅kh+N2⁡(z)Kh2+f2⁡(y)m2K2,
where Ω expresses the local frequency so that explicit space and time dependencies are only due to the large-scale (mean) flow, vertical variations of stratification, and horizontal variations of the Coriolis parameter. In the absence of dissipation, the development of the GW field, given the mean flow, is determined by
⁡(∂∂t+cg⋅∇)k=−∇xΩand
∂A∂t+∇⋅(cgA)=0,
while the GW effect on the mean flow is described by
⁡(∂U∂t,∂Θ∂t)gw=[−1ρ¯ ∇⋅(ρ¯v′u′¯)+fθ¯ ez×u′θ′¯,−∇h⋅u′θ′¯].
Here, Θ is the mean-flow potential temperature deviation from the reference-atmosphere potential temperature θ¯⁡(z), ρ¯⁡(z) is the reference-atmosphere density, v and u denote the full wind vector and the horizontal wind vector, respectively, and ∇h stands for the horizontal components of ∇. The GW momentum fluxes ρ¯v′u′¯ and horizontal potential temperature flux u′θ′¯ can be calculated from k and A. Clearly, Eq. (5) does not account for the energy deposition by GWs, which has an important thermal effect in the mesosphere and lower thermosphere (MLT) (e.g., Becker 2017). It also implies that the representation of this effect is left out from the current study and will have to be incorporated in the future.

2) Pseudomomentum approximation

In the spirit of a stepwise implementation of the most general theory, for the time being, MS-GWaM does not use Eq. (5) in its full complexity. Instead, resting on considerations by Andrews and McIntyre (1976, 1978) and following the procedure of all present-day GWPs, Eq. (5) is replaced by
⁡(∂U∂t,∂Θ∂t)gw=[−1ρ¯ ∇⋅(c^gkhA),0],
where kh is the horizontal part of the wavenumber vector, c^g=cg−U is the intrinsic group velocity, and c^gkhA is the GW Eliassen–Palm or pseudomomentum flux. An advantage of this approximation is that no GW potential temperature fluxes are required. The latter would enter via their horizontal convergence, which one is inclined to avoid in single-column GWPs. Wei et al. (2019) discussed this approximation in detail. Equation (6) basically assumes that the large-scale flow is in geostrophic and hydrostatic balance. When this is not the case, errors can occur outside of the tropics whenever near-inertial GWs are involved with ω^ close to f. In future work it is intended to drop both the pseudomomentum approximation and the single-column approximation.

3) Single-column approximation

The single-column approximation is taken in present-day GWPs for the sake of efficiency, and we do so here as well. One neglects in the GW–mean-flow interaction all horizontal derivatives and also neglects in the wave-action equation all horizontal group-velocity components so that, using the pseudomomentum approximation as well, the approximated dynamics is described by
⁡(∂∂t+cgz ∂∂z)⁡(kh,m)=(0,−∂Ω∂z),
∂A∂t+∂∂z ⁡(cgzA)=0,and
⁡(∂U∂t)gw=−1ρ¯∂∂z ⁡(cgzkhA),
where cgz is the vertical group velocity. This approximation neglects all effects of horizontal GW propagation. Note that the pseudomomentum-flux convergence in the right-hand side of Eq. (9) can be written as (e.g., Achatz et al. 2017)
−1ρ¯∂∂z⁡ (cgzkhA)=−1ρ¯[∂∂z ⁡(ρ¯u′w′¯)−fez×∂∂z⁡(ρ¯u′θ′¯dθ¯/dz)],
as is also known from derivations from generalized Lagrangian-mean theory (Andrews and McIntyre 1978b).

4) Steady-state approximation and its implications

The final step taken in present-day GWPs for the sake of efficiency is the assumption that the GW field adjusts instantaneously to a given mean-flow distribution. This way GW effects are propagating within one time step from a source to the model top and bottom. One neglects in the prognostic equations for the GW field all time derivatives so that Eqs. (7) and (8) are replaced by
∂∂z ⁡(kh,m)=(0,−1cgz∂Ω∂z)and
∂∂z ⁡(cgzA)=S.
Here we have introduced a source or sink S on the right-hand side of the wave-action-density equation. This is decisive. One sees that in the steady-state approximation the horizontal wavenumber is a constant so that without any source or sink there would be no GW forcing of the mean flow in Eq. (9). Hence, in this approximation, GW dissipation—for example, by GW breaking or close to critical layers—is indispensable for a GW effect on the mean flow, while the explicit description of GW transience as in Eqs. (7) and (8) also allows GW impacts on the mean flow via nondissipative wave–mean-flow interactions.

Consequences of applying the steady-state approximation instead of the transient GW-model Eqs. (7)–(9)—and thus neglecting nondissipative GW–mean-flow interactions—have been studied by Bölöni et al. (2016) in a highly idealized setup using wave-resolving simulations as a reference. They achieved a reliable evolution of the GW energy and the mean flow only using the transient model. In case of using the steady-state equations, important features of the GW–mean-flow interactions were not captured: the GW packet propagated way too fast until static instability set in and its induced mean flow did not agree with the results from wave-resolving simulations. Using a Fourier-ray model (Broutman et al. 2006) and high-resolution WRF (Skamarock et al. 2019) simulations, Kruse and Smith (2018) found that, in the interaction of mountain waves with the mean flow, both dissipative and nondissipative forcings of the mean flow seem to play an important role. The natural question of how important are nondissipative GW–mean-flow interactions in the context of global dynamics has motivated the present study.

b. Spectral treatment of transient GW distributions

Although the consideration of locally monochromatic GW fields is helpful for deriving the prognostic system of Eqs. (3)–(5) or its single-column pseudomomentum approximation in Eqs. (7)–(9), real-world GW fields are made up of a full spectrum of components. Even if one starts out from a locally monochromatic initial condition, GW–mean-flow interactions tend to quickly lead to caustics where more than one wavenumber is observed at a single location. Correspondingly, attempts to solve the above discussed equation sets directly on a computer most often fail due to numerical instabilities near caustics. As shown by Muraschko et al. (2015), this can be avoided by considering a spectral wave-action density in wavenumber-position phase space (e.g., Hertzog et al. 2002) instead:
N⁡(x,k,t)=∫ℝ3d3 βAβ⁡(x,t)δ⁡(k−kβ),
where β is a three-dimensional index field and each combination of Aβ and kβ satisfies Eqs. (3) and (4) or Eqs. (7) and (8) separately. If the corresponding wave amplitudes are weak enough, a superposition of these solutions is a WKB solution of the basic dynamical equations as well, assuming that the required scale separation between the various spectral components and the large-scale mean flow still holds, and one can derive the prognostic equation
DrNDt≡∂N∂t+cg⋅∇xN+k˙⋅∇kN=0.
Here, cg(x, k, t) = ∇kΩ is again the group velocity defined in Eq. (2) for wavenumber k, and k˙⁡(x,k,t)=−∇xΩ is the rate of change of the wavenumber k as it appears on the right-hand side of Eq. (3); Dr/Dt is a material derivative along trajectories in phase space, so-called rays, tangential to the phase-space velocity ⁡(cg,k˙). Along these rays the phase-space wave-action density is conserved. The GW impact on the mean flow is the sum of the impact of all spectral components so that, with the pseudomomentum approximation,
⁡(∂U∂t)gw=−1ρ¯ ∇⋅∫d3k c^gkhN,
with d3k = dk dl dm. Similar expressions can be formulated also without the pseudomomentum approximation (Wei et al. 2019). We note in passing that in the absence of background winds, Eq. (14) agrees with the radiative transfer equation without wave–wave interactions that has been used in the oceanic context for GWPs (Olbers et al. 2019, and references therein). There, however, the shape of the GW spectrum is prescribed, whereas in our implementation it develops without constraints.
In the single-column approximation, one again neglects all horizontal derivatives as well as the horizontal group velocity in Eq. (14), resulting in
DrDt=∂∂t+cgz ∂∂z+m˙ ∂∂mand⁡(cgz,m˙)=(∂Ω∂m,−∂Ω∂z)
in the system
DrNDt=0and
⁡(∂U∂t)gw=−1ρ¯∂∂z∫d3k cgzkhN.
The corresponding rays, along which N is conserved, are given by
DrDt ⁡(xh,z)=(0,cgz)=[0,−m⁡(ω^2−f2)ω^K2]and
DrDt ⁡(kh,m)=(0,m˙)=(0,−kh⋅∂U∂z−NKh2ω^K2∂N∂z).
Moreover, partial integration of Eq. (17), using Eq. (16), also yields
∂∂t∫dm N+∂∂z∫dm cgzN=0,
which is the equivalent to Eq. (8).

In a steady-state approximation, one again neglects the time derivatives in the wave-action density Eq. (17) and in the ray Eqs. (19) and (20). Hence kh is again a constant and Eq. (18) yields together with the steady-state version of Eq. (21) the nonacceleration result ∂U/∂t = 0; that is, the mean flow is unaffected by GWs, unless Eq. (21) is supplemented by sources or sinks.

3. Implementation in a high-top atmosphere model

Our single-column pseudomomentum-approximation subgrid-scale GW model applying the transient Eqs. (17)–(20), extended by a saturation scheme, has been named MS-GWaM. It has been implemented into the Icosahedral Nonhydrostatic (ICON) model (Zängl et al. 2015) in its upper-atmosphere configuration UA-ICON (Borchert et al. 2019), allowing numerical studies over a wide altitude range from Earth’s surface to the lower thermosphere. For the sake of simplicity and clear traceability of causes and consequences, the current orographic GWP in UA-ICON, based on Lott and Miller (1997), has been left untouched, and MS-GWaM only replaces the nonorographic GWP there, based on Orr et al. (2010).

As a reference and a representative of currently available GWP schemes, in addition to the transient implementation, two steady-state versions of MS-GWaM have also been implemented to UA-ICON. The first one excludes nondissipative GW–mean-flow interactions through the steady-state approximation but shares all other parameterization components with the transient MS-GWaM, such as GW sources and the saturation scheme. The other one differs from MS-GWaM in its saturation scheme as well, i.e., instead of an integrated treatment of the GW breaking criterion, it applies a monochromatic approach (see the details in section 3b). Throughout the paper, the implementation of the transient MS-GWaM into UA-ICON will be referred to as TR, while the two steady-state implementations will be called ST and STMO, respectively.

a. Transient scheme

In a global implementation the interaction equations would have to be rewritten in spherical coordinates. The single-column approximation, however, eliminates any horizontal changes of the GW field and all metric terms, which amounts to treating the parameterization equations in local Cartesian coordinates on an f plane.

1) GW propagation and interaction with the mean flow

Following Muraschko et al. (2015), we define Lagrangian ray volumes as carriers of the GW fields’ wave-action density and simply trace their positions in phase space. Because of Eq. (17), their spectral wave-action density is conserved, unless wave dissipation is active. Each ray volume is six-dimensional, and its horizontal cross section is given by that of the corresponding ICON column. In the single-column approximation, it does not change so that we suppress it in the following notation. Likewise, horizontal wavenumbers do not change either, but because of the source formulation below we keep track of the ray-volume extent in the corresponding directions.

As illustrated in Figs. 1a and 1b, each ray volume has an extent Δz in z direction and extents Δk, Δl, Δm in the three-dimensional wavenumber space. They move, expand, or shrink in the z and m directions. From ∂cgz/∂z+∂m˙/∂m=0, the phase-space content of each ray volume, and hence in our discretization Vp = ΔzΔm, do not change. To achieve this, first, changes in the vertical extent of ray volumes are calculated via Eq. (19) as Δ˙z=cgzt−cgzb where cgzt and cgzb stand for the vertical group velocities at the top (z = zt) and the bottom (z = zb) of the ray volume, respectively. Second, the displacement of the ray-volume center point is calculated via Eq. (19) as z˙c=⁡(cgzt+cgzb)/2 in the vertical direction and, via Eq. (20), as m˙c=m˙⁡(kh,mc) in the m direction. For the latter, the resolved dynamical fields N, ∂zU, and ∂zN are interpolated to the ray-volume center point z = zc. Last, the ray-volume extent in the m direction is updated by Δm = Vp/Δz.

Fig. 1.
Fig. 1.

A schematic of the GW field representation by ray volumes in the transient MS-GWaM: (a) representation in z–m space (see the text for explanations on how the subgrid-scale GW pseudomomentum fluxes are projected to the vertical grid) and (b) representation in spectral space, showing that the ray volume center-point mc and the extent in the m direction changes prognostically.

Citation: Journal of the Atmospheric Sciences 78, 4; 10.1175/JAS-D-20-0065.1

Next, the acceleration of the resolved horizontal wind is calculated via Eq. (18). The pseudomomentum fluxes (PMF) are calculated on the half levels of the ICON vertical grid; that is, at the half level at z = zi+(1/2), the integral on the right-hand side of Eq. (18) is approximated as
PMFzi+(1/2)=⁡(∫d3k cgzkhN)zi+(1/2)≈⁡(∑j=1NiΔzijδzi cgzjkhjNjΔkjΔljΔmj)zi+(1/2),
where Ni is the number of ray volumes overlapping the vertical layer zi+1 < z ⩽ zi with a thickness of δzi and j is the index over those ray volumes. As can be seen in Fig. 1a, the ratio Δzij/δzi represents the pseudomomentum-flux fraction of the jth ray volume contributing to the layer centered at z = zi+(1/2). After the calculation of the pseudomomentum fluxes at half levels, the mean-flow tendency
⁡(ΔU/Δt)gwzi
at the full level at z = zi is calculated via Eq. (18) by centered finite differences. The resolved horizontal wind is updated as
Uzinew=Uzi+δt⁡(ΔU/Δt)gwzi,
where a time step δt = 60 s is used. To ensure that the ray volumes do not jump over strong, shallow shear layers—and thus describe reflection and critical-layer filtering properly—a fourth-order Runge–Kutta sub–time stepping is used for the integration of Eqs. (19) and (20) with a time step of δts = 30 s. Note that the development of z and m via Eqs. (19) and (20) depends on the stratification and the wind shear, i.e., on Uzi, which—among others—includes the GW induced wind contribution.

2) GW breaking

The phase-space wave-action density N is conserved along rays until GWs break. In that case, turbulence is generated that damps the GWs via turbulent viscosity and diffusivity. Hence N decreases, which generates pseudomomentum-flux convergence additionally to the nondissipative GW–mean-flow interactions described in the previous section. In MS-GWaM, following Lindzen (1981), this is taken into account by diagnosing whether the GW field can turn the flow into a statically unstable state. Once this is the case in a given layer, the wave-action density of all overlapping ray volumes is reduced so that static instability cannot occur anymore. Following Bölöni et al. (2016), the static instability criterion for a quasi-monochromatic wave is given by m2|B|2 > N4, where |B|2=2AN4Kh2/⁡(ρ¯ω^K2) is the squared complex GW buoyancy amplitude and | | denotes the modulus. Applying the phase-space concept to represent the full spectrum, this reads
2N4ρ¯∫d3k m2N Kh2ω^K2>N4.
The single-column discretization of this—for example, in the layer with thickness δzi centered at z = zi+(1/2)—is
⁡(2N4ρ¯ ∑j=1NiSj>N4)zi+(1/2)withSj=Δzijδzi Nj mj2Khj2ω^jKj2 ΔkjΔljΔmj,
where again, all variables with a j -index denote known properties of ray volumes overlapping the layer, and those without are resolved variables at z = zi+(1/2). Whenever static instability is diagnosed via Eq. (24), the saturation scheme is called, and the turbulent diffusivity and viscosity are determined so that they exactly counteract the amplitude increase of the contributing GWs that would cause Eqs. (23) or (24) to be satisfied. In the case of buoyancy, for example, the turbulence effect is then captured via ∂tb=⋯+K⁡(∂x2b+∂y2b+∂z2b) with the turbulent diffusivity K. After a Fourier transformation in space, the corresponding buoyancy change over a short time Δt is Δ|b˜|2=⋯−2KΔt|b˜|2⁡(Kh2+m2) or, for the amplitude,
Δ⁡(d|B|2dm dm)=2N2ρ¯ Δ⁡(m2ω^Ndm)=−2KΔt 2N2ρ¯ m2⁡(Kh2+m2)ω^Ndm.
After simplifying Eq. (25), requiring the diffusivity and viscosity to be just strong enough to prevent Eqs. (23) and (24) from being satisfied, and returning to discretized variables, one is left with
Nsatj=Nj⁡(1−KKj2)⁡(j=1,…,Ni),
where Nj is the phase-space wave-action density one would have directly from wave-action conservation without the turbulence impact, and Nsatj is the saturated wave-action density corresponding to the equality sign in Eqs. (23) and (24). The local turbulent diffusion coefficient hence is
K⁡(z)=∑j=1NiSj−ρ¯/2∑j=1NiSjKj2.
The vertical wavenumber dependence of the saturation Eq. (26) is such that small-scale GWs are damped most strongly. The diffusivity estimated as above could be used to predict corresponding frictional heating as well as modifications of the GWenergy deposition (e.g., Becker 2017). For the time being, however, these effects are here not taken into account. Furthermore, the effect of GW damping due to the heat diffusion (i.e., downward heat flux) is also ignored. We also note that the above-described saturation scheme, similar to Lindzen (1981), assumes that the Prandtl number is approximately 1, that is, that momentum and potential temperature are equally effected by turbulent diffusion. This assumption is to be revisited in the future as, e.g., Fritts and Dunkerton (1985) suggest that the Prandtl number should be very large for breaking GWs.

3) GW source representation

A simple representation of the nonorographic GW sources has been chosen. Instead of parameterizing the GW sources associated, e.g., with convection and jets and fronts, it is assumed that the superposition of all nonorographically emitted GWs obeys the universal Desaubies spectrum (VanZandt 1982; Fritts and VanZandt 1993). Following Scinocca (2003), the corresponding GW launch momentum-flux spectrum, projected onto the horizontal propagation direction of each spectral component, defined by the azimuth angle ϕ ∈ [0, 2π) so that k = Kh cosϕ, l = Kh sinϕ, is
ρ¯F0⁡(c˜,ω˜,ϕ)=ρ¯Cm*3 c˜N3ω˜1−pN4+m*4c˜4,
where ω˜=NKh/|m| is the nonrotational and hydrostatic approximation of the intrinsic frequency and c˜=N/|m| is the respective intrinsic phase speed. Note that upward GW propagation corresponds to m < 0, implying ω˜=−NKh/m at the source. In the current study, four azimuthal angles have been used in defining GW propagation directions toward east, north, west, and south; that is,
ϕ=(0,π/2,π,3π/2)⇒k=Kh⁡(1,0,−1,0),l=Kh⁡(0,1,0,−1).
The launch spectrum is characterized in terms of intrinsic phase speeds as c˜∈(c˜min,c˜max]=(0,36 ] m s−1 in each of the four directions, with an equidistant spacing and thus equally large spectral elements Δc˜=⁡(c˜max−c˜min)/nc˜, where nc˜=6 is the number of spectral elements. In terms of intrinsic frequency, a range
ω˜∈[ω˜min,ω˜max]=[10−4,5×10−4] s−1
is considered again with an equidistant spacing and equally large spectral elements Δω˜=⁡(ω˜max−ω˜min)/nω˜, where nω˜=2 is the number of elements. A characteristic vertical wavenumber m* = 2π/(2 km) is used, while the value of p is set to 5/3 on the basis of Warner and McIntyre (1996) and Fritts and Lu (1993). The factor C is a tuning parameter enabling to set a desired launch-level pseudomomentum-flux magnitude M, so that
∫c˜minc˜max∫ω˜minω˜maxρ¯F0⁡(c˜,ω˜,ϕ) dω˜ dc˜=M
for each azimuthal direction ϕ. To account for the seasonal variability of nonorographic GW sources emitted by jets and fronts, time and latitudinal dependence of M has been introduced as
M⁡(φ,t)=Mbs⁡(φ)+β⁡(t)[Mbw⁡(φ)−Mbs⁡(φ)],
where φ is the latitude in degrees, β(t) = {1 + cos[2π(t − t0)]}/2 is a time-dependent function with t0 being 0000 UTC 22 December of the given year, and Mbw and Mbs are respectively the boreal winter and summer flux-magnitude profiles given as
Mbw⁡(φ)=[1−α⁡(φ)]Mmin+α⁡(φ)Mmaxand
Mbs⁡(φ)=[1−α⁡(φ)]Mmax+α⁡(φ)Mmin,
with
α⁡(φ)=[1+tanh⁡(φ/s)]/2
being a function with a smooth transition between 0 and 1 with s = 11°. The resulting meridional launch flux profile is plotted in Fig. 2 for the northern summer and winter solstices with the choice of (Mmin, Mmax) = (1.5, 2.5) mPa, which are the actual values chosen for the implementation.
Fig. 2.
Fig. 2.

Latitudinal profile of launch momentum-flux magnitudes (mPa) in the nonorographic GW source at Northern Hemispheric winter (blue) and summer (green) solstices.

Citation: Journal of the Atmospheric Sciences 78, 4; 10.1175/JAS-D-20-0065.1

To express the spectral distribution of the source in terms of the wavenumbers (k, l, m) as needed by MS-GWaM, the c˜,ω˜,ϕ-dependent pseudomomentum-flux spectrum Eq. (28) is transformed via the sequence of Jacobian transformations
ρ¯F1⁡(m,ω˜,ϕ)=J0ρ¯F0⁡(c˜,ω˜,ϕ)withJ0=∂c˜∂m=Nm2,
ρ¯F2⁡(m,Kh,ϕ)=J1ρ¯F1⁡(m,ω˜,ϕ)withJ1=∂ω˜∂Kh=−N|m|,and
ρ¯F3⁡(m,k,l)=J2ρ¯F2⁡(m,Kh,ϕ)withJ2=∂(Kh,ϕ)∂(k,l)=1Kh,
where the magnitude of the horizontal wave vector is always calculated through the dispersion relation as Kh=ω˜|m|/N. After the transformation, using a typical N at launch level, the launch spectrum spans λz ∈ [0.8, 8] km with an increasing resolution in m toward large vertical wave lengths (corresponding to large group velocities) and λx,y ∈ [46, 1036] km with an increasing resolution in k, l toward large horizontal wave scales. Because
∫d3k ρ¯F3kh/Kh=∫d3k khcgzN,
the phase-space wave-action density of ray volumes at launch level z = zl (=300 hPa in this study) is initialized as
[Nj=ρ¯F3⁡(mj,kj,lj)Khjcgzj]zl,
where j = 1, …, Nl, is the ray-volume index with Nl = 4ncnω being the total number of spectral elements launched at a time in the four azimuthal directions and mj, kj, lj denoting the wavenumber values at the ray-volume centers. The spectral extent of the jth ray volume in the m direction is calculated as Δmj=Δc˜jmj2/N, which results in decreasing ray volume extents toward large vertical wave lengths. As shown in Fig. 3, the spectral extents in k and l directions are defined as Δkj=ΔKhj and Δlj=KhjΔϕ for eastward- and westward-propagating waves and as Δkj=KhjΔϕ and Δlj=ΔKhj for northward- and southward-propagating waves, where ΔKhj=Δωj|mj|/N and Δϕ = π/2 is the azimuthal angle difference between propagation directions.
Fig. 3.
Fig. 3.

A schematic showing the attribution of horizontal wavenumbers to ray volumes in the nonorographic GW source (see the text for more explanation).

Citation: Journal of the Atmospheric Sciences 78, 4; 10.1175/JAS-D-20-0065.1

In the transient framework discussed in this section, the GW emission by nonorographic sources is implemented as a lower boundary condition for Eqs. (17)–(20). This requires that the GW ray volumes are emitted continuously at the launch level for the whole spectrum, so that the total pseudomomentum flux ∫d3k ρ¯F3⁡(k)kh/Kh is maintained at all times. To achieve this, an elaborate ray-volume launching procedure is applied as demonstrated by Fig. 4. Below the launch level zl, a ghost layer is defined with a thickness δzg, and at launch time t = t0 the ray volumes are initialized via Eq. (36) with their top matching the bottom of the ghost layer at z = zl − δzg (see Fig. 4a). The five ray volumes sketched in the figure are located in adjacent spectral positions in m. At time t = t0 + Δt all ray volumes have propagated vertically, but to different extents (see Fig. 4b). To preserve the shape of the spectrum until the launch level is reached, all ray volumes with a center point below z = zl propagate without refraction, that is, with cgz=constant, m˙=0,Δz=constant, and Δm=constant. As soon as a ray volume is completely in the ghost layer (i.e., its bottom has passed the height z = zl − δzg), a new ray volume is launched so that its top matches the bottom of the previous ray volume in the same spectral position. In Fig. 4b, this happens at the spectral positions 4 and 5, where the “old” ray volumes launched at t = t0 are denoted by a black center point and boundaries while the “new” ray volumes launched at t = t0 + Δt have red center points and boundaries. Two additional features are demonstrated at the spectral position 5: 1) here, the old ray volume has traveled so much within Δt, that even the new ray volume’s bottom ends up at z = zl − δzg, which allows the emission of a second new ray volume right away at the same launch time t = t0 + Δt, and 2) the old ray volume’s center point has passed z = zl so that the full set of GW–mean-flow interaction Eqs. (17)–(20) begins to act, leading to refractions (displacement of the center point in the m direction) and deformations (changes of Δz and Δm). It is repeated here that the above-described GW source is kept simple and nonintermittent on purpose in order to allow a clear separation of transient propagation effects from those implied by intermittent sources.

Fig. 4.
Fig. 4.

A schematic of the representation of the nonorographic GW emission as lower boundary condition in terms of sequential ray-volume launches (see the text for more explanation).

Citation: Journal of the Atmospheric Sciences 78, 4; 10.1175/JAS-D-20-0065.1

b. Steady-state schemes

In this section the steady-state implementations of the single-column GW–mean-flow interaction Eqs. (17)–(20) are presented. In the steady-state context it is assumed that the GWs propagate instantaneously from any source to model bottom and model top and that they instantaneously assume an equilibrium with the resolved mean flow and the source distribution. This equilibrium remains unchanged until source or resolved flow change, when the GW distribution again adjusts instantaneously. As a consequence, GWs cannot influence the resolved flow, unless wave dissipation is active. The mean-flow acceleration by GWs is hence realized exclusively via GW breaking and critical-layer filtering, that is, by diagnosing at what height the equilibrium breaks down as a result of dissipative processes, leading to corresponding pseudomomentum-flux convergences. The next few sections describe the steady-state implementations of MS-GWaM in detail.

1) GW source representation

The spectral characteristics and the magnitude of the nonorographic GW sources are identical to the transient implementation presented in section 3a(3); that is, the GW launch-level pseudomomentum flux is distributed among monochromatic spectral elements characterized in the very same way in spectral space as in the transient case (λx,y ∈ [47, 1036] km, λz ∈ [0.8, 8] km), with the very same values ρ¯F3⁡(m,k,l), so that the total GW pseudomomentum flux at each launch time is ∫d3k ρ¯F3⁡(k)kh/Kh. The corresponding wave-action contribution by each spectral element at the launch level z = zl can be calculated as Ai⁡(zl)=A⁡(zl,ki)=ρ¯F3⁡(ki)/⁡(Khicgzi)ΔkiΔliΔmi, where i = 1, …, Nl is the spectral-element index with Nl = 4ncnω.

2) Equilibrium profile

One can easily convince oneself that the steady-state version of Eq. (21) holds also componentwise so that we have for each spectral element cgzi⁡(z)Ai⁡(z)=cgzi⁡(zl)Ai⁡(zl)=constant, where both cgzi⁡(zl) and Ai⁡(zl) are known from the GW source. In this way, one is left with equilibrium profiles of the wave-action flux, which entirely determine the GW dynamics after adjustment to the resolved flow. To diagnose GW breaking altitudes, critical layers, or reflection layers, it is not sufficient to have the products cgzi⁡(z)Ai⁡(z) as known quantities, but in addition, one needs the corresponding Ai⁡(z) and cgzi⁡(z) profiles individually. The vertical group velocity profiles are obtained from the dispersion relation in Eq. (1) as
cgzi⁡(z)=−mi⁡(z)[ω^i2⁡(z)−f2]ω^i⁡(z)[Khi2+mi2⁡(z)],⁡(i=1,…,Nl),
with
mi⁡(z)=−Khi2[N2⁡(z)−ω^i2⁡(z)]ω^i2⁡(z)−f2,⁡(i=1,…,Nl),
where ω^i⁡(z) are the intrinsic frequency profiles of the adjusted GW field’s spectral elements. The key to calculate ω^i⁡(z), needed in Eqs. (37) and (38), is the Eikonal frequency equation DrΩ/Dt = ∂Ω/∂t = k ⋅ (∂U/∂t) that one can derive directly from the definitions cg = ∇kΩ and k˙=−∇xΩ. Hence, in the steady-state case extrinsic frequencies ωi(zl) are unchanged along a ray. This means that after diagnosing ωi(zl) at the launch level, the intrinsic frequency profiles can be calculated based on the known wind profile as ω^i⁡(z)=ωi⁡(zl)−ki⋅U⁡(z). Using this in Eq. (37), the wave-action profile of each spectral element can be calculated as
Ai⁡(z)=cgzi⁡(zl)Ai⁡(zl)cgzi⁡(z),⁡(i=1,…,Nl).

3) Critical-layer filtering and reflection

At critical layers, the intrinsic frequency approaches f and the vertical wavenumber diverges, see, for example Eqs. (1) or (38). With decreasing vertical wavelength, a GW eventually becomes unstable and dissipates. In the steady-state picture, critical layers are diagnosed at the lowest altitude z = zc where ω^⁡(zc)=ω⁡(zl)−k⋅U⁡(zc)≤|f|, that is, where the Doppler-shift term turns the wave intrinsic frequency to a smaller value than the Coriolis frequency. Accordingly, for each spectral element i, we set the pseudomomentum-flux profile PMFi(z) to zero at z ≥ zc.

When wave reflection occurs, the intrinsic frequency approaches N and m changes sign so that the group velocity is reverted. In the steady-state versions of MS-GWaM this is taken into account by diagnosing the height of potential reflection by finding the lowest altitude z = zr where ω^⁡(zr)≥N. If a reflection layer is diagnosed at z = zr for a spectral element, its corresponding pseudomomentum-flux profile PMFi(z) is set to zero above zr. Unless GW breaking has changed the equilibrium profile below the reflection layer, the pseudomomentum-flux profile PMFi(z) vanishes also for z < zr as well, because under reflection the pseudomomentum-flux changes sign so that the contributions from the upward- and downward-propagating components cancel each other when the mean flow is in a steady state. However, if GW breaking takes place (see the next section) at any altitude z = zb < zr, the pseudomomentum fluxes carried by the upward- and downward-propagating spectral elements do not completely cancel; thus, at altitudes z < zb a residual PMFi⁡(z)=PMFupi⁡(z)−PMFdowni⁡(z) is maintained.

4) GW breaking

In the steady-state setups of MS-GWaM the instability criterion Eq. (23) is used as well. The two steady-state implementations (ST and STMO) differ, however, in the way this is done and how the GW amplitudes are adjusted whenever wave breaking is diagnosed.

A simple approach—most often applied in present-day nonspectral GWPs—is to treat each spectral element independently from each other, that is, applying Eq. (23) for each element separately, leading to a saturation amplitude
Asati⁡(z)=ρ¯2 ω^[1m(z)i2+1Khi2],⁡(i=1,…,Nl).
Wave breaking is diagnosed at an altitude z = zb if Ai⁡(zb)>Asati⁡(zb), and static stability is then reinforced by setting Ai⁡(zb)=Asati⁡(zb). Given the monochromatic treatment of the saturation process, this implementation is called steady-state monochromatic MS-GWaM, or shortly STMO.
An integrated treatment of wave breaking proceeds completely in line with the treatment in the transient MS-GWaM, i.e., wave breaking is diagnosed at an altitude z = zb if Eq. (23) is fulfilled there, with the integral taken over all spectral components, so that
2N4⁡(zb)ρ¯⁡(zb) ∑i=1NlPi⁡(zb)>N4⁡(zb)withPi⁡(zb)=Ai⁡(zb) mi2⁡(zb)Khi2ω^i⁡(zb)Ki2⁡(zb).
Then static stability is reinforced by reducing the wave-action densities via
Ai⁡(zb)→Asati⁡(zb)=Ai⁡(zb)[1−K˜cgzi Ki2⁡(zb)]⁡(i=1,…,Nl),
with
K˜⁡(zb)=∑i=1NlPi⁡(zb)−ρ¯⁡(zb)/2∑i=1NlPi⁡(zb)Ki2⁡(zb)/cgzi,
which is proportional to an altitude-dependent turbulent diffusivity. Since diffusivity cannot be applied in terms of time increments in the steady-state framework, wave breaking and the resulting state of stability is reinforced in terms of vertical increments. This in turn introduces cgz in Eqs. (42) and (43) [cf. Eqs. (26) and (27)] so that, for a given vertical distance and diffusivity, slowly propagating waves tend to dissipate more than those propagating fast. This implementation is called steady-state MS-GWaM, or ST. Given that it shares the treatment of the GW sources and GW breaking with the transient implementation, the only difference between ST and TR is how the propagation is accounted for, that is, GW transience.
Both STMO and ST account for the case of multiple wave breaking in the course of the adjustment to the equilibrium profile. This is achieved by calculating the equilibrium profile sequentially from layer to layer, that is, solving Eq. (39) as
Ai⁡(zi−1/2)=cgzi⁡(zi+1/2)Ai⁡(zi+1/2)cgzi⁡(zi−1/2),⁡(i=1,…,Nl).
This allows for applying the GW saturation Eq. (40) or Eqs. (41)–(43) within the vertical adjustment process and eventually using the replacement Ai[zi+(1/2)]→Asati[zi+(1/2)] if GW breaking is diagnosed at the half-level zi+(1/2).

As in the transient implementation, the present study does not take corresponding effects on frictional heating and GW energy deposition into account.

5) Mean-flow forcing

After having calculated the equilibrium profile and having taken into account dissipative processes, the total pseudomomentum flux at half levels is calculated as
PMFzi+(1/2)=∑i=1Nlkhicgzi[zi+(1/2)]Ai[zi+(1/2)],
and the GW drag at full levels is obtained exactly as in the transient case.

c. Stability measures and computational aspects

To facilitate GW studies in a large altitude range, our model top within UA-ICON has been set to 150 km. In UA-ICON and ICON in general a sponge layer prevents spurious wave reflections from the model top, based on a Rayleigh damping applied to the vertical wind (Zängl et al. 2015). In the setup used here the bottom of the sponge layer is at 110 km. Several measures had to be taken in MS-GWaM to prevent numerical instabilities in the sponge, because of excessive mean-flow accelerations by insufficiently controlled GW pseudomomentum fluxes.

1) Molecular viscosity

Molecular viscosity, inversely proportional to density, is taken into account by
Nj⁡(t+Δt)=Nj⁡(t) exp[−2Kj2⁡(t)Δt ηρ¯],
with the temperature-dependent dynamic viscosity η = η[T(z, t)] based on Sutherland’s viscosity law (see, e.g., Atkins and Escudier 2013). In the steady-state implementations the same prescription is used, but proceeding from layer to layer, and with Δt=Δz/cgzi; that is,
Ai⁡(z+Δz)=Ai⁡(z) exp[−2Ki2⁡(z) Δzcgziηρ¯].

2) Scale height correction

The WKB theory applied by Achatz et al. (2017) predicts that in case of a clear scale separation between GWs and a resolved flow, the former obey the Boussinesq GW dispersion relation in Eq. (1). In the numerical implementation (i.e., TR, ST, and STMO), however, the scale separation does not always hold. Vertical GW wavelengths can grow by refraction and eventually reach values similar to the scales of vertical variations of the resolved mean flow. An ideal treatment of such a situation would be to somehow “transfer” the large-scale GW to the resolved flow and stop treating it as a subgrid-scale wave. A theory for such a procedure, however, is not known to us, and the problem is complicated further by the possibility of such a wave still being unresolved in the horizontal.

Based on our experience during the implementation of TR, the primary problem arising from large vertical GW scales is that using Eq. (1) the vertical group velocity gets too large and leads to excessively strong pseudomomentum fluxes PMF=∫d3k cgzkhN. This is now avoided by using
ω^=±N2Kh2+f2⁡(m2+Γ2)K2+Γ2
in all calculations, where Γ = [1/(2Hρ)] − (1/Hθ) is the pseudoincompressible scale height with Hρ=−[⁡(1/ρ¯)⁡(dρ¯/dz)]−1, Hθ = [(1/θ)(dθ/dz)]−1 being the density scale height and the potential temperature scale height, respectively. By assuming that the atmosphere is locally isothermal, the pseudoincompressible scale height in TR, ST, STMO is further simplified to Γ = [1/(2Hρ)][(1/2) − (R/cp)] with R being the ideal gas constant and cp being the specific heat capacity at constant pressure. From the modified dispersion relation, one obtains for the vertical group velocity
cgz=−m⁡(ω^2−f2)ω^⁡(K2+Γ2),
and the prognostic equation for m becomes
DrmDt=−kh⋅∂U∂z−1ω^⁡(K2+Γ2)[NKh2 ∂N∂z+(f2−ω^2)Γ ∂Γ∂z].
The scale height correction begins to matter at vertical GW scales where the squared vertical wavenumber m2 becomes small enough so that it is only an order of magnitude larger than Γ, that is, when 0.1 × m2 ~ Γ2. By substituting m = 2π/λz, Hρ ≈ 5–8 km, and R/cp ≈ 2/7, one arrives at a vertical wavelength of λzcorr ≈ 45–75 km. Although, in the launch spectrum used in this paper, the vertical wavelength is not larger than λz = 8 km, the transient implementation has been observed to lead to vertical GW scales as large as λzcorr. Thus, to be on the safe side, and for being fully consistent between the transient and steady-state schemes, all implementations of MS-GWaM have been based on the discretized versions of Eqs. (48)–(50) to describe the evolution of the GW field.

3) Pseudomomentum-flux smoothing

In the TR implementation, because of unavoidable local undersampling of ray volumes, pseudomomentum-flux profiles can get noisy so that the GW impact on the resolved flow can exhibit undesired spikes. Thus, a crucial numerical aspect to stabilize TR simulations has been to apply a vertical smoothing on the pseudomomentum fluxes after the projection via Eq. (22) and before calculating the resolved wind tendencies. The smoothing is using the zeroth-order filter of Shapiro (1975), which removes noise with length scales of 2δz but leaves larger-scale structures mostly unaffected.

4) Controlling the total number of ray volumes

To prevent excessive computational costs, the total number Nc of ray volumes per column is limited to a value Ncmax. It has been found that in terms of the time-averaged zonal-mean circulation, a numerical convergence of the TR simulations has been achieved by using Ncmax = 2500 if using a source with Nl = 4 × nc × nω = 4 × 6 × 2 = 48 [see section 4b(4)]. The practical implementation is simple: each time step before the call to the ray-volume emission at the launch level, it is checked columnwise whether Nc > Ncmax. If this is the case in a column, Nc − Ncmax of lowest-energy ray volumes are removed.

5) Computational costs

Table 1 shows the computational costs of TR, ST, STMO, and the operational GW drag scheme used in ICON for NWP purposes (Orr et al. 2010). The computational costs are presented in terms of 1) ttot, that is, total run times of 1-month simulations with UA-ICON using the different GW schemes (see the Table 1 caption for the grid spacing) and 2) tav, that is, average time spent on a single call of the subroutines corresponding to the different parameterizations. The TR scheme is ~5 times as expensive as ST in terms of tav, which leads to about a factor ~2.5 of overhead costs in terms of ttot. This is what transience costs. If the wave breaking scheme is monochromatic (STMO), and as such simpler, accelerations by a factor ~2.3 in terms of tav, and by a factor ~1.3 in terms of ttot can be achieved. There is a further acceleration by factors ~4.5 and ~1.2 between STMO and the operationally used (Orr et al. 2010) scheme in terms of tav and ttot, respectively. Hence TR is ~50 and ~4.1 times more costly than the operational scheme in terms of tav and ttot, respectively.

Table 1.

Computational costs of the different GW parameterizations coupled to UA-ICON on 960 CPUs with a horizontal grid spacing of ~160 km (R2B4 grid) and with 120 vertical levels up to 150 km with the same distribution as described by Borchert et al. (2019).

Table 1.

With regard to the costs in memory, TR simulations use 2% more memory than ST simulations, where 100% stands for the memory cost of the ST simulations. This is not negligible but is relatively small.

4. Results

a. Experimental setup

The first step in order to validate the implementation of MS-GWaM was to reproduce the idealized one-dimensional cases of Bölöni et al. (2016) in UA-ICON. This technical step has been followed by global simulations using TR, ST, STMO, with a horizontal grid spacing of ~160 km (R2B4 grid1). A stretched vertical grid has been used with layer thicknesses gradually increasing with height, with a typical thickness of a few tens of meters in the boundary layer, 700–1500 m in the stratosphere, and a maximum of ~4 km in the lower thermosphere. Similar to Borchert et al. (2019), a model top at ~150 km has been used with a sponge layer acting above 110 km. As initial condition, operational Integrated Forecasting System (IFS) of the European Centre for Medium-Range Weather Forecasts (ECMWF) (https://www.ecmwf.int/en/publications/ifs-documentation) analyses have been used. They have been interpolated to the ICON grid at altitudes covered by IFS/ECMWF and extrapolated toward a simple climatology above. The first few weeks’ simulations after each initialization have been discarded from any scientific analysis to make sure that the adjustment process from the climatology toward the actual realization of the circulation at higher altitudes is excluded.

b. Mean circulation

The first proof of concept for MS-GWaM in a global modeling framework was to validate the zonal-mean circulation it generates as coupled to UA-ICON. For this validation the UARS Reference Atmosphere Project (URAP) data (Swinbank and Ortland 2003) have been used as a reference, because this dataset involves zonal-mean climatologies up to rather high altitudes, i.e., 85 km for temperature and 110 km for zonal wind.

1) Zonal-mean wind and temperature

Simulations with TR, ST, and STMO have been run for the 8 URAP years (1991–98) for December and June (initialized on the 1 November and 1 May, respectively). Figures 5 and 6 respectively show the time-averaged zonal-mean zonal winds and temperatures from the TR and ST simulations as well as the reference URAP data. In general, both the TR and ST simulations produce a very similar zonal-mean circulation (results from the STMO simulations are not shown due to their similarity to ST), which compares reasonably well with URAP. Both models capture the reversal of the summer mesospheric jet although somewhat too low in altitude both in December and June and too weak in June. The corresponding summer mesopause is too cold by ~20–30 K, which might be explained by the fact that the thermal effects of energy deposition (e.g., Becker 2017) of GWs are ignored for the time being. The polar night jet is reasonably well placed, but its magnitude is overestimated in both the TR and ST simulations in both months, especially in June. The stratospheric easterly jet cores are placed too much equatorward in both models and both months. Based on this qualitative comparison, the similarity between the TR and ST suggests that transience does not play a very important role in terms of seasonal-mean and zonal-mean circulation. This does not come as a surprise, as indeed, spatial and time averaging should hide local, short-lived transient effects and eventually reflect a quasi-steady-state circulation of the respective months. At a second look, a sharp eye will spot already nonnegligible differences between TR and ST simulations in Figs. 5 and 6. For instance, the magnitude of the June polar night jet is overestimated to a larger extent, while the June lower-thermospheric jet magnitude in Northern Hemisphere (NH) is underestimated to a larger extent in the ST simulation.

Fig. 5.
Fig. 5.

Zonal-mean zonal wind (m s−1) averaged over (top) Decembers and (bottom) Junes in the period 1991–98 for (a),(d) URAP data; (b),(e) TR; and (c),(f) ST. Solid black isolines are drawn between −120 and 120 m s−1 with a spacing of 30 m s−1.

Citation: Journal of the Atmospheric Sciences 78, 4; 10.1175/JAS-D-20-0065.1

Fig. 6.
Fig. 6.

As in Fig. 5, but for zonal-mean temperature (°C). Solid black isolines are drawn between −150° and 30°C with a spacing of 30°C.

Citation: Journal of the Atmospheric Sciences 78, 4; 10.1175/JAS-D-20-0065.1

2) Residual circulation and zonal-mean GW drag

The residual circulation of UA-ICON with MS-GWaM is presented in Fig. 7 by plotting the residual-mean mass streamfunction along with the corresponding meridional velocity υ* in the transformed Eulerian mean (TEM) equations [Andrews and McIntyre 1978; Hardiman et al. 2010, their Eq. (19)]. Both TR and ST simulations result in a qualitatively similar circulation as presented by, for example, Smith (2012) and Becker (2017). It appears that ST simulations lead to a stronger υ* in the upper mesosphere in comparison with TR simulations, implying that the vertical branch of the residual circulation near the poles is also stronger in ST simulations. This is in line with the somewhat colder temperatures at the summer mesopause regions in ST simulations as compared to TR simulations, because a stronger residual circulation corresponds to stronger cooling in the summer upper mesosphere and heating in the winter lower mesosphere. The difference in the residual circulations of TR and ST can be explained by the zonal-mean GW drag (Fig. 8) from both simulations. The structure of GW drag is well matched with that of υ* in the mesosphere, demonstrating the impact of GWs on the residual circulation. The GW drag of ST in the MLT is larger than that of TR by ~80 m s−1 day−1 in December and by ~160 m s−1 day−1 in June. This corresponds well to the differences found in the strength of the residual circulation between TR and ST and reflects that adding transience to a GWP has important implications on the mean circulation and the heat budget.

Fig. 7.
Fig. 7.

Residual circulation averaged over (a),(b) Decembers and (c),(d) Junes in the period 1991–98 for (a),(c) TR and (b),(d) ST. Filled contours give residual-mean meridional velocity in the transformed Eulerian mean equations υ* (m s−1). Black contour lines give residual mass streamfunction (kg s−1) shown by solid or dashed lines for positive or negative values, respectively, for the magnitudes of 104–1010 kg s−1, with two contours per 10 times increase.

Citation: Journal of the Atmospheric Sciences 78, 4; 10.1175/JAS-D-20-0065.1

Fig. 8.
Fig. 8.

As in Fig. 7, but for zonal-mean net gravity wave drag (m s−1 day−1). Black contours are drawn between the intervals [−360, −40] (dashed) and [40, 160] (solid) with an interval of 40.

Citation: Journal of the Atmospheric Sciences 78, 4; 10.1175/JAS-D-20-0065.1

3) Perpetual runs

A more comprehensive appreciation of the differences between the simulations with the transient and the steady-state GW schemes has been enabled by running long perpetual December simulations with TR, ST, and STMO. The perpetual runs have been achieved by imposing a constant radiative and surface forcing, corresponding to 22 December 1992, including a diurnal cycle. The simulations have been run for 24 months of which the last 12 months have been used for comparison. Mean wind differences between the TR and ST simulations (Fig. 9a) are larger in magnitude, and more statistically significant, than those between the two steady-state simulations ST and STMO (Fig. 9b). This shows that the impact of GW transience is somewhat larger than that of the change in the saturation scheme between ST and STMO [see section 3b(4)], even in the context of the time averaged zonal-mean circulation.

Fig. 9.
Fig. 9.

Zonal-mean zonal wind differences (m s−1) averaged over the last 12 months of a 24-month perpetual December simulation, showing the (a) difference ST − TR, (b) difference STMO − ST, and (c) difference TRx2 − TR, where TRx2 stands for the transient MS-GWaM with a doubled number of ray volumes per column, i.e., Ncmax = 5000 instead of Ncmax = 2500. The solid black isolines are drawn for 0 m s−1. Red stippling denotes points where the differences are statistically significant with 95% confidence, based on t tests taking into account time correlation of the sample. The numbers of statistically significant points are 334 for ST − TR, 313 for STMO − ST, and 63 for TRx2 − TR.

Citation: Journal of the Atmospheric Sciences 78, 4; 10.1175/JAS-D-20-0065.1

4) Numerical convergence

As a validation of the employed maximum number of ray volumes per column Ncmax = 2500, we show in Fig. 9c the mean-wind difference between perpetual December TR simulations using Ncmax = 2500 and Ncmax = 5000. These differences are clearly lower in magnitude and less statistically significant than those between the ST and TR. This demonstrates that the effect of transience is much larger than the effect of doubling the amount of ray volumes in the TR simulations. It confirms both that the TR simulations using Ncmax = 2500 are numerically converged and that the difference due to transient GW propagation (ST − TR) is robust; that is, it reflects a physical feature and not a numerical uncertainty.

c. GW pseudomomentum fluxes

Apart from the time averaged zonal-mean circulation, temporal and spatial variability of the GW pseudomomentum fluxes is of interest. As will be shown, the modulation of the GW spectrum through transient propagation leads to fundamentally different pseudomomentum-flux magnitudes and spatial structures as compared to the steady-state GW schemes.

1) Intermittency and variability

A simple quantification of GW intermittency is the histogram of pseudomomentum fluxes, i.e., the probability of occurrence of various pseudomomentum-flux values at given geographical locations. Following Hertzog et al. (2012), histograms of GW absolute zonal pseudomomentum fluxes have been plotted for TR, ST (Fig. 10), and STMO (not shown but similar to ST) with a similar spatial and temporal sampling as in the above-mentioned paper (see figure captions). The difference between TR and ST is obvious, showing a much better fit of the TR simulations to the observed histograms based on the Vorcore superpressure balloons and the HIRDLS satellite (see Fig. 2 in Hertzog et al. 2012). The low intermittency of the ST simulations is not surprising, since steady-state schemes with a nonintermittent source—such as used here—are known to underestimate the occurrence of high pseudomomentum fluxes. Due to the fact that in the steady-state approximation only dissipative effects—due to wave breaking or close to critical layers—can lead to pseudomomentum-flux variations, no higher values can occur than the launch-level pseudomomentum-flux magnitudes. With the GW source used in this study, the launch-level absolute zonal pseudomomentum-flux magnitude in October is ~4 mPa. Indeed, in the ST simulations no higher values occur than that. In contrast, in the TR simulations at z ≈ 20 km, pseudomomentum fluxes of 60 mPa occur with a nonzero probability, which means that fluxes happen to grow by a factor of 15 at this altitude with respect to their launch values. Figure 10 also shows that, up to flux values of ~30 mPa, the probability of large fluxes decreases with altitude, which is in line with the findings of, for example, de la Cámara et al. (2016) in this respect. The probability of occurrence for flux values larger than ~30 mPa shows a vertical dependence that has never been found in steady-state GWPs: it is increasing with altitude between z ≈ 20 km and z ≈ 40 km, and then it drops down significantly above.

Fig. 10.
Fig. 10.

Histogram of absolute zonal pseudomomentum-flux occurrences sampled at φ ∈ [−65°, −50°], λ ∈ [−180°, 180°] over the Octobers of 1991–98 for TR (solid) and ST (dashed) at different altitudes: z ≈ 20 km (black), z ≈ 40 km (red), z ≈ 60 km (blue), and z ≈ 80 km (green).

Citation: Journal of the Atmospheric Sciences 78, 4; 10.1175/JAS-D-20-0065.1

To understand the vertical dependence of GW intermittency in the TR simulations and to further illustrate the large difference between the TR and ST simulations, Hovmöller diagrams of absolute pseudomomentum fluxes are shown in Fig. 11. Obviously, in the ST simulation pseudomomentum-flux magnitudes decrease monotonically with altitude, while in the TR simulation slanted stripes of increased values with time and altitude demonstrate that GW packets gain pseudomomentum flux in a nondissipative manner in the course of their propagation up to the altitude of 50–70 km and then they dissipate due to saturation. The only way the nondissipative increase can happen—kh being constant—is via variation of ∫dm cgzN. This can originate from local variations of N via Eq. (17), from a local increase of the vertical group velocity via Eq. (20), or by both effects together. Because the Hovmöller diagrams and the histograms in Fig. 10 have been sampled for locations beneath the southern hemispheric polar night jet, it is to be expected that variations described by the nondissipative GW–mean-flow interaction Eqs. (17)–(20) are primarily driven by a resolved wind shear ∂U/∂z > 0 between the launch level and z ≈ 55–60 km. Then, given m < 0, this shear tends to shift westward-propagating (eastward propagating) GWs toward larger (smaller) vertical wavelengths and thus larger (smaller) vertical group velocities in this altitude range. Hence the effect of local vertical group-velocity increase can only play a role for the westward-propagating GWs. Separate Hovmöller diagrams for the westward and eastward pseudomomentum fluxes (not shown), however, reflect similar levels of intermittency as for the absolute values (Fig. 11). Thus, the variation of ∫dm N seems to be the dominant cause, which is the process that can act only in transient dynamics, while group velocity variations are also possible in a steady-state framework. A critical reader will note that we are here at the limits of WKB theory. While WKB assumes the time scale of the wave amplitudes to be significantly longer than the GW periods, this is not really the case here. Had we only the derivations of the theory on paper this would be a worry. Fortunately, however, we know from comparisons between wave-resolving simulations and transient MS-GWaM that the WKB theory still works surprisingly well even in this range. Bölöni et al. (2016) show that their WKB code—a “toy model” version of transient MS-GWaM—can reproduce GW behavior at reflection and critical levels, and it also shows similar short-time-scale variations of GW energy (their Fig. 6) that are strictly beyond the validity of WKB but still in good agreement with the wave-resolving LES. Hence one can have confidence in the simulated wave packets that we are seeing here. The question arises now over which time scales these transient effects survive and thus make a difference as compared to steady-state schemes.

Fig. 11.
Fig. 11.

Hovmöller diagram of absolute pseudomomentum fluxes at φ ≈ −60°, λ ≈ −150° for 1 Jun 1998 for (a) TR and (b) ST. The base-10 logarithm of the fluxes in millipascals is presented.

Citation: Journal of the Atmospheric Sciences 78, 4; 10.1175/JAS-D-20-0065.1

2) Zonal and time mean

An interesting consequence of nondissipative GW–mean-flow interactions is that the nondissipative pseudomomentum-flux convergence is reflected not only locally and for short periods, but also in the time averaged zonal mean. This is illustrated in Figs. 12a–f, where monthly-mean (Octobers of 1991–98) zonal-mean pseudomomentum fluxes from TR simulations turn out to be larger than those obtained from ST simulations everywhere below z ~ 40 km. This is the mean effect of the transient flux changes shown in Fig. 11, which—as explained above—should be due to local variations of ∫dm N. All this suggests that transient effects do not average out completely even in the zonal-mean over monthly time scales, or in other words, the steady-state approximation does not really hold even over these time scales.

Fig. 12.
Fig. 12.

Zonal-mean pseudomomentum fluxes (mPa) averaged over the Octobers of 1991–98, showing vertical cross sections over the Southern Hemisphere for (a),(d) eastward, (b),(e) westward, and (c),(f) absolute fluxes as simulated by (top) TR and (middle) ST. The spacing of the solid black lines is 0.25 mPa in (a), (b), (d), and (e) and 0.5 mPa in (c) and (f). (bottom) Horizontal cross sections of absolute fluxes over the Southern Ocean as simulated by (g) TR and (h),(i) ST, where in (i) the values from the ST simulations are multiplied by 1.3.

Citation: Journal of the Atmospheric Sciences 78, 4; 10.1175/JAS-D-20-0065.1

The pseudomomentum-flux differences between the TR and ST simulations can be put in the context of the missing drag—a general underestimation of the GW forcing at about 60°S by GCMs (McLandress et al. 2012). In particular, Jewtoukoff et al. (2015) showed that the relatively high-resolution operational IFS/ECMWF analyses are underestimating the GW momentum fluxes by a factor of 5 over the Southern Ocean at about 20-km altitude, as compared with superpressure balloon observations. In addition, de la Cámara et al. (2016) showed that the parameterized GW fluxes in the Laboratoire de Météorologie Dynamique Zoom (LMDz) model agree with those resolved by the operational IFS/ECMWF to a good degree, indicating that some state-of-the-art GCMs suffer from an underestimation of GW pseudomomentum fluxes by about a factor of 5. Several studies suggested that part of this underestimation originates from the lack of orographic drag due to small islands not represented in the topographic databases of GCMs (McLandress et al. 2012; Alexander et al. 2009; Alexander and Grimsdell 2013; Garfinkel and Oman 2018) and others proposed that some of the underestimation is due to the lack of horizontal GW propagation in GWPs (Sato et al. 2009; Ehard et al. 2017) or the misrepresentation of nonorographic sources (Hendricks et al. 2014; de la Cámara et al. 2016). It appears, however, that the lack of transience in present-day GWPs might also be responsible for a small but nonnegligible fraction of the missing drag. This is demonstrated in Figs. 12g–i, where horizontal maps of absolute pseudomomentum fluxes are plotted at z ≈ 20 km above the Southern Ocean from the TR (Fig. 12g) and the ST (Fig. 12h) simulations. As expected from the cross sections in Figs. 12c and 12f, the absolute pseudomomentum-flux values from the ST simulations are smaller than those from TR, and as shown in Fig. 12i, if ST fluxes are multiplied by a factor of 1.3, a relatively close match with TR is achieved. Transience thus brings an increase of 30% in terms of absolute momentum fluxes, which is, however, still very far from the missing 500% reported by, e.g., Jewtoukoff et al. (2015). The difference between TR and ST simulations can also be expressed in terms of the zonal GW drag. The drag averaged over φ ∈ [−65°, −55°], z ∈ [20, 50] km and over the Octobers of 1991–98 is −0.291 m s−1 day−1 from ST and −0.476 m s−1 day−1 from TR simulations. Hence transience seems to increase the drag by about 60%. The nonnegligible effect discussed above shows that the transience does matter even over monthly time scales. It is also recalled that all differences presented between ST and TR simulations in this paper are due to the nonorographic fluxes only, in a completely nonintermittent GW source setup.

d. Contribution of different wavelengths to the GW signal

Given that MS-GWaM is a spectral scheme, a decomposition of the GW momentum fluxes and drag into the contributions from different wavelengths is straightforward. Such a decomposition could be of interest for validation purposes against observations if GW sources were realistically taken into account. This is yet not the case here; however, even with the simple GW source used in this study, a decomposition by scales is useful to get a simple first guess about the required horizontal and vertical resolutions for GW resolving simulations. The decomposition is based on the TR implementation of MS-GWaM given its additional realism as compared to ST, i.e., given transience and the prognostic treatment of the vertical wavenumber spectrum. The contribution of GWs with different spatial scales to the pseudomomentum fluxes has been diagnosed by calculating Eq. (22) for a subset of the ray volumes j = 1, …, Ni for which certain conditions hold with respect to their horizontal (λh) or vertical wavelengths (λz). The corresponding drag contribution has been calculated via Eq. (18) (its discretized form) just like for the full drag. These diagnostics have all been achieved in an offline mode, meaning that the resolved flow has been forced with the total drag imposed by the total pseudomomentum fluxes.

1) Decomposition results

The contribution of GWs with different spatial scales to the total absolute flux and drag is shown in Fig. 13. Figure 13a shows the zonal-mean total absolute pseudomomentum flux and drag averaged over Junes of 1991–98. Figures 13b–e suggest that excluding horizontal wavelengths smaller than 50, 100, 200, and 250 km leads to signal losses of ~20%, 50%, 75%, and 75%–85%, respectively, both with respect to fluxes and the drag. The contribution of GWs with different vertical scales can be seen by comparing the total signal with Figs. 13f–i, where vertical wavelengths smaller than 1, 2, 5, and 10 km are excluded, respectively. Here the drag signal is much less affected; namely, no loss can be seen if having contributions from waves with λz > 5 km, and only ~25% is lost if waves with λz < 10 km are excluded. In terms of fluxes, however, the loss of signal below z ≈ 40 km is larger than ~50% if GWs with λz < 2 km are excluded, which increases to a loss of ~75%–80% if GWs with λz < 10 km do not contribute.

Fig. 13.
Fig. 13.

Zonal-mean absolute pseudomomentum fluxes from TR simulations (mPa; filled red contours) and zonal GW drag (gray contours) averaged over Junes of 1991–98. The spacing of fluxes is 0.25 mPa in the interval [0, 1] mPa and 1 mPa in the interval [1, 9] mPa. Contours of GW drag are drawn by 20 m s−1day−1 over [−180, −20] m s−1day−1 (dashed) and by 10 m s−1day−1 over [10, 80] m s−1day−1 (solid). Shown are (a) the total, i.e., the contribution from all horizontal and vertical scales; contributions from scales λh > (b) 50, (c) 100, (d) 200, and (e) 250 km; and contributions from scales λz > (f) 1, (g) 2, (h) 5, and (i) 10 km.

Citation: Journal of the Atmospheric Sciences 78, 4; 10.1175/JAS-D-20-0065.1

The contribution of GWs with different scales to the total intermittency has been examined as well (not shown). It turns out that at z = 20 km, occurrence of large momentum fluxes (≳10 mPa) is completely lost if waves with horizontal scales smaller than 100 km are excluded, leading to similarly unrealistic intermittency curves as obtained from ST simulations. If only small-scale GWs with horizontal scales λh < 50 km are left out, most of the total intermittency is reproduced, leaving us with a loss of at most ~30% for all flux values. Excluding the smallest vertical scales (λz < 2 km) does not affect intermittency, but leaving out even larger-scale waves (λz < 5 or 10 km) reduces the occurrence of fluxes between 5 and 40 mPa significantly.

2) Consequences for GW resolving simulations

Explicitly resolving GWs instead of parameterizing them is recently of increasing interest even in global simulations. In the light of the above, a simple estimate of the required spatial resolution can be given: to get most of the GW signal one needs to resolve horizontal scales of 50 km or smaller and vertical scales of 2 km or smaller. Because in NWP models and GCMs the effective resolution of a given spatial scale λ requires 7–10 grid points per λ, the necessary horizontal and vertical grid spacings to be used for GW resolving simulations can be estimated as Δx < 5 km and Δz < 200 m, respectively. This estimate has to be treated with caution because it does not take into account that resolving the generation (GW source mechanisms), and dissipation of GWs might require even higher spatial resolution than is suggested by the scale decomposition applied here.

5. Summary and conclusions

This paper describes the first implementation—to the best of our knowledge—of a transient subgrid-scale GW parameterization into a state-of-the-art GCM and NWP model. This parameterization is called Multiscale Gravity Wave Model (MS-GWaM). It does not rely on the steady-state approximation and therefore enables both dissipative and nondissipative GW–mean-flow interactions, whereas standard GW parameterizations assume an instantaneous equilibrium between GWs, mean flow, and sources, thereby leaving room only for dissipative forcing. For an estimate of the GW-transience impact, a steady-state version of MS-GWaM (ST), using exactly the same GW saturation scheme, and coupled to the same GW source, has been implemented and used as reference for the transient GW parameterization (TR). The TR implementation of MS-GWaM differs in several respects from other GWPs in the literature that use ray tracing. Song and Chun (2008) as well as Amemiya and Sato (2016) have implemented somewhat similar GWPs into state-of-the-art GCMs. They have, however, kept the steady-state assumption in the prediction of the wave amplitudes via wave-action conservation. As compared with earlier transient implementations (Senf and Achatz 2011; Ribstein et al. 2015), one main difference is that TR MS-GWaM allows a feedback from the resolved mean flow to the subgrid-scale GW field through the ray equations, which is especially not the case for Senf and Achatz (2011). Also, MS-GWaM applies the phase-space representation (section 2b), which, so far, is the only viable solution to avoid numerical problems that arise as a result of caustics. Ribstein and Achatz (2016) already used a fully coupled ray tracer including the phase-space approach, however, not in a GCM but in a more-simple tidal model, similar to Senf and Achatz (2011). Last, but not least, the wave breaking scheme of TR MS-GWaM is also a point that makes an important difference with respect to other GWPs, in that the saturation is diagnosed with a contribution from the full GW spectrum represented by the parameterization at a given altitude at a given time.

The time averaged zonal-mean circulation turned out to be broadly similar in TR and ST simulations, both of them agreeing reasonably well with observations (URAP data by Swinbank and Ortland 2003). Closer inspection shows, however, that in some aspects TR yields slightly better results than ST. By excluding interannual variability via perpetual runs, it has also been shown that the effect of transience is larger than that of varying the saturation scheme in the steady-state implementation, especially in the mesosphere and lower thermosphere. That the summer mesopauses are too cold both in TR and ST simulations is likely a consequence of ignoring thermal effects of energy deposition by GWs. Having a leading-order thermal effect in the MLT (e.g., Becker 2017), this process will have to be included into MS-GWaM. Another finding in the same context is that temperature errors at summer mesopauses are smaller in TR simulations than in ST simulations, which is explained by the weaker residual circulation driven by weaker zonal-mean net GW drag in the MLT region. This is a sign that transient effects do not average out completely and may have important implications on the mean zonal and meridional circulations.

Even more evident differences between TR and ST simulations are found in terms of GW pseudomomentum-flux variability. As expected from earlier studies (e.g., de la Cámara et al. 2016), ST simulations strongly underestimate the intermittency of GW pseudomomentum fluxes (occasional occurrence of large values), while TR simulations lead to considerably more realistic results. The reason for this is that the steady-state assumption only allows dissipative effects to change GW-pseudomomentum fluxes, and hence only allows them to decrease as compared to the source, while nondissipative GW–mean-flow interactions can also lead to an increase of these fluxes. This effect is not only visible locally and over short time scales, but it also affects monthly averages of zonal means: mean pseudomomentum fluxes in the lower stratosphere are ~30% larger in TR simulations than in ST. In the Southern Hemisphere, this is where a missing GW drag has been diagnosed by several studies (McLandress et al. 2012; Jewtoukoff et al. 2015). Hence the neglect of transient GW–mean-flow interactions in standard GW parameterizations might contribute to this issue in a modest extent, beside the lack of lateral propagation (Sato et al. 2009; Ehard et al. 2017), the misrepresentation of nonorographic sources (Hendricks et al. 2014; de la Cámara et al. 2016) or the lack of orographic drag due to missing islands in the insufficiently detailed model topographies (Alexander et al. 2009; McLandress et al. 2012; Alexander and Grimsdell 2013; Garfinkel and Oman 2018).

Increasing the realism of GW parameterizations by including transient wave–mean-flow interactions is seen by us as only a first step. Lateral GW propagation will have to be included as well, which—on the basis of Senf and Achatz (2011); Kalisch et al. (2014); Ribstein et al. (2015); Amemiya and Sato (2016)—changes several aspects of the GW distribution and its impact on the mean flow. Corresponding work has just begun, after a six-dimensional (6D)2 version of MS-GWaM has been successfully implemented into the same f-plane pseudoincompressible flow solver as used by Bölöni et al. (2016) and Wei et al. (2019). More realistic source schemes are also an issue. In Part II of this study (Kim et al. 2021), we report on the effects of coupling MS-GWaM in ICON to a convective GW-source scheme, and more improvements with regard to mountain waves and GWs due to jets and fronts will have to follow. As pointed out by Plougonven et al. (2020), one should always be aware that a realistically looking large-scale circulation is no proof that the parameterization is correct. Instead, the parameterized processes will have to be studied by measurements and wave-resolving simulations as well, and it will have to be made sure that all parts of the parameterization reproduce the properties identified therein. Only then can we have a guarantee that the GWP will be reliable even in a changing climate.

The reader might wonder whether the computational cost of a Lagrangian ray-tracing approach as suggested here is not too overwhelming. As summarized in Table 1 and in section 3c(5), according to the strictest measure (tav), including transient effects increases the computational costs by a factor of ~5 with respect to the ST implementation of MS-GWaM, and by a factor of ~50 with respect to Orr et al. (2010)—the current operational scheme used in the NWP configuration of ICON. The discrepancy in computational costs by a factor of ~10 between the steady-state scheme ST and Orr et al. (2010)—which should perform calculations of similar complexity—suggests that MS-GWaM’s efficiency in general (both TR and ST) could probably be improved by means of code optimization. On the basis of this assumption, an optimized transient MS-GWaM should be about a factor of ~5 more costly than state-of-the-art GW schemes. For the time being it cannot be excluded that lateral GW propagation might increase the costs further, although there is no reason to expect that more ray volumes will be needed per column than are already used in the present MS-GWaM implementation. Keeping in mind other potential overhead costs, such as the MPI communication of ray volumes, a safe estimate for a 6D version of MS-GWaM is approximately a factor-of-10 increase of computational costs, as compared with standard steady-state GW parameterizations. This might be seen as a large increase in costs; however, relating it to costs of other alternatives—such as GW-resolving simulations—might quickly change one’s perspective. As also suggested in section 4d, GW resolving simulations would require a horizontal grid spacing of 5 km (or smaller, e.g., 1 km) and a vertical grid spacing of 200 m. If this requirement were to be satisfied with respect to the horizontal resolution alone, the computational costs (in terms of ttot) would increase by a factor of ~30 000 (for 5 km) or ~5 million (for 1 km). The vertical resolution increase to 200 m everywhere above the troposphere would lead to a cost increase of a further factor of ~8, ending up with something between a factor of 240 000 and a factor of 40 million. Therefore, already in its present state, ICON/MS-GWaM can be a useful tool for research purposes, allowing much less costly simulations than those resolving GWs globally and more realistic than achievable by standard GCM resolutions with classic steady-state parameterizations. Once flow-dependent sources for GWs from orography and jet–frontal systems have been implemented, it will be ready, for example, to accompany field campaigns and help in interpreting their results. The long-term goal of eventually using ICON/MS-GWaM in climate simulations and weather forecasting, however, is also not to be left out of sight.

Acknowledgments

The authors thank the German Research Foundation (DFG) for partial support through the research unit Multiscale Dynamics of Gravity Waves (MS-GWaves) and through Grants AC 71/8-2, AC 71/9-2, AC 71/10-2, AC 71/11-2, AC 71/12-2, BO 5071/2-2, BO 5071/1-2, and ZA 268/10-2. Calculations for this research were conducted on the supercomputer facilities of the Center for Scientific Computing of the Goethe University Frankfurt. This work also used resources of the Deutsches Klimarechenzentrum (DKRZ) granted by its Scientific Steering Committee (WLA) under Project bb1097.

Data availability statement

The ICON software is freely available to the scientific community for noncommercial research purposes under a license from DWD and MPI-M. Potential users who would like to obtain ICON can contact icon@dwd.de. The MS-GWaM code and its module for an implementation in ICON have been developed at Goethe-Universität Frankfurt am Main. Please contact Prof. Ulrich Achatz (achatz@iau.uni-frankfurt.de) for these. The URAP wind and temperature data are available online (https://www.sparc-climate.org/data-centre/data-access/reference-climatology/urap/), as are the ERA5 reanalysis data (https://cds.climate.copernicus.eu).

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1

For the ICON model, by RnBk a global grid is denoted that originates from an icosahedron whose edges have been initially divided into n parts, followed by k subsequent edge bisections.

2

Three + three dimensions in physical and spectral space.

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