## 1. Introduction

Gravity waves (GWs) have been observed to be intermittent (e.g., Hertzog et al. 2008; Alexander et al. 2010). The intermittency of GWs is an important factor that affects the ratio of local amplitudes to the mean (total) amplitude of waves within a given area and time period, both essential variables in GW parameterization (GWP) (Kim et al. 2003; Alexander et al. 2010). The mean amplitude is related to the total amount of (pseudo)momentum carried by GWs, while the transfer of momentum to the mean flow can depend on the local wave amplitude when nonlinear processes, such as wave breaking (Lindzen 1981), are involved. In this regard, a tunable parameter, the so-called efficiency factor (or intermittency factor), has often been introduced in GWPs to prescribe that ratio (e.g., McFarlane 1987; Garcia et al. 2007). Meanwhile, it has been observed that the degree of intermittency varies significantly depending on the season and region as well as on the source of waves (Hertzog et al. 2012; Plougonven et al. 2013).

Nonorographic GWPs that employ the Lindzen saturation scheme or its variants for describing the GW breaking process (e.g., Garcia et al. 2007; Alexander and Dunkerton 1999; Richter et al. 2010; de la Cámara et al. 2014; Choi et al. 2018) apply the wave-saturation criterion separately to each spectral component of GWs. This monochromatic approach assumes that GWs with various characteristics do not coexist within the same area at wave-breaking altitudes, while this assumption may implicitly require some degree of wave intermittency. It is therefore worth investigating GW intermittency both to confirm this picture and to help constraining the aforementioned efficiency factor. Various observational efforts to assess the GW intermittency have been made recently (Hertzog et al. 2008, 2012; Wright et al. 2013; Plougonven et al. 2013; Ern et al. 2014; Cao and Liu 2016).

Factors that affect wave intermittency may include the intermittency of wave sources and dispersion/focusing of GWs in space and time during the propagation. By introducing flow-dependent sources of waves into GWP, it can explicitly take into account the sources’ variability including the source intermittency as well as other longer-term variability (de la Cámara and Lott 2015; Bushell et al. 2015; Kang et al. 2017). On the other hand, the spatiotemporal dispersion/focusing of GWs is a transient and local phenomenon not represented by conventional GWP where the steady-state assumption is used; i.e., each time step an instantaneous equilibrium between GWs, their sources, and the mean flow is calculated throughout each column. To fully reproduce GW dispersion, a prognostic model is required, where the propagation of GWs in their simultaneous interaction with the resolved flow is described explicitly.

In the companion paper by Bölöni et al. (2021, hereinafter Part I), a prognostic GW model based on the Wentzel–Kramers–Brillouin (WKB) theory (Grimshaw 1975; Achatz et al. 2017), the Multiscale Gravity Wave Model (MS-GWaM) (Muraschko et al. 2015; Bölöni et al. 2016; Wilhelm et al. 2018; Wei et al. 2019), is described in detail and applied to a global model as a GWP. MS-GWaM employed in Part I is a one-dimensional (single column) version of the model but it allows GWs to propagate with finite group velocities and to interact with temporally and vertically varying mean flows throughout their propagation. A flow-independent source varying with season is used there to represent GWs from all nonorographic sources. Despite this source varying only slowly, substantial variations are found in the GW activity on various time scales, from short scales relevant to the GW intermittency to the monthly scale. This demonstrates the effects of transient evolutions of GWs and mean flow on the variability of the modeled GWs (Part I).

The present paper focuses on the intermittency of parameterized GWs using the same GW model as that used in Part I. To include the source intermittency and variability, we introduce the convective source, among others, into the model, which has been known as the major source of GWs in the tropics (e.g., Tsuda et al. 1994; Dunkerton 1997; Kim et al. 2003). Convection is highly intermittent by nature, and it generates a very broad spectrum of GWs, with variations in wave activity on diurnal to decadal time scales, depending on the region (Richter et al. 2010; Lott and Guez 2013; Kim et al. 2013; Kim and Chun 2015; Kang et al. 2017). As will be shown, these aspects can result in the parameterized GW intermittency to be quite different from that simulated in Part I with the flow-independent source. We also investigate the effect of prognostic GW modeling on the simulated intermittency, as the model can explicitly simulate the dispersion and focusing of waves in the vertical direction and in time, while effects of horizontal GW propagation are not taken into account.

Section 2 describes the method to couple the convective sources to MS-GWaM and the experimental setup. In section 3a, the simulated mean circulations are briefly examined and compared with those presented in Part I. The investigation of simulated GW intermittency follows in section 3b. Summary and discussion are given in section 4.

## 2. Methods

### a. Experiments

The experimental setup is the same as in Part I, except for the GW sources. The upper-atmosphere extension of the Icosahedral Non-Hydrostatic (ICON) general circulation model (UA-ICON; Borchert et al. 2019) is used with a horizontal grid spacing of ~160 km and 120 vertical layers from the surface to the model lid at *z* = 150 km (the sponge layer starts from 110 km). The model is integrated for 2 months for each of 16 cases initialized on 1 November and on 1 May for the years of 1991–98, and results for the second month each (December and June) are used for the analyses.

**x**,

**k**,

*t*) depending on position

**x**with vertical coordinate

*z*, wavenumber

**k**=

**k**

_{h}+

*m*

**e**

_{z}decomposed into its horizontal part

**k**

_{h}and the vertical wavenumber

*m*, and time. The corresponding prognostic equation is

*c*

_{gz}is the vertical group velocity of spectral component and

**U**are the density and horizontal wind of the resolved flow. The numerical discretization uses a decomposition of that part of phase space, spanned by

**x**and

**k**, with nonzero

Three experiments are conducted following those in Part I: One uses MS-GWaM as described above, but also takes wave breaking into account by a spectral saturation approach. In this approach, breaking is diagnosed wherever the constructive interference of all spectral components can lead to static instability. A turbulent diffusivity is determined and applied at that location, reducing

The wave breaking is represented in a different way in the two steady-state approximations of MS-GWaM. The first (ST) uses exactly the same spectral saturation scheme as TR. In the second (STMO), following the classic approach of many GWPs (e.g., Garcia et al. 2007), wave breaking is diagnosed if a single discretized spectral component leads to static instability, and the turbulent diffusivity is applied only to that component. This requires larger wave amplitudes for breaking than the spectral approach. Further details on MS-GWaM and its numerical implementation are given in Part I.

All three experiments employ the convective GW source as described in section 2b, in addition to a flow-independent seasonally varying source accounting for other nonorographic sources such as jets and fronts in the extratropics. The spectrum and high-latitude (50°–90°) amplitudes of the prescribed source follow those used in section 3a(3) of Part I, but in the current experiments this source is reduced to zero within 20°N–20°S, with a smooth transition of amplitudes between 50° and 20° latitudes (see Fig. 1 for its latitudinal profile). We refer to the GWs from this source as background GWs (BGWs). Currently, orographic sources are not coupled to MS-GWaM implemented into ICON, and thus ICON’s operational orographic GWP (Lott and Miller 1997) is used in all the three experiments. Therefore, orographic GWs are not included in the analysis of GW intermittency in this study. The model outputs are sampled at 3-h intervals for the analysis in section 3 (except those used in Fig. 2; see below).

### b. Coupling of convective sources to the parameterizations

#### 1) Formulation

*S*=

*Q*/

*T*due to the latent heat

*Q*(

*x*,

*y*,

*z*,

*t*) associated with a single convective updraft in a basic flow with vertically varying temperature

*T*(

*z*). The spatiotemporal distribution of the forcing is assumed to have the form

*s*

_{0}is its peak magnitude,

*ζ*(

*z*) = max{0, 1 − [(

*z*−

*z*

_{m})/

*d*]

^{2}} is a quadratic function of altitude, with its center at

*z*

_{m}and half depth

*d*, and

*G*(

*x*,

*y*,

*t*) is a Gaussian in both horizontal directions and time that has an approximate radius

*δ*

_{h}and half duration

*δ*

_{t}, moving with a velocity

**c**

_{q}[for the expression of

*G*, refer to

*q*/

*q*

_{0}in Eq. (30) in SC05]. This configuration is the same as in SC05, except that in SC05 the forcing was configured in terms of heat (i.e.,

*Q*=

*q*

_{0}

*ζG*, where

*q*

_{0}is the peak magnitude of

*Q*) rather than entropy, and entropy was expressed as

*Q*/

*T*

_{0}with a constant temperature

*T*

_{0}in their governing equation [their Eq. (3)].

^{1}

*z*

_{t}=

*z*

_{m}+

*d*), the horizontally and temporally integrated momentum flux as a function of phase speed

*c*and horizontal direction

*φ*is derived as

*q*

_{0}by

*s*

_{0}and omitting

*T*

_{0}in their formulations because of the above modification). Here,

*ρ*

_{0}is the reference density,

*m*is the vertical wavenumber calculated from the dispersion relation [

*N*and

*z*

_{t}, and

*c*

_{p}being the specific heat capacity at constant pressure and

*N*

_{q}being the reference Brunt–Väisälä frequency below

*z*

_{t}. Here,

*X*is a complex function given by Eq. (19) in SC05, which depends on the vertical structure of the forcing (

*d*and

*z*

_{m}) and the background flow, and Θ

_{s}is

*c*

_{q,φ}≡

**c**

_{q}⋅ (cos

*φ*, sin

*φ*). Note that the momentum flux is equal to the pseudomomentum flux for the nonrotating-flow solution, which is valid at the typical scales of GWs generated by deep convective heating and in the tropics.

*m*, by multiplying Eq. (4) by ∂

*c*/∂

*m*, yielding

*m*,

*φ*), the horizontal-wavenumber (

*K*

_{h}) spectrum of the (pseudo) momentum flux has the form

*A*is a normalization factor) [see Eq. (29) of SC05]. This spectrum has its peak at the wavenumber

#### 2) Parameters for convection properties

As we consider the latent heat associated with a convective updraft, which typically has horizontal scales of ~10 km, the convection properties defined above are retrieved from the heating profiles calculated in the subgrid-scale convection parameterization in ICON (Bechtold et al. 2008). In this study, convective GWs (CGWs) are generated only in grid cells where deep convection is diagnosed by that parameterization, with its top altitude more than 5 km above the surface.

First, the profile of diabatic heating due to phase changes in convective updrafts is extracted from various terms of the convective forcing (which also include, for example, subgrid-scale adiabatic cooling/heating by updrafts/downdrafts) and divided by the temperature profile to obtain the gridcell mean entropy forcing *z*_{m} and *d* are obtained by fitting *ζ*(*z*), resulting in *z*_{m} = *σ*(*z*)/*σ*(1) and *d*^{2} = 5*σ*[(*z* − *z*_{m})^{2}]/*σ*(1), where *z*_{0} and *z*_{1} being the altitudes at which

*ϵ*

_{q}on the horizontal plane is taken into account so that the magnitude of the entropy forcing within a single updraft is determined as

*ϵ*

_{q}and Eq. (4), the gridcell mean (pseudo) momentum flux of GWs generated from all local updrafts

*M*

_{1}⟩⟩ given in Eq. (9) is derived in the appendix. Note that

*s*

_{0}. The

*ϵ*

_{q}is a tunable parameter, which may be considered to be less than 10%, as suggested by observations (e.g., Dorrestijn et al. 2015), unless the gridcell area Δ

*A*is too small; that is,

*ϵ*

_{q}= 1%.

It is notable that parameterized CGW fluxes will be dependent on the model resolution even when a constant value is used for *ϵ*_{q} across the resolutions. The heating magnitudes retrieved from subgrid-scale convection parameterization must decrease with increasing resolutions, as the fraction of explicitly resolved convective heating increases. The decrease of subgrid-scale heating (

As in SC05, *δ*_{h} and *δ*_{t} are set to 5 km and 20 min, respectively, considering typical scales of convective latent heat release. The **c**_{q} is determined to be the low-level (0–3 km) averaged horizontal wind (Choi and Chun 2011), and *ρ*_{0} is the density at *z*_{m}. This configuration of parameters results that the (pseudo) momentum fluxes calculated in our experiments have a peak intrinsic frequency ^{−1} in their global mean spectrum, and the spectrum tends to decrease quickly as the frequency increases (not shown). The decrease of the spectrum from the frequency far lower than *N* may justify the hydrostatic approximation used in SC05, given that the approximation is valid for GWs with

#### 3) Discretization of spectra and the GW-launch process

*m*spectrum of the (pseudo) momentum flux (9) is discretized based on the corresponding

*N*/|

*m*|): 22 values of

^{−1}with varying intervals (Δ

*c*= 2 m s

^{−1}at

^{−1}and Δ

*c*= 4 m s

^{−1}at

^{−1}).

^{2}The bin widths of the discretized

*m*are calculated by

*φ*=

*π*/2) are determined such that one of them is the same as the direction of

**c**

_{q}−

**U**(

*z*

_{t}), where

**U**is the horizontal wind vector, because (pseudo) momentum-flux spectra of CGWs tend to have peaks in that direction (e.g., SC05; Choi and Chun 2011). Only one horizontal wavenumber is used for a given pair of (

*m*,

*φ*) in this study; that is,

*K*

_{h}=

*K*

_{h,1}[given in Eq. (7)] is used as a representative value. The wavelength corresponding to

*K*

_{h,1}varies between about 10 and 250 km, depending on

*c*(or

*m*). In total, spectra are discretized into 88 elements. Last, for a discrete ray volume, the wave-action density at the top of the heating (

*z*

_{t}, referred to hereinafter as the cloud top) is determined from

*k*and Δ

*l*are constant in the single-column formulation and eventually cancel in the equations so that they need not be specified.

Once deep convection is diagnosed by the convection parameterization in a grid cell, the GW spectrum calculated as described above is launched continuously until the convection parameterization is called again. The launch level is the cloud top (*z*_{L} = *z*_{t}), and the launch procedures are described in detail in section 3a(3) of Part I.

#### 4) Cloud-top (pseudo) momentum flux in the simulations

Figure 1 shows the monthly and zonal mean of the cloud-top (pseudo) momentum fluxes for CGWs calculated by Eq. (9) in December and June in the TR, ST, and STMO experiments, projected onto the eastward and westward directions, along with the prescribed launch-level pseudomomentum flux for BGWs, which is isotropic. In both months in all experiments, the latitudinal distribution of the CGW (pseudo)momentum fluxes exhibits a dominant peak in the tropics for eastward waves and for westward waves peaks of comparable magnitudes are found in the tropics and the midlatitudes of each hemisphere. The flux magnitudes and latitudes of the tropical peaks are very similar to those presented, for both the eastward and westward CGWs in both months, by Richter et al. (2010, their Fig. 2), which resulted from a different parameterization of CGWs.

Relative to the results of previous studies that used SC05-based source parameterizations (e.g., Kang et al. 2017, Fig. 1; Choi et al. 2018, Fig. 5), a difference is found in the latitudinal distribution of the cloud-top (pseudo)momentum fluxes: the fluxes in the winter midlatitudes are much less pronounced in Fig. 1, whereas in those studies, the cloud-top CGW fluxes appear to have their dominant peak in these latitudes. The latitudinal structure of cloud-top CGW fluxes is determined by multiple factors, such as latitudinal variations in the height and strength of convections, basic-state stability, and vertical structure of winds (SC05; Kang et al. 2017). Further detailed investigation will be needed to find out the cause for the difference in the flux distribution, beyond the scope of this paper.

The magnitudes of the CGW fluxes seem reasonable in the tropics, given that the absolute pseudomomentum fluxes averaged over 15°N–15°S at 20 km (refer to Figs. 9d and 10d, described in more detail below) have magnitudes (3.7–3.9 mPa in both months in all experiments) that are comparable to the observed ones (e.g., Jewtoukoff et al. 2013). In addition, differences in the cloud-top pseudomomentum flux between the experiments seem to be small (Fig. 1), which enables the assessment of the effects of transient versus steady-state dynamics on the GW intermittency, excluding the source effects.

## 3. Results

Figure 2 illustrates the time series of pseudomomentum fluxes for CGWs in a grid column in the TR and ST experiments to demonstrate differences in the behavior of parameterized fluxes between the transient and steady-state models. The selected grid column is at ~3.5°N, 116°E over Borneo, a convectively active region, and the time series are presented for the first 3 days of the simulation initialized on 0000 UTC 1 May 1998, with 5-min intervals. The occurrences of subgrid-scale deep convection are indicated by red marks at cloud-top heights in each experiment, which reveal a pronounced diurnal cycle with more convection at 0600–1500 UTC (1400–2300 LST). The evolution of the pseudomomentum flux in the TR experiment shows that, first, it can take several hours or more for the CGWs to propagate into the mesosphere from the troposphere, in contrast to the instantaneous vertical propagation shown in the ST experiment. For example, at *z* ~ 50 km, the daily maxima of the westward flux tend to occur at 1500–2100 UTC in the TR experiment (Fig. 2b), about 6–9 h later than the occurrences of convection. Such long propagation is often associated with a vertical shear of the wind: in this case, the strong easterly shear at 20–25 km as seen by the wind profile in Fig. 2c.

Second, the fluxes are largely dispersed in time in the TR experiment, which results from a broad spectrum of vertical group velocities of CGWs. Accordingly, in the upper stratosphere and mesosphere, the occurrence probability of CGWs is higher and the daily fluctuations of the fluxes tend to be smaller in amplitude in the TR experiment than those in the ST experiment. This may result in lower intermittency of CGWs in the TR experiment, as will be quantified in section 3b. It can be inferred that this feature may depend on the variability of sources. In the case of GWs that have a steady source (Part I), the transient model produces more fluctuations than the steady-state model where the background flow is varying.

Another potential consequence of the differences between the two models might involve the modification of large-scale waves, such as atmospheric tides. Senf and Achatz (2011) and Ribstein and Achatz (2016) showed that temporal evolutions of the background flow due to atmospheric tides change the frequencies of GWs, which in turn affect the tidal amplitudes, when the wave transience by the flow evolution is considered in the ray-tracing model with a uniform source in time. It can be expected that CGWs interact with tides more effectively than the waves from a uniform source, because convection typically has diurnal variations. The temporally lagged and dispersed CGW fluxes in the TR experiment in contrast to the instantaneous, localized fluxes in the ST experiment, as shown in Fig. 2, may make differences in simulated GW–tide interactions. Investigation on this will have to involve a thorough analysis of the simulated tidal waves as well as GWs, which merits future study. In this paper, we restrict our focus to the assessment of the GW intermittencies represented by the transient and steady-state models.

### a. Mean circulation

Since the convective sources are included here and the background source is reduced in the tropics, the simulated mean flows are altered from those shown in Part I. Figure 3 presents the monthly-mean zonal-mean zonal winds in December and June in the TR, ST, and STMO experiments, averaged for the 8 simulations (1991–98) for each month. Relative to the results using the transient MS-GWaM with the uniform source only (Figs. 5b,e in Part I), in both months the peak magnitude of the polar night jet is smaller by about 30 m s^{−1} and its peak altitude is lower by ~5 km in the TR experiment with convective sources (Fig. 3), showing better agreement with the *UARS* Reference Atmosphere Project (URAP) wind (see Figs. 5a,d in Part I, for the URAP wind). The weakening and lowering of the peak of the jet may result from the additional westward momentum transported by CGWs at around 40°–60° of the winter hemisphere in the TR experiment (Fig. 1).

In the summer hemisphere, the wind reversal near the mesopause occurs at lower altitudes in the TR experiment with convective sources than that in Part I in both months (by approximately 3 km). As also noted in Part I, the location of the easterly-jet maximum is different between the model and URAP: in the model, its maximum is in the subtropics, rather than at high latitudes as in URAP. Note that this difference is common with other general circulation model results (e.g., Richter et al. 2010; de la Cámara and Lott 2015; Bushell et al. 2015).

The overall zonal-mean zonal wind in both the ST and STMO experiments seems to be qualitatively similar to that in the TR experiment, while differences are found in the altitude of the wind reversal in the summer hemisphere and the westerly magnitude above the reversal (Fig. 3). Relative to the TR experiment, in December, the peak magnitude of the westerlies at ~88 km in the ST experiment is 4 m s^{−1} larger, and that in the STMO experiment is about 25 m s^{−1} larger. The altitude of the wind reversal at 25°–60°S is ~2 km higher in both the ST and STMO experiments than in the TR experiment. These differences in the summer hemisphere are less clear in June. Figure 4 shows the wind differences between the experiments: ST minus TR and STMO minus TR, with statistical confidence levels higher than 95% based on Welch’s *t* test. The wind difference around the Southern Hemisphere mesopause in December between the ST or STMO and TR experiments is respectively about 10–20 or 15–30 m s^{−1}, with vertically alternating signs, which is associated with the aforementioned vertical shift of the wind reversal as well as the difference in the westerly magnitude. Except this region in December, the differences between the ST and TR experiments as well as those between the STMO and TR experiments are rather moderate, with typical magnitudes of 5–10 m s^{−1} in the mesosphere in both months. In the stratosphere, the differences in the monthly-mean wind between the three experiments are small everywhere (Fig. 4), which will allow us to exclude potential effects of difference in the mean flow on the simulated GW intermittency in its comparisons between the experiments in the next section.

As mentioned above, the westerlies at ~88 km in the summer hemisphere in December are notably stronger in the STMO experiment than in both the TR and ST experiments (Fig. 3), which also results in the large wind differences in STMO minus TR (Fig. 4). The wind differences between the two experiments using the steady-state models, ST and STMO, were generally small everywhere except this westerly region (not shown). In the STMO experiment, due to the monochromatic application of the saturation criterion, GW amplitudes required for their saturation are larger than those in the ST experiment. This results in larger momentum fluxes in the upper mesosphere, due to less saturation/dissipation below this level, and thus leads to larger GW drag where they break, as indicated by the stronger westerlies, relative to the ST experiment.

### b. Intermittency of GWs

#### 1) Probability density functions

*F*) of GWs are constructed to investigate the distributions of flux magnitudes. For a given PDF, the contribution of fluxes that are larger than a certain threshold

*a*to the total flux budget is defined as

*P*being the PDF as a function of

*F*′. Here, we calculate and plot the PDFs with a logarithmic axis, i.e.,

*F*′ = log

_{10}

*F*and

*a*′ = log

_{10}

*a*. Figures 5a and 5b show the PDFs for the fluxes sampled in the tropics (15°N–15°S) at five altitudes from 20 to 100 km in December during the 8 years in the TR and ST experiments, respectively. Note that in the tropics, the parameterized GWs are solely from the convective sources (Fig. 1). On each PDF curve, the lower bound of the largest fluxes that contribute to 99% of the total flux budget,

*r*

^{−1}(0.99) ≡

*F*

_{1}, is indicated by the filled circle to confine our focus to the fluxes with significant magnitudes. The mean flux

At 20 km, CGWs parameterized in the TR experiment have a broad distribution of *F*, with the largest probability of occurrence in the logarithmic representation at *O*(10 mPa) (Fig. 5a). The mean flux at this level is 3.8 mPa, and the total flux budget is mostly (99%) accounted for by fluxes larger than ~0.62 mPa, as indicated by *F*_{1} (red circle). As the altitude increases, the PDF shifts as a result of wave dissipation to smaller *F*. Interestingly, however, the shapes of the PDFs, with respect to their *F* > *F*_{1}. For example, the location of peak *F* (

In the ST experiment (Fig. 5b), the PDFs for CGW fluxes resemble the lognormal distributions (within the *F* range shown in the figure), especially in the stratosphere. At 20 km, both the mean and peak *F* values of the PDF (3.7 and 8.9 mPa, respectively) are close to those in the TR experiment while *F*_{1} (2.2 mPa) is 3–4 times as large and very large fluxes (~10^{2} mPa) appear more frequently than in the TR experiment. As the altitude increases, the PDF tends to be left skewed; i.e., *F* = 0). Thus the entire PDFs differ from the lognormal distributions (see also Fig. 5c around *F* ~ 0).

The PDFs at 20 km have also been constructed on the linear scale, for comparisons with observations from previous studies. The PDFs in the tropics in the two experiments (Fig. 5c) are quite similar to those for the GW momentum fluxes measured from the superpressure balloons (SPBs) during the PreConcordiasi campaign in February–May 2010 (Jewtoukoff et al. 2013). For instance, the occurrence probabilities at 10, 30, and 100 mPa (±0.5 mPa) in the TR experiment are 1.2 × 10^{−2}, 1.3 × 10^{−3}, and 4.8 × 10^{−3}, respectively, each of which is within the observed range for the corresponding flux magnitude shown in Fig. 16 in Jewtoukoff et al. (2013) (see the PDFs for balloons 1 and 2, which flew within 15°N–12°S). The occurrence probabilities in the ST experiment are higher at *F* > 30 mPa than those in the TR experiment (as also shown in Figs. 5a,b), and are still within the range of SPB observations at

A difference between the modeled and observed PDFs in the tropics exists at *F* ~ *O*(1 mPa), where both experiments exhibit much smaller occurrence probabilities than the observations. This might be because our source parameterization neglects some weak convection by applying the criterion for the cloud-top height (section 2a) and/or the observations would capture signals of GWs with horizontal wavelengths of hundreds of kilometers, which could be emitted by large-scale convective systems but are not considered in the current parameterization. However, given that these flux magnitudes are an order smaller than

*F*

_{1}(

*p*

_{c}) can be used as a measure of intermittency, so that

*F*

_{1}is larger and the

*P*peak is smaller in the ST experiment, resulting in smaller 1 −

*p*

_{c}, relative to those in the TR experiment. The

*p*

_{c}values are presented in Table 1 and indicate that the flux budget is accounted for by about 20% and 40% of samples in the ST and TR experiment, respectively. The lower intermittency in the TR experiment may result from the dispersion of fluxes in time by the transient modeling of GWs, as expected from Fig. 2. In each experiment,

*p*

_{c}does not change significantly with altitude (Table 1).

Percentile of the lower bound of the largest pseudomomentum fluxes that contribute to 99% of the total flux budget in the tropics in December in the three experiments (see Figs. 5a,b).

Figure 6 shows PDFs in the southern midlatitudes (25°–65°S) in December for CGWs and BGWs each and for their sum. In the extratropics, convective activities are more intermittent than those in the tropics, and thus CGWs are also more intermittent. This is reflected in the PDFs in both experiments by much lower probabilities everywhere at *F* > *F*_{1} (thus, smaller 1 − *p*_{c}) in the midlatitudes, relative to those in the tropics (Figs. 6a,b, left; Figs. 5a,b). Also, this latitudinal difference is much emphasized in the ST experiment than in the TR experiment: the peak probabilities in the midlatitudes are only 10%–20% of those in the tropics in the ST experiment. This may imply that the temporal dispersion of waves in the transient model can mitigate the GW intermittency where the source intermittency is high. The *p*_{c} values in the midlatitudes are presented in Table 2.

As in Table 1, but in 25°–65°S for the CGW, BGW, and total pseudomomentum fluxes (see Figs. 6a,b).

BGWs in the southern midlatitudes in the ST experiment have very high and sharp peaks in their *F* PDFs in the stratosphere (Fig. 6b, center), as expected from the nearly invariant launch-level flux in time. In this experiment, the wave dissipation in the stratosphere occurs very selectively for the largest fluxes, while above 60 km, it occurs over a broad range of fluxes. The BGWs in the TR experiment (Fig. 6a, center) exhibit broader and smoother distributions of *F* than in the ST experiment, and are close to the lognormal distributions below 100 km. In some rare cases, their *F* magnitudes become larger than ~10 mPa in the stratosphere, while the launch-level flux is 6 mPa at most (at the latitudes higher than 50°), as discussed in Part I. The PDFs for the total GW fluxes (Figs. 6a,b, right) are dominated at moderate or small *F* magnitudes (e.g., *F* tails. In both experiments, the probabilities at the PDF peaks decrease gradually with altitude, which results in increase of *p*_{c} (Table 2, lower). Therefore, the intermittency of the parameterized GWs in the midlatitudes is found to increase with altitude. This will be illustrated more quantitatively by Gini coefficients in the next section.

Figures 6c and 7 show the 20-km PDFs in the midlatitudes (25°–65°S) and in 50°–65°S, respectively, on the linear scale. The latter latitude band has been targeted in several observational and modeling studies of GW intermittency (e.g., Hertzog et al. 2012; de la Cámara and Lott 2015). In the midlatitudes, the PDFs for the total GW fluxes show a large difference between the two experiments at 5 < *F* < 30 mPa (Fig. 6c, right), which is attributed mostly to differences in the BGW fluxes. The PDF in the TR experiment seems qualitatively similar to the PDF simulated by de la Cámara et al. (2014, their Fig. 4b), who used a stochastic method to parameterize GWs with convective and background sources. The contributions of the fluxes larger than the 90th and 99th percentiles (7.2 and 32 mPa, respectively) to the total flux budget (*r*_{90th} and *r*_{99th}) are 52% and 17%, respectively, which are similar to *r*_{90th} (49%) and a bit larger than *r*_{99th} (11%) simulated by de la Cámara et al. (2014) at the same altitude. In the high latitudes (Fig. 7), due to weak convective activity, the contribution of CGW fluxes to the total GW fluxes is relatively weak for moderate or small flux magnitudes (*r*_{90th} (*r*_{99th}) in the TR experiment is 36% (8.6%), which is comparable to that obtained from the SPB observations of the Vorcore campaign and High-Resolution Dynamics Limb Sounder (HIRDLS) observations, i.e., 33%–45% (8%–10%), both reported by Hertzog et al. (2012, Fig. 5).

Our analysis of PDFs simulated by MS-GWaM in the tropics and Southern Hemisphere in December shows that the lognormal distributions of GW momentum-flux magnitudes observed in previous studies (e.g., Hertzog et al. 2012) can be simulated by the transient modeling of GWs with the combination of invariant and convective sources (Fig. 6a). The CGWs contribute to the high-*F* tails of the PDFs in the middle and high latitudes. Results in the northern midlatitudes and those in June (not shown) are qualitatively similar to the results shown above.

#### 2) Gini index

*concentration ratio*, for a set of quantities based on their distribution, and this index was proposed as a measure of GW intermittency by Plougonven et al. (2013). The index

*I*

_{g}is defined as

*n*nonnegative quantities, where

*f*

_{i}is the

*i*th quantity normalized by the sum of all quantities in the set, after sorting them in ascending order. Here, we calculate the Gini index for the absolute pseudomomentum flux (

*F*), i.e.,

*I*

_{g}= 0 if the quantities in the set are all the same and nonzero (

*f*

_{i}=

*n*

^{−1}),

*I*

_{g}= 1 if there exists only one nonzero element in the set (

*f*

_{i}= 0 for all

*i*<

*n*and

*f*

_{n}= 1:

*perfect concentration*), and 0 ≤

*I*

_{g}≤ 1. A large value of the index indicates that a small number of large-amplitude events account for a large portion of the total momentum-flux budget, which means high intermittency.

Figure 8a presents the Gini index for *F* from all GWs (CGW + BGW), and Fig. 8b shows the monthly-mean flux *I*_{g} is close to 0.5 in convectively active regions with large mean CGW fluxes (~10 mPa), i.e., southern Africa, South America, and near-equator regions on the Pacific and Atlantic Oceans. These *I*_{g} values are very similar to those observed by SPBs in the tropics (0.5–0.6; see Jewtoukoff et al. 2013).

Outside those regions, *I*_{g} tends to be inversely proportional to *I*_{g} values in the regions of small mean fluxes should be interpreted carefully. It only takes into account the GWs generated by subgrid-scale deep convection, as our CGW source describes only those and thus it is not straightforward to compare the modeled values with observations where the subgrid-scale convection is weak. As discussed in the previous section, GWs generated by shallow convection, as well as large-scale GWs, may also appear in the real atmosphere and, where subgrid-scale deep convection is relatively rare, these GWs may contribute to the momentum budget, resulting in smaller *I*_{g} in the observed GW fields than shown here (~0.8–0.9). Also note that in the real atmosphere, where convection and, hence, GWs appear sparsely, the occurrence frequency of GWs could increase through dispersion of waves in the horizontal directions as well as in time, which might mitigate the wave intermittency, while the horizontal propagation is currently not modeled. This is discussed further in section 4.

Global maps of the Gini index for the total GW flux are shown in Fig. 9a, at 20, 50, and 80 km in the TR and ST experiments, along with the 20-km monthly mean CGW flux in Fig. 9b, in December. Relative to *I*_{g} in the tropics shown above, it is lower in the extratropics at 20 km, where 0.3 < *I*_{g} < 0.7 in the TR experiment, with larger values in midlatitudes than in high latitudes (Fig. 9a). This distribution is actually very similar to that of *I*_{g} for BGWs only (not shown) outside 30°N–40°S. Around 60°S, one has *I*_{g} = 0.4–0.5 in the TR experiment, which is similar to values obtained from SPBs in nonorographic regions at high latitudes in December, i.e., 0.4–0.5 in the Vorcore campaign (Plougonven et al. 2013) and 0.36 in the Concordiasi campaign (Jewtoukoff et al. 2015). As the altitude increases, *I*_{g} tends to increase in the extratropics (e.g., *I*_{g} > 0.6 at 80 km), while it does not change in the tropics (i.e., for CGWs) between 20 and 80 km (Fig. 9a). At 80 km, the latitude dependence of *I*_{g} is much reduced in the TR experiment.

Significant differences in *I*_{g} are found between the TR and ST experiments (Fig. 9a): *I*_{g} for CGWs is overestimated in the ST experiment at all altitudes by about 0.2 in the convectively active regions (southern Africa, South America, and near-equator regions) and by ~0.1 elsewhere in the tropics, whereas *I*_{g} in the middle to high latitudes, in which BGWs dominate over CGWs, is largely underrepresented in the ST experiment. The results found from Fig. 9a, regarding the difference in *I*_{g} between the two experiments and the vertical change of *I*_{g}, are consistent with the results obtained from the PDFs in the previous section.

Figure 9c presents the Gini index in all the three experiments, calculated as a function of latitude by sampling *F* over all longitudes. Here, the fluxes at all grid points, including those where *F* = 0 during a whole month (gray in Fig. 9a), are used to calculate the index, resulting in very large values of *I*_{g} in the tropics (Fig. 9c). As explained above, these values might be reduced if large-scale convective GWs or GWs from weak convection were taken into account. The Gini index in the STMO experiment is generally very similar to that in the ST experiment in the stratosphere and mesosphere, while some differences are found between the two (by about 0.1) at 80 km in the northern high latitudes. At these latitudes (*z* ≤ 80 km, as can be seen in Fig. 3. Thus, the difference in the GW intermittency should be attributed to the different saturation schemes used in the two experiments, with rather similar mean flows. The investigation of PDFs for *F* in the northern high latitudes (not shown) revealed that large fluxes (

The Gini index in June, along with the 20-km altitude CGW flux, is presented in Fig. 10. In general, the results for the GW intermittency are qualitatively similar to those in December, considering the hemispheric change in season and the difference in regions of active convection. In June, the convectively active regions with *I*_{g} in these regions is close to 0.5 at all altitudes (except near the central United States: ~0.8), which is about 0.2 smaller than that in the ST experiment.

The substantially larger intermittency of extratropical BGWs in the TR experiment, compared to the ST experiment (Figs. 9 and 10), demonstrates an effect of transient dynamics of GWs, given the fact that the three experiments have the same GW sources as inputs of the GWPs (Fig. 1) and, in the stratosphere, the simulated mean flows are also not significantly different in the monthly mean between the experiments (Figs. 3 and 4). This effect may be associated with temporal variations of the background flow which can cause GWs to converge/diverge in the vertical direction and time in the transient model [see Fig. 1c in Muraschko et al. (2015) for an illustration of this process]. In addition, once the GWs have converged or diverged vertically, nondissipative GW forcing occurs, which provides the possibility to generate additional variations of the flow if the forcing is large enough. On the other hand, the vertical increase in the intermittency of extratropical BGWs, shown in all experiments, may also be associated with the filtering effect by vertical variations of the background flow (e.g., Hertzog et al. 2012).

The smaller intermittency of tropical CGWs in the TR experiment, compared to the ST experiment, demonstrates another effect of transient dynamics of GWs, i.e., temporal dispersion of waves, as explained in the previous section. It is notable that the dispersion of tropical CGWs occurs already at very low altitudes (~20 km, Fig. 2), so that differences in the CGW intermittency between the experiments arise even in the lower stratosphere.

## 4. Summary and discussion

The prognostic GW model MS-GWaM has been developed in recent years to represent the propagation of GWs, and their interaction with the resolved flow, in a more realistic manner than in conventional GWPs. In particular, the model simulates the evolution of GWs in phase-space (Muraschko et al. 2015) and describes the direct interaction between propagating GWs and the mean flow, thereby allowing both nondissipative and dissipative GW forcing (Bölöni et al. 2016). In the companion paper (Part I), the implementation of this model into the global model ICON as a transient subgrid-scale parameterization is described, and effects of the transient evolution of GWs and mean flow, which have been ignored in conventional GW parameterizations, on the variability of parameterized GWs are demonstrated. The current paper focuses on the assessment of intermittency of GWs parameterized by MS-GWaM and investigates the effects of transient GW dynamics on intermittency.

To introduce spatiotemporal variability and intermittency in GW sources, convective sources are coupled to MS-GWaM, based on the formulation of SC05, and a flow-independent source is added in the extratropics to take into account GWs from other nonorographic sources. Orographic sources are currently not coupled to MS-GWaM in ICON (the ICON’s operational orographic GWP is used for the simulations), thus orographic GWs are not included in our intermittency analysis.

MS-GWaM simulates the temporal dispersion of convective GWs above the source level. The dispersion appears even in the lower stratosphere (~20 km), which implies a broad spectrum of vertical group velocities for convective GWs. The dispersion results in doubling of the occurrence probability of GWs (thus, lowering of the intermittency) in the tropics throughout the whole stratosphere and mesosphere, compared to that revealed in the control experiment using the steady-state GW parameterization with instantaneous vertical propagation (Table 1). GW intermittency, measured by the Gini index for wave pseudomomentum fluxes, is found to be similar to that from SPB observations in the lower stratosphere. In addition, the intermittency of convective GWs does not change much with altitude in our simulations.

In the extratropics, where nonconvective GWs dominate over convective GWs, the intermittency of GWs is much lower than that in the tropics throughout the stratosphere and much of the mesosphere, although it increases largely with altitude. By comparison with the control experiment, it is found that the transient modeling of GWs using MS-GWaM significantly increases the intermittency of GWs from the uniform source, as also seen in Part I. The result on the intermittency of extratropical GWs (including its vertical variation) might change if GW sources such as orography and jets/fronts, which have their own variability, were explicitly coupled to the parameterizations.

The pronounced impact of the temporal dispersion in lowering the convective GW intermittency suggests a possibility that it could be even lower if horizontal dispersion would also be reproduced via modeling of the three-dimensional GW propagation. Such horizontal dispersion would perhaps tend to reduce the intermittency more at higher altitudes away from the wave sources, while the effect of lateral propagation on the intermittency is still quite unknown. Implementation of the three-dimensional version of MS-GWaM into a global model and investigation of its impacts on the simulated GW intermittency, among many others, remain future work.

The changes in the simulated mean circulation by the transient modeling of GWs are found to be substantial in the mesosphere (changes of up to ~20 m s^{−1} in the zonal mean wind) and small in the stratosphere in the monthly mean field. The impact of transient GW dynamics on the mean flows involving shorter time scales (e.g., stratospheric sudden warming, planetary waves and tides) could be even larger. This merits further investigations.

## 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 (CSC) 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 of DWD and MPI-M. To obtain ICON, please 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 data are available online (https://www.sparc-climate.org/data-centre/data-access/reference-climatology/urap/).

## APPENDIX

### Conversion to Gridcell Mean Flux Using the Areal Fraction of Heating

*N*

_{c}updrafts within a gridcell area Δ

*A*during a period Δ

*t*and using the horizontal and temporal structure of a single heat source

*G*, the vertical maximum of gridcell mean entropy forcing can be expressed as

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^{1}

This modification helps to apply the SC05 formulations to modeled fields in that the ambiguity in determining *T*_{0} from the vertically decreasing temperature profiles is eliminated and that the vertical structures of *ζ*(*z*) better than those of

^{2}

We confirmed that inclusion of intrinsic phase speeds higher than ~60 m s^{−1} in the launch spectra does not change the results of the current analysis because their contribution to the GW momentum flux and drag is small (not shown).