Compensation between Resolved Wave Forcing and Parameterized Orographic Gravity Wave Drag in the Northern Hemisphere Winter Stratosphere Revealed in NCEP CFS Reanalysis Data

Compensation between the resolved wave (RW) forcing and the parameterized orographic gravity wave drag (OGWD) accompanying barotropic/baroclinic (BT/BC) instability in the realistic atmosphere is investigated using Climate Forecast System Reanalysis data in the Northern Hemisphere winter stratosphere. When sufficiently narrow and/or strong negative OGWD drives instability, RWs are generated in situ, providing positive Eliassen–Palm flux divergence that compensates for the parameterized OGWD enhancement; this is consistent with the findings of previous studies based on the idealized general circulation models. However, dependence of the compensation rate on RW forcing differs from the nearly complete compensation in the previous studies, implying that an additional mechanism operates for the compensation: the refractive-index modification by BT/BC instability. The negative meridional gradient of the quasigeostrophic potential vorticity leads to the negative refractive index squared for RWs with phase speeds less than the zonal-mean zonal wind. This prevents RWs from entering the destabilized areas, resulting in the divergence of Eliassen–Palm fluxes that cancels out the parameterized OGWD perturbation. Although both mechanisms act simultaneously, the refractive-index modification plays an important role in the compensation processes in the stratosphere where RWs are dominated by the planetary-scale waves.

Keywords: Gravity waves; Planetary waves; Middle atmosphere; Instability

Planetary waves (PWs) and gravity waves (GWs) are the primary drivers of the middle-atmospheric circulation. While PWs mostly generated by large-scale mountains and land–sea contrasts are sufficiently resolved in general circulation models (GCMs), small-scale GWs forced by small-scale mountains, convection, and jet–front systems cannot be fully resolved in most GCMs and require parameterization ([7]; [13]).

Mutual interactions between the resolved waves (RWs) and the parameterized GWs have been identified. PWs embedded in the mean flow modify the propagation and dissipation of GWs ([23]), while GWs influence PWs by altering the large-scale flow, resulting in PW propagation modification ([22], hereafter [22]) or barotropic/baroclinic (BT/BC) instability, causing the in situ generation of RWs ([4], hereafter [4]; [19]). Zonally asymmetric GW drag (GWD) can directly force PWs as a nonconservative source of the potential vorticity (PV) equation ([14]; [24]).

The interesting phenomenon in their interactions is probably the compensation between RW forcing and the parameterized GWD. It was first reported by [14] with a quasi-linear quasigeostrophic (QG) model: the total momentum forcing of orographic GWs (OGWs) and stationary PWs in the mesosphere remains nearly constant, although the individual forcings are artificially varied. [4] addressed this in the context of the Brewer–Dobson circulation (BDC) by employing an idealized GCM (IGCM). They suggest that compensation is an inevitable response of RWs to stabilize the stratosphere when the stratosphere is driven toward instability by strong and/or narrow OGW drag (OGWD). Such phenomenon is identified even from the GW-resolving GCM ([20]). [22] showed that compensation is robust in a comprehensive climate model. They suggested, however, that change in the refractivity index of PWs induced by the weakened zonal winds above the subtropical jet by OGWD is responsible for the compensation. Such compensation was also identified by [15] and [21], who incorporated extra GWD in the Southern Hemisphere stratosphere to alleviate the cold-pole bias. [5], hereafter [5]) summarize three compensation mechanisms: PV mixing inside the surf zone (a major breaking region for PWs), stability constraints ([4]), and the refractive-index modification ([22]). They showed that while these mechanisms are not mutually exclusive, the prominent mechanism is determined by the location of GWD with respect to the surf zone and interaction time scale.

In view of these compensative interactions, the conventional downward control approach (Haynes et al. 1991) that linearly separates the contributions of the different scale waves in driving the middle-atmospheric circulations may overlook the entire physical processes. It can be reflected in the broad spread between the comprehensive-climate models in the relative contributions of the various wave components to BDC, despite the agreement in the total circulation (Butchart et al. 2011). This raises the importance of identifying the mechanisms responsible for compensation, quantifying the compensation rate, and specifying the compensation properties depending on the mechanisms based on the controlled experiments ([4]; [5]; [22]).

Through these modeling results, one can ask whether such compensation occurs in the real atmosphere. To address this question, we examine the compensative interactions between RWs and the parameterized OGWs, particularly focusing on the BT/BC instability mechanism using National Centers for Environmental Prediction (NCEP) Climate Forecast System Reanalysis data (CFSR; [18]). Data, analysis method, and climatologies of wave forcings and instability are described in section 2. The criteria for identifying compensation, especially from the reanalysis data, are established in section 3. The compensation characteristics are examined through a composite analysis in section 4, and an additional mechanism that has not been elucidated in previous studies is proposed. Section 5 discusses some aspects that require further studies on the compensation and section 6 provides a summary and conclusions.

Six-hourly CFSR data with a horizontal resolution of 1° × 1° (longitude × latitude) and 17 vertical levels from 1000 to 1 hPa covering 32 years (1979–2010) are employed. We use wind, temperature, geopotential height, and the parameterized OGWD ([12]). Analyses focus on the Northern Hemisphere (NH) winter months [December–February (DJF)], the period of maximal interaction between PWs and GWs.

Fluxes and forcings of RWs are examined by Eliassen–Palm flux (EP flux) and its divergence (EPFD), respectively ([1]):

(1) $\mathbf{F}=(F^{\varphi },F^z)={\rho }_{0}a\cos \varphi \{-\overline{u^{\prime }{\upsilon }^{\prime }}+{\overline{u}}_z\frac{\overline{{\upsilon }^{\prime }{\theta }^{\prime }}}{{\overline{\theta }}_z},\hspace{0.17em}\left[f-\frac{1}{a\cos \varphi }{(\overline{u}\cos \varphi )}_{\varphi }\right]\frac{\overline{{\upsilon }^{\prime }{\theta }^{\prime }}}{{\overline{\theta }}_z}-\overline{u^{\prime }{w}^{\prime }}\},$

(2) $\mathbf{\nabla }\cdot \mathbf{F}=\frac{1}{a\cos \varphi }\frac{\partial }{\partial \varphi }(F^{\varphi }\cos \varphi )+\frac{\partial F^z}{\partial z},$

where Fϕ and Fz are the meridional and vertical components of EP flux F, respectively; (1/ρ0a cosϕ)∇ ⋅ F is the EPFD; ϕ and z are the latitude and log-pressure height, respectively; a is the mean radius of Earth; f is the Coriolis acceleration; ρ0 is the reference density; u, υ, and w are the zonal, meridional, and vertical wind components, respectively; and θ is the potential temperature. The overbar and prime denote the zonal mean and the departure from the zonal mean, respectively. All results in this study are based on the daily averages.

Figures 1a and 1b describe the NH winter climatologies of the zonal-mean zonal component of OGWD and EP fluxes overlaid on EPFD, respectively. Above major mountainous areas (30°–50°N), OGWs predominantly deposit negative forcings at two locations: just above the subtropical jet (30°–40°N and 40–100 hPa) and 40°–50°N in the upper stratosphere (∼1 hPa). In the former region, weak horizontal winds act as a critical level for OGWs, leading to substantially localized OGWD, whereas the latter region is associated with OGW's saturation. Despite the coarse vertical resolution of CFSR (17 levels), EP fluxes and EPFD show sufficient similarity to those calculated by the model-level (40 levels) MERRA-2 reanalysis data (not shown). PWs originate primarily from the extratropical troposphere and exert negative EPFD dominantly in the extratropical upper troposphere and mid-to-high-latitude stratosphere with amplitudes larger than OGWD. Figures 1c and 1d show the daily standard deviation (STD) of OGWD and EPFD, respectively. Interestingly, OGWD and EPFD have STD greater than their respective mean values, indicating highly intermittent properties of each wave forcing. OGWD has the largest variabilities in the maximum-OGWD region, with a STD (∼2.2 m s−1 day−1) twice its maximum value. Regarding that narrow and/or strong GWD is more likely to induce instability ([4]), such intermittent and strong OGWD bursts are expected to cause instability. Consistent with OGWD, EPFD exhibits considerable daily variations, but its maximum STD (∼20 m s−1 day−1) appears at ∼70°N.

Graph: Fig. 1. Latitude–height cross sections of the (a) zonal-mean orographic gravity wave drag (OGWD) and (b) Eliassen–Palm flux (EP flux; green arrow) overlaid on their divergence (EPFD) averaged over 32 winters in the Northern Hemisphere [from December to February (DJF)] and the standard deviations of (c) OGWD and (d) EPFD. (e) The 32-year-averaged meridional gradient of the quasigeostrophic potential vorticity (q¯y) and (f) occurrence rate of the barotropic/baroclinic (BT/BC) instability. The black contours overlaid on (a)–(f) are the 32-yr-averaged zonal-mean zonal winds. The solid (dashed) lines denote westerlies (easterlies).

To examine BT/BC instability, the meridional gradient of QG PV ( ${\overline{q}}_y$ ) is considered ([1]):

(3) ${\overline{q}}_y=\beta -{\overline{u}}_{yy}-\frac{1}{{\rho }_o}{\left({\rho }_o\frac{f^2}{N^2}{\overline{u}}_z\right)}_z,$

where β denotes the meridional derivative of f and N is the buoyancy frequency. A necessary condition for BT/BC instability is that ${\overline{q}}_y$ becomes negative ([9]). Figures 1e and 1f present the ${\overline{q}}_y$ climatology and the instability occurrence rate. The highest occurrence of instability appears within the area of remarkably small ${\overline{q}}_y$ equatorward of the polar night jet above the subtropical jet (Figs. 1e,f).

To demonstrate the compensative interactions between RWs and the parameterized OGWs, [4] conducted idealized simulations in which the parameterized OGWD was arbitrarily perturbed in the stratosphere. They found a remarkable compensation of the negatively perturbed OGWD by a positive EPFD response accompanying the in situ PW generation. The authors suggested that compensation is achieved as PWs adjust the destabilized flow initiated by the perturbed OGWD to a stable state. They also found that this instability is more likely when the GWD perturbations have large amplitudes (A), meridional scales (L) smaller than the Rossby radius of deformation (Ld), and vertical scales (H) larger than the density scale height (Hρ).

In the present study, findings from previous simulations are examined under the realistic atmosphere using the reanalysis data. This is accomplished by building our own criteria to identify the compensative interactions between OGWD and EPFD via instability regarding (i) the perturbation of the parameterized OGWD (ΔOGWD) and the response of EPFD (ΔEPFD), (ii) the analysis area, and (iii) the ΔOGWD threshold value required to cause instability. Unlike in controlled IGCM experiments, it is not possible to systematically perturb either the parameterized GWs or RWs in this study. Therefore, ΔOGWD and the corresponding ΔEPFD are simply defined as diurnal variations (i.e., the difference from one day to the next day), considering the highly transient nature of wave forcings. Accordingly, instability is examined based on the daily ${\overline{q}}_y$ .

The sign and amplitude of the perturbations of wave forcings are evaluated based on area-averaged values. Analysis is conducted in the area of 30°–60°N latitudes and 20–1 hPa altitudes, where OGWD exhibits significant temporal variations (Fig. 1c). The entire analysis area is divided into five subregions with 10° meridional widths, allowing 5° overlaps. The 10° meridional width is slightly shorter than Ld (15°–20° in 30°–45°N and 10°–15° in 45°–60°N), as the meridional extent of the parameterized OGWD, determined from the spatial extent of the subgrid-scale mountain height and surface winds, has a shorter scale than Ld ([4]). Note that the surf zone (30°–50°N) where PV mixing is more favorable for compensation than instability ([5]) is included in the analysis area.

The critical amplitude Ac for ΔOGWD to induce instability is investigated using Eq. (13) of [4]:

(4) $A_c>\frac{Q_y{L}^4{H}_{\rho }}{{\tau }_r{L}_d^2{H}^2},$

where Hρ = 7 km and Qy is the scale of the background ${\overline{q}}_y$ in the stable stratosphere. Considering that the damping time in the extratropical stratosphere is much faster than 40 days used in [4] ([10]), the relaxation time τr is determined as 10 days in this study, resulting in Ac 4 times greater than that in [4]. If we suppose Qy scales as β at 45°N, then we get the critical amplitude Ac ≈ 0.1 m s−1 day−1 with the meridional (L = 10° latitude ≈ 1112 km) and vertical (H ≈ 21 km) extents of the subdivided analysis layer and the Newtonian relaxation time τr = 10 days. Therefore, in the present study, Ac is assumed to be 0.1 m s−1 day−1.

Although we adopt Ac derived based on the steady-state QG equations with simple Newtonian relaxation ([4]), the threshold value is thought to be available even in a time-dependent state considered in this study according to the following QG equation ([1]):

(5) $\frac{\partial }{\partial t}({\overline{q}}_y)=-{({\rho }_0{}^{-1}\mathbf{\nabla }\cdot \mathbf{F})}_{yy}-{(\overline{{\text{GWD}}_x})}_{yy}=-{(\overline{G})}_{yy},$

where $\overline{{\text{GWD}}_x}$ is the zonal component of GW forcing and $\overline{G}$ is the total wave forcing. Let us consider the zonal-mean QG PV of the stable initial state ${\overline{q}}_0$ such that ${\overline{q}}_{0y}\ge 0$ and that of the destabilized state ${\overline{q}}_1$ such that ${\overline{q}}_{1y}<0$ . Then the above Eq. (5) becomes

(6) $\frac{{\overline{q}}_{1y}-{\overline{q}}_{0y}}{\mathrm{\Delta }T}=-{(\overline{G})}_{yy}\hspace{0.17em}\Rightarrow \hspace{0.17em}{\overline{q}}_{1y}={\overline{q}}_{0y}-\mathrm{\Delta }T{(\overline{G})}_{yy}<0.$

The necessary condition for instability is achieved when the meridional curvature of the wave forcing $\overline{G}$ multiplied by time difference ΔT overwhelms the existing ${\overline{q}}_{0y}$ . Given that ΔT is positive, the meridional curvature should have the positive sign. Using standard analysis technics (e.g., [2]), let the scale of background ${\overline{q}}_{0y}$ as Qy, T be the time scale, A and L be the amplitude of wave forcing and the meridional scales on which the wave forcing varies, respectively. Then, each term in (6) is scaled as follows:

(7) $\frac{Q_y}{T}<\frac{A}{L^2}\hspace{0.17em}\Rightarrow A>\frac{Q_y{L}^2}{T}.$

With Qy scales as β at 45°N, T = 1 day, and L ≈ 1112 km, we can obtain a rough estimate of the wave forcing A ≈ 10 m s−1 day−1. Considering that EPFD almost accounts for the required wave forcing in the analysis layer (Figs. 1b,d), the enhanced OGWD could be responsible for this instability despite their relatively small magnitude (0.1 m s−1 day−1).

In summary, within a subregion of 10° meridional width and vertical range of 20–1 hPa, if

• the area-averaged ΔOGWD is negative with an amplitude greater than 0.1 m s−1 day−1,
• there exists any location that is originally stable ( ${\overline{q}}_y>0$ ) but becomes unstable ( ${\overline{q}}_y<0$ ) within one day accompanying the in situ generation of RWs, and
• the averaged ΔEPFD has the opposite sign (i.e., positive) to the averaged ΔOGWD,

then the day is selected as a compensation case through instability. Here, the local generation of RWs is evaluated by the positive EPFD of the transient component of RWs, regarding the prevalent transient nature of RWs induced by the BT/BC instability ([20]). The transient EPFD is calculated as follows: RWs defined as the zonally varying anomalies from the zonal mean of the dependent field variables (wind, temperature, etc.) are separated into stationary and transient components based on the time series analysis. Over the 31-day windows moving at a 1-day interval, stationary waves are defined as the 31-day-averaged anomalies assigned to a center of the 31-day window, while transient waves are defined as the departures from the 31-day moving averages in each time window. Hereafter, the EPFD by the transient waves is denoted as TEPD. Note that the response of TEPD (ΔTEPD) almost accounts for that of EPFD (ΔEPFD) (not shown).

The OGWD enhancement [criterion (i)] is met ∼40% of the entire days, and more than 80% of these cases satisfy the instability condition [criterion (ii)]. However, only ∼30% of these cases meet the EPFD condition [criterion (iii)], resulting in the low occurrence rate (6%–10%): out of 2798 days, 172, 243, 263, 230, and 195 cases are selected in the 30°–40°N, 35°–45°N, 40°–50°N, 45°–55°N, and 50°–60°N latitudinal bins, respectively. There are two possible reasons for the low probability: (i) natural variability in EPFD involved in the reanalysis data can overwhelm the instability-induced changes in EPFD, and (ii) area averages applied to wave forcing can reduce the opportunity for detecting compensations occurring on a smaller meridional and vertical scales. This implies that more frequent compensating interaction may occur in realistic atmosphere than those identified in this study.

Figure 2 illustrates one representative case in January 1984. From 29 to 30 January, the parameterized OGWD is considerably enhanced (Figs. 2a,b), resulting in a negative ΔOGWD with the maximum of ∼7 m s−1 day−1 (Fig. 2g). This ΔOGWD causes a positive meridional curvature of the zonal-mean zonal winds through decelerating the westerlies, yielding barotropic instability (Fig. 2d). In the instability region, a positive EPFD indicating the local generation of RWs appears (Fig. 2f), providing the positive ΔEPFD (Fig. 2h) that nearly cancels out the negative ΔOGWD.

Graph: Fig. 2. Latitude–height cross sections of (a),(b) OGWD, (c),(d) q¯y , and (e),(f) EP flux (vectors) overlaid on EPFD on (a),(c),(e) 29 and (b),(d),(f) 30 Jan 1984, respectively, and changes in (g) OGWD (ΔOGWD) and (h) EP flux (ΔEP flux; vectors) and EPFD (ΔEPFD) of RWs; (i) −2 × ΔOGWD × ΔEPFD used for the calculation of degree of compensation (DOC) between ΔOGWD and ΔEPFD. The black contours overlaid on (a)–(f) are the zonal-mean zonal winds. To emphasize the negative q¯y , all positive values are shaded as white in (c) and (d).

To quantify the compensation rate, we calculate the degree of compensation (DOC) defined in [4]:

(8) $C=1-\frac{{\displaystyle \sum _i{[P(x_i)+R(x_i)]}^2}}{{\displaystyle \sum _i{P}^2(x_i)}+{\displaystyle \sum _i{R}^2(x_i)}}=-\frac{2{\displaystyle \sum _i[P(x_i)R(x_i)]}}{{\displaystyle \sum _i{P}^2(x_i)}+{\displaystyle \sum _i{R}^2(x_i)}}.$

Here, C is DOC, P and R denote the OGWD perturbation and EPFD response, respectively, and xi is a generic spatial coordinate. Because DOC measures the spatial covariance between perturbation and response, it is the best metric for a time-independent compensation. Nevertheless, we decide to adopt this parameter in the present study considering simultaneous daily variations in each wave forcings. Perfect compensation (C = 1) is achieved if the response is equal and opposite to the perturbation (i.e., R = −P), while no compensation occurs if there is no response (i.e., R = 0) or if R is not correlated with P (C = 0). When R = P, the system amplifies the perturbation. To avoid ambiguous noise in the response in regions where the perturbation is weak ([4]), the summation is constraint to the area in which |P(xi)| > 0.1 max |P(xi)| inside the analysis layer (35°–45°N). Figure 2i describes the spatial structure of product of ΔOGWD and ΔEPFD multiplied by −2, which is [−2P(xi)R(xi) in the numerator in Eq. (8)]. It shows that not all ΔOGWD is compensated inside the analysis layer, and this is because the ΔEP fluxes emerging from the instability converge in 40°–45°N (Fig. 2h), reducing the anticorrelation between the two wave forcings and even amplifying ΔOGWD in 40°–45°N latitudes. The DOC obtained from Eq. (8) averaged over the whole analysis domain (35°–45°N) is 0.26, while that over 35°–40°N is 0.6, as expected from Fig. 2i.

To examine the time scale of OGWD enhancement and EPFD response through instability, we show in Fig. 3 the evolution of ${\overline{q}}_y$ , OGWD, EPFD, and TEPD averaged inside the analysis layer over 35°–45°N and 20–1 hPa from 29 to 31 January 1984 using 6-hourly CFSR data. The negative OGWD strengthens rapidly from 0000 UTC 29 to 0000 UTC 30 January, and then its magnitude is slightly decreasing after that. Corresponding to the enhancement of OGWD, instability develops with the considerable reduction in ${\overline{q}}_y$ until 0000 UTC 30 January when OGWD stops increasing. This supports that the strengthened OGWD is responsible for the emergence and development of ${\overline{q}}_y$ . From 1200 UTC 29 January, slightly increased negative EPFD (red solid) starts to decrease significantly until 0600 UTC 30 January, and a gentler decrease follows until it changes the sign to positive at 0600 UTC 31 January. Such changes in EPFD are almost attributed to those in TEPD (red dashed). In particular, TEPD is predominantly positive from 1800 UTC 30 January, implying in situ generation of the transient RWs by instability. The apparent anticorrelation between OGWD and TEPD implies that the compensation processes can occur instantaneously within a day. Such a rapid cancelation is related to the stable nature of the stratosphere, supporting the validity of the analysis method used in the present study that deals with the difference in daily wave forcings. From 30 January, ${\overline{q}}_y$ gradually increases and almost returns to the stable condition at 1800 UTC 31 January accompanying the continuous in situ generation of the transient RWs. This is consistent with the hypothesis of [4] that the resolved waves respond to stabilize the background atmosphere driven toward instability by the perturbed OGWD.

Graph: Fig. 3. The time evolution of q¯y (black), OGWD (blue), EPFD (red), and EPFD by the transient component of RWs (TEPD; red dashed) averaged inside the analysis layer (35°–45°N and 20–1 hPa) from 29 to 31 Jan 1984. Gray horizontal line denotes the zero drag.

To examine the characteristics of the compensating interactions, a composite analysis is carried out. Figure 4 shows the composited ΔOGWD, increase rate of instability, ΔEPFD, and the normalized product of the two wave forcings multiplied by −2, which is $-2[P(x_i)R(x_i)]/[{\displaystyle {\sum }_i{P}^2(x_i)}+{\displaystyle {\sum }_i{R}^2(x_i)}]$ in Eq. (8), hereafter called the normalized product. The normalized product is devised to complement DOC by visualizing the spatial correlation between ΔOGWD and ΔEPFD. The value written in the lower-left corner of each panel denotes DOC, the summation of the normalized product inside the analysis layer.

Graph: Fig. 4. Latitude–height cross sections of the composited (from left to right) ΔOGWD, increase rate of instability (ΔInstability), ΔEPFD, and normalized product (refer to the detail explanation in the manuscript) in the (a) 30°–40°N, (b) 35°–45°N, (c) 40°–50°N, (d) 45°–55°N, and (e) 50°–60°N latitudinal bin. The value written in the lower-left corner of each panel in the rightmost column denotes DOC, the summation of the normalized product inside the analysis layer.

The composited ΔOGWD reveals a parabolic shape. The maximum value at the center of parabolic ΔOGWD is the largest in the 40°–50°N latitude bin (greater than 1 m s−1 day−1). Despite the wider meridional scales (15°–20°) than the assumed width, all the composited ΔOGWD exhibit evident positive curvature with a sufficient magnitude required for causing instability as expected from Eq. (5). The instability occurrence associated with these ΔOGWD is examined by assessing the increase rate in the number of days with negative ${\overline{q}}_y$ . This prevents the problem associated with overlapping positive and negative ${\overline{q}}_y$ values, which makes ${\overline{q}}_y$ composites hard to interpret. After OGWD is enhanced, the incidence of instability increases predominantly by more than 10%–50%, supporting the occurrence of instability driven by OGWD. The composited ΔEPFD exhibits a broader meridional structure and greater amplitude than those of ΔOGWD, with an increasing trend with latitude: the maximum is ∼2 (7) m s−1 day−1 in 30°–40°N (50°–60°N). Given that we already select ΔOGWD that satisfies the instability condition, ΔEPFD magnitude is expected to be nearly equal to that of ΔOGWD if almost perfect compensation occurs through the in situ PW generation, according to [4]. However, ΔEPFD reveals latitudinal variations resembling those of the EPFD STDs (Fig. 1d), without evident correlation with ΔOGWD. Accordingly, DOC is the maximum at 40°–50°N and decreases away from this latitude.

This is more clearly identified from Fig. 5, which presents bar charts of the compensation occurrence frequency, the composite of ΔOGWD and ΔEPFD averaged in the 10° latitudinal bins, and DOC with respect to different ΔOGWD magnitude ranges. Regarding the different case numbers (Fig. 5a), a bootstrap analysis is conducted to estimate composite values ([6]). We generate 10 000 bootstrap samples containing the same numbers of members in each of the 20 cases. The STDs of the average bootstrapped values denoted by error bars are insignificant, guaranteeing the bootstrap estimation accuracy.

Graph: Fig. 5. (a) Number of compensation cases and composited (b) ΔOGWD magnitude, (c) ΔEPFD, and (d) DOC within 10° latitudinal bins in 30°–60°N with respect to different ΔOGWD magnitude ranges. The error bars correspond to one standard deviation in each variable.

While ΔOGWD is less than 1 m s−1 day−1 (Fig. 5b) in most cases, ΔEPFD ranges from 1 to 10 m s−1 day−1 (Fig. 5c). Accordingly, DOC exhibits an increasing trend as ΔOGWD magnitude increases: for example, in 40°–50°N, DOC is greater (less) than 0.8 (0.3) for ΔOGWD magnitude greater (less) than 1 (0.25) m s−1 day−1. Furthermore, ΔEPFD increases with latitude, while only ΔOGWD greater than 1 m s−1 day−1 is maximized in 40°–50°N. Therefore, DOC for ΔOGWD less than 1 m s−1 day−1 decreases with latitude except for 0.1–0.25 m s−1 day−1 range, whereas that for ΔOGWD greater than 1 m s−1 day−1 is maximized in 40°–50°N (Fig. 5d). Overall features differ from the theoretical inferences in [4], in which compensation is nearly complete. We cannot entirely rule out the possibility that the natural variabilities are included in ΔEPFD, likely resulting in the weak compensation rate.

To understand the differences between our results and those of [4], we conduct composite analyses of EP fluxes and EPFD of the transient RWs before and after F6 the OGWD enhancement in Fig. 6. According to EP flux budget analysis, [5] identified a clear sign for the local generation of RWs in the compensation through instability: EPFD changes sign from negative to positive, leading to the poleward (equatorward)-directed momentum flux at the pole (equatorial) side and less heat flux from below. However, the composited EP fluxes in this study tell a different story: After OGWD enhancement, (i) the negative TEPD changes the sign to positive, resulting in equatorward fluxes that develop along their propagation inside the analysis layer. The resultant pole-side fluxes are directed equatorward rather than poleward. (ii) The positive ΔTEPD is mostly induced by ΔEP fluxes that occur along the incident PW's propagation path, which is primarily vertical in the high latitudes (50°–60°N) and becomes equatorial in the lower latitudes (30°–40°N). Accordingly, (iii) the changes in the vertical component of EP fluxes (ΔEPFz) contribute more to ΔTEPD at higher latitudes. The features described in (i) are generally consistent with [4], while those in (ii) and (iii) are more similar to the concept of compensation through PV mixing in the surf zone proposed by [5]. However, as all the selected cases in this study accompany instability without mixing, an additional mechanism must be operated for the present compensation processes.

Graph: Fig. 6. Latitude–height cross sections of the transient component of EP fluxes (vectors) and EPFD (TEPD, shading) (left) before and (center) after OGWD enhancement and (right) the differences of EP fluxes (ΔEP fluxes) and TEPD (ΔTEPD) between the two in the (a) 30°–40°N, (b) 35°–45°N, (c) 40°–50°N, (d) 45°–55°N, and (e) 50°–60°N latitudinal bin. ΔEP fluxes are multiplied by 5 for better representation.

Previous studies have surmised instability as an in situ source for PWs. However, BT/BC instability indicated by negative ${\overline{q}}_{\varphi }$ can also inhibit PW propagation. This can be easily demonstrated by the refractive index squared, indicating the affinity for PW propagation ([1]):

(9) $n^2=\left[\frac{{\overline{q}}_{\varphi }}{a(\overline{u}-C)}-{\left(\frac{k}{a\text{cos}\varphi }\right)}^2-{\left(\frac{f}{2NH}\right)}^2\right]a^2.$

PWs can propagate in regions where the refractive index squared is positive ([11]). However, as inferred from Eq. (9), negative ${\overline{q}}_{\varphi }$ makes the refractive index squared negative for PWs having the Doppler shifted phase speed C less than $\overline{u}$ , thereby preventing those waves from propagating toward the instability region. Westward propagation with respect to the mean flow is a typical characteristic of Rossby waves. Moreover, PWs with C greater than the prevalent westerlies (greater than ∼10 m s−1) inside the analysis layer during the NH wintertime account only for 10%–15% (not shown), implying that the majority of PWs are prevented from entering the unstable area. From this point of view, increase in the occurrence of instability can be reinterpreted as a decrease in the incident PWs toward the destabilized analysis layer. This perspective has not been mentioned in the previous IGCM studies ([4]; [5]).

In light of the refractive-index modification associated with instability, the positive ΔEPFD is attributable to the decrease in the negative EPFD, as incident RWs avoid entering the destabilized area. As shown in Fig. 1b, EP fluxes reveal dominant meridional propagations in the low latitudes (30°–40°N), while vertical propagation becomes more significant in the high latitudes (50°–60°N). The intensity of EP fluxes and corresponding EPFD also increase with latitude. Such latitudinal dependence of RW's behavior suggests that the meridional location of ΔOGWD (i.e., the region where OGWD-induced instability prevents the incident PWs) is an important factor in determining the response of PWs. In lower (higher) latitude, the reduction of fluxes induced by PWs avoiding the destabilized area is more attributed to the decrease in the meridional (vertical) component of EP fluxes. The amounts of ΔEP fluxes and ΔEPFD increase with latitudes, proportional to those of incident PWs. Such EP-flux changes and the equatorward-directed pole-side fluxes have also been identified in the compensation through the refractive-index modification in [22].

Then why does the refractive-index modification dominate in the compensation process via instability in the present study? Based on the instability hypothesis, instability associated with the meridionally narrow OGWD is expected to be small, generating small-scale waves. [5] also identified the abundance of RWs having zonal wavenumber (ZWN) 7–9 excited by the OGWD-induced instability. On the other hand, the refractive-index modification is associated with the rearrangement (propagation and breaking) of the preexisting PWs. This suggests that the horizontal scales of RWs related to the two different mechanisms could be different. It is explored in Fig. 7, which shows the spectral analysis of ΔEP fluxes and ΔEPFD. For simplicity, RWs are decomposed as PWs of ZWN 1–3 and relatively smaller-scale waves (SWs) with ZWN greater than 3. The composited ΔEPFD by PWs (ΔEPFDPW) is similar to that by RWs and the aforementioned RWs' behaviors are also mainly attributed to PWs (not shown). This implies that PWs prevail the response of RWs to the OGWD-induced instability through the refractive-index modification. There exists in situ PW generation unexpectedly based on the instability hypothesis, although ΔEPFD by this local PW excitation would have the magnitude equivalent to ΔOGWD ([4]) that can be overwhelmed by ΔEPFD attributed to the refractive-index modification. The magnitude of ΔEPFD by SWs (ΔEPFDSW) is almost negligible except for a 30°–40°N latitude, making it difficult to evaluate relative importance of the two mechanisms.

Graph: Fig. 7. Latitude–height cross sections of ΔEP fluxes (vectors) and ΔEPFD (shading) of the (left) resolved waves, (center) planetary-scale waves (PWs; ZWN 1–3), and (right) smaller-scale waves (SWs; ZWN > 3) between the before and after OGWD enhancement in the (a) 30°–40°N, (b) 35°–45°N, (c) 40°–50°N, (d) 45°–55°N, and (e) 50°–60°N latitudinal bin.

Meanwhile, the barrier to equatorward EP fluxes arising from the OGWD-induced instability leads to the apparent negative ΔEPFD in adjacent (north as well as south) regions. Such phenomenon was also identified in [22], although the refractive-index modification process inducing compensation in their study is caused by the weakened zonal-mean zonal winds in the upper flank of the subtropical jets due to increased OGWD, rather than to negative ${\overline{q}}_{\varphi }$ . They argued that the increased EPFD in the high latitude coinciding with the reduced EPFD in the midlatitude accounted for the lack of high-latitude compensation as the increase in OGWD was weak at high latitudes. [15] also found a similar behavior of PWs: while arbitrarily introduced OGW sources around 60°S resulted in the OGWD strengthening accompanied by the reduction in EPFD at 60°S, EPFD in the north and south of 60°S increased, suggesting the latitudinal spreading of the EPFD instead of compensation. This implies that the compensation accompanying the alternation of PW propagation can yield the nonlocal effects in other latitudinal regions.

In the present study, the compensation between the parameterized OGWD and RWs accompanying instability revealed in the idealized simulations ([4]; [5]) are extended to more realistic atmosphere with the reanalysis data. This is achieved by constructing the criteria for identifying the compensative interactions via instability based on the findings from the previous IGCM studies and theoretical inferences, considering inherent properties of each wave forgings and instability (Fig. 1). Nevertheless, there are a number of things that require further studies in the context of compensation.

First, this study examined the compensation processes in which the reinforced OGWD causes instability, leading to a local generation of RWs simultaneously within a day, regarding the strongly transient nature of each wave forcings. Instantaneous compensation in a day identified in Fig. 3 supports that this approach can capture the compensation processes in the real atmosphere satisfactorily. However, as shown in [20], the enhancement in GWD, subsequent destabilization, and the consequent excitation of PWs appear sequentially with a time lag. In their study, the time lag was about 3 days (Fig. 10 in [20]). [17] also identified the in situ generation of PWs signals in the next 4-day averages after instability emerged. In addition, if instability persists for several days, the response of RWs can also last for more than a day. The suppression of PWs entering the destabilized area as well as the local generation of RWs are expected to continue until the instability is alleviated. This implies that the response of RWs associated with both compensation mechanisms could have a time scale equivalent to the duration of instability. In this regard, we can further speculate that the amount of EPFD response could be related to the strength of OGWD perturbation even in the compensation through the refractive-index modification.

While the parameterized OGWD is intensively examined in association with the compensation via instability, the parameterized nonorographic GWD (NOGWD) has not been discussed in the present study. The major finding of [4] is that strong and/or meridionally narrow GWD is more likely to drive instability. In general, the parameterized NOGWD is meridionally broadened by NOGWs forced by both local sources such as convection and nonlocalized sources such as jets/fronts ([16]). Furthermore, in the NH, NOGWD is dominated by OGWD inside the analysis layer above the major mountain areas (Fig. S1 and section S1 in the online supplementary material). Such a meridionally gentle and relatively weak magnitude of NOGWD, at least in the stratosphere, is not suitable for causing instability. [4] also suggested the wide meridional structure of NOGWD as a reason for the low compensation rate in the interaction between NOGWs and resolved waves. However, in the upper stratosphere/mesosphere in which the magnitude of NOGWD might be significant, the compensation associated with NOGWD via instability may possible, despite their wide meridional extent. This remains for future research topic.

This study first demonstrates that the compensating interactions between RWs and parameterized OGW forcings via BT/BC instability identified in previous IGCM studies also appear under very realistic atmosphere. Although the number of cases inevitably depends on the criteria for identifying compensative interactions, the key processes in the interaction in [4] also appear in this study: OGWs with narrow meridional widths and/or strong drag sufficiently destabilize the stratosphere, leading to the in situ generation of PWs that eventually compensates for the parameterized OGWD.

However, this study also found that such interactions occur even within one day as well as inside the surf zone as the highly transient and strong OGWD generates PV reversals, thereby driving BT/BC instability. In addition, the compensation rate shows a remarkable dependency on the response of EPFD rather than on the OGWD perturbation that has not been recognized in any previous IGCM studies. As a plausible mechanism responsible for this phenomenon, we suggest the refractive-index modification by instability that reduces PW propagation toward the unstable region. In this mechanism, the key factor determining the compensation rate is found to be the meridional location of OGWD-induced instability. In situ generation and the refractive-index modification occur simultaneously in the compensation processes via instability. In the NH winter stratosphere where PWs are prevalent, the refractive-index modification of the preexisting PWs plays an important role in determining the compensation processes.

Most previous studies have discussed the refractive-index modulations by the background wind and stability changes separately. However, consideration of the refractive index in relation to the necessary condition for BT/BC instability provides a more comprehensive and direct interpretation of PW propagation. This enables us to identify the alternation of PW's propagation condition that occurs in the compensating interactions through BT/BC instability.

This work was supported by a National Research Foundation of Korea (NRF) grant funded by the South Korea government (MSIT) (2021R1A2C100710212). The first author is supported by the Global Ph.D. Fellowship Program (2019H1A2A1077307). We would also like to thank Min-Jee Kang and In-Sun Song for helpful discussions.