### 3.1 Synoptic evolution

This subsection presents an overview of the cyclogenesis during the period of the experiment. Figure 1a,b,c,d,e shows the horizontal distributions of temperature (color) and geopotential height (contour) at 950 hPa for the entire computational domain at 12-h intervals. The explosive cyclogenesis is captured by this simulation as follows. At *t*=12 h, a cyclone starts to develop over Japan. The horizontal gradient of temperature becomes steeper, leading to frontogenesis. At *t*=36 h, the cyclone moves to the east of Hokkaido Island, at which time the central pressure drops to the lowest in the cyclone’s life. At *t*=48 h, the cyclone moves to the east of the Kamchatka Peninsula, and the sharp core of the cyclone starts to decay. The model domain covers the cyclone throughout the period of the experiment. The domain also covers an anticyclone to the east of the cyclone until *t*=24 h and another anticyclone to the west from *t*=24 h. Asymmetry between the cyclone and the anticyclones is large; the cyclone is deeper and compact, and the anticyclones are wider. Although this asymmetry may create some bias in domain-averaged statistics on the TIL, the main purpose of the paper is to describe the time evolution of the TIL with fine-scale structures, and global or zonal mean statistics are beyond the scope of this study.

Figure 1f shows a time series of minimum sea level pressure within the computational domain. As the cyclone develops, the pressure starts to decrease from *t*=10 h. The pressure becomes the lowest, 965 hPa, at around *t*=36 h and then starts to increase. The pressure decreases nearly 35 hPa in 24 h between 35 and 45°N, which satisfies the criterion for a ‘bomb’ cyclone by put forward by Sanders and Gyakum ([1980]). As the position and the central pressure of the cyclone at *t*=36 h are almost the same as those in the JMA analysis ([Japan Meteorological Agency 2009]), we consider that the model simulates the cyclone realistically. In this paper, we define the development stage of the cyclone as the period from 10 to 36 h; its typical snapshot is at *t*=24 h, and the mature stage is at *t*=36 h.

Figure 2 shows vertical cross sections along *x*=131 in the model (almost meridional sections crossing the center of the surface cyclone) of the zonal wind (*u*, color) and potential temperature (*θ*, contour) at *t*=36 h. The tropopause detected according to the definition by the World Meteorological Organization (WMO) is shown by the dots. The axis of the subtropical jet is located around 1,600 km from the southern boundary (145°E, 35°N), 300 hPa. The center of the surface cyclone is located at 2,700 km (147°E, 44°N), where the zonal wind reverses. The tropopause is located above 100 hPa at equatorward latitudes of about 45°N, and around 300 to 500 hPa at latitudes poleward of the jet. Near the jet, a multiple tropopause appears, exhibiting tropopause folding.

Figure 3a shows the horizontal distributions of the tropopause height according to the WMO definition (*Z*
_{TP}, color) and the wind speed at 300 hPa (contour) at *t*=36 h. When multiple tropopauses are detected at the same horizontal grid point, the lowest level is used in this plot. The contour line for 80 m s ^{−1}, which shows the jet axis, lies from 132°E, 32°N to 154°E, 38°N. The gap of *Z*
_{TP} is located around the jet axis, *Z*
_{TP}>14 km to the south and *Z*
_{TP}<14 km to the north. The lowest *Z*
_{TP} appears around 140 to 150°E, 40 to 45°N, where the center of the surface cyclone is located. In the following analyses, only regions where *Z*
_{TP}<14 km are used.

Figure 3b shows the vertical component of relative vorticity at *Z*
_{TP} (hereafter *ζ*
_{TP}) at *t*=36 h. Horizontal winds at *Z*
_{TP} are also shown by arrows. A spiral pattern of positive and negative vorticities associated with the cyclone is discernible. The highest *ζ*
_{TP} is obtained around 145 to 150°E, 40 to 45°N, which is located to the north of the jet exit region and to the east of the surface cyclone. The lowest *ζ*
_{TP} is obtained around 160 to 165°E, 45°N. The region of *ζ*
_{TP}>0 along the northern edge of the jet streak (southern edge of the analysis region) is caused by the horizontal wind shear of the jet streak. A spiral band of *ζ*
_{TP}>0 extending to the northwestern corner of the domain coincides with the northeastern edge of strong westerlies at the tropopause level. Within the analysis region, the area of positive *ζ*
_{TP} is greater than that of negative *ζ*
_{TP}. Thus, data are binned by *ζ*
_{TP} and normalized by the sample size to ensure that the conclusions are not biased by the difference of positive and negative *ζ*
_{TP} areas.

### 3.2 Relationship between TIL and relative vorticity at tropopause

As previous studies reported the existence of a negative correlation between *ζ*
_{TP} and the strength of the TIL, we will examine the relationship between these two quantities. In order to examine the dependence on *ζ*
_{TP}, 22,631 temperature profiles within regions with *Z*
_{TP}<14 km at *t*=36 h are binned by *ζ*
_{TP} and averaged in each bin (Figure 4a). The bin width is 5×10^{−5} s ^{−1}. The vertical coordinate is the height above the tropopause. The mean profiles for negative *ζ*
_{TP} (dark blue lines) show clear temperature inversions with amplitudes of 4 to 5 K and thicknesses of 1.5 to 2 km just above the tropopause, whereas the profiles for positive *ζ*
_{TP} (red lines) do not show clear inversions. This result is consistent with previous observational and numerical studies (e.g., Randel et al. [2007]; Son and Polvani [2007]). Note that the absence of inversion layers in the mean temperature profile for *ζ*
_{TP}≥1.5×10^{−4} s ^{−1} seems partly due to the vertical resolution of the model.

Figure 4b is the same as Figure 4a but for the buoyancy frequency squared (*N*^{2}). The tropospheric *N*^{2} (4 km below the tropopause) is about 1.5×10^{−4} s ^{−2}, whereas the stratospheric *N*^{2} (4 km above the tropopause) is about 3 to 4×10^{−4} s ^{−2}. The profiles for *ζ*
_{TP}<0 show a sharp local maximum just above the tropopause and a sharp local minimum just below the tropopause (except for −5×10^{−5}≤*ζ*
_{TP}<0 s ^{−1}), whereas the profiles for *ζ*
_{TP}>0 show a smooth transition from the troposphere to the stratosphere without sharp peaks. As *ζ*
_{TP} increases, the sharpness of the *N*^{2} profiles decreases. Following [Birner et al. (2006]), the maximum *N*^{2} within the temperature inversion (denoted by {N}_{\text{max}}^{2}) is used as a measure of the strength of the TIL in the following analyses. In this study, {N}_{\text{max}}^{2} is searched for within 4 km above the local tropopause.

The horizontal map of {N}_{\text{max}}^{2} at *t*=36 h is shown in Figure 3c. Regions with high {N}_{\text{max}}^{2} are located to the north of the surface cyclone, whereas regions with low {N}_{\text{max}}^{2} are located around the south edge of the analysis region. The distribution of {N}_{\text{max}}^{2} around the cyclone is similar to that in the idealized simulation with dry dynamics by [Wirth and Szabo (2007]) (their Figure three), implying common mechanisms between these two simulations. A comparison of Figure 3b,c shows that regions with high (low) {N}_{\text{max}}^{2} coincide with negative (positive) *ζ*
_{TP}. Thus, in general, a negative correlation exists between *ζ*
_{TP} and {N}_{\text{max}}^{2} in the horizontal distributions at *t*=36 h. However, the regions of *ζ*
_{TP}<0 in the eastern part of the domain coincide with {N}_{\text{max}}^{2} greater than 8×10^{−4} s ^{−2} (red regions in Figure 3c), whereas those with similar negative values of *ζ*
_{TP} in the western part coincide with {N}_{\text{max}}^{2} less than 7×10^{−4} s ^{−2} (green and blue regions in Figure 3c), implying the existence of other factors that determine {N}_{\text{max}}^{2}.

The relationship between *ζ*
_{TP} and {N}_{\text{max}}^{2} is further examined along the life cycle of the cyclone. Figure 5 shows two-dimensional histograms of *ζ*
_{TP} (the horizontal axis) and {N}_{\text{max}}^{2} (the vertical axis) at 6-h intervals. From *t*=0 to 18 h, the negative correlation between *ζ*
_{TP} and {N}_{\text{max}}^{2} is not clear. The correlation coefficient between *ζ*
_{TP} and {N}_{\text{max}}^{2} is −0.28 at *t*=0 h and increases to −0.64 at *t*=18 h. At the initial time (*t*=0), the TIL might be underestimated due to the coarser resolution of the NCEP FNL, which is used as the initial condition. However, it is difficult to distinguish the spin-up of the TIL from the evolution of the TIL due to the synoptic evolution. Thus, we mainly analyze *t*=24 to 36 h.

From *t*=24 to 42 h, a clear negative correlation between *ζ*
_{TP} and {N}_{\text{max}}^{2} is obtained with a correlation coefficient of about −0.7. The linear regressions of {N}_{\text{max}}^{2} as functions of *ζ*
_{TP} (black lines in Figure 5) show steeper slopes, and the regressed values become about 7×10^{−4} s ^{−2} around *ζ*
_{TP}=−1×10^{−4} s ^{−2}, and about 5 × 10^{−4} s ^{−2} around *ζ*
_{TP}>2 × 10^{−4} s ^{−1}. The proportionality coefficient between *ζ*
_{TP} and the mean {N}_{\text{max}}^{2} for *ζ*
_{TP}<0 is about −1, which is somewhat smaller than that for the reference run in [Erler and Wirth (2011]), presumably due to the coarser spatial resolution of our model. The deviations from the regression lines are partly due to small-scalel disturbances such as gravity waves, which will be discussed in the following subsections.

From *t*=42 to 48 h, the negative correlation between *ζ*
_{TP} and {N}_{\text{max}}^{2} becomes weaker, and the correlation coefficient becomes about −0.6 at *t*=48 h. The linear regression slope becomes gentler. In summary, the negative correlation between *ζ*
_{TP} and {N}_{\text{max}}^{2} is clearer during the development and mature stages of the cyclone, whereas it is unclear before and after that period.

### 3.3 Analysis of gravity waves around the tropopause

Figure 6 shows *∂* *w*/*∂* *z* at 260 hPa every 3 h from *t*=24 to 36 h. Red colors show vertical convergence, and blue colors show vertical divergence. An arc-shaped wave packet with east-west wave fronts over the Sea of Okhotsk-Northwestern Pacific propagates northward. At *t*=24 h, the wave fronts extend eastward from the southern tip of Sakhalin. The wave fronts tilt gradually toward the southwest-northeast direction, and at *t*=36 h, they extend almost in a south-north direction off the east coast of middle Sakhalin. In the region of the arc-shaped wave packet, {N}_{\text{max}}^{2} is the highest, where *ζ*
_{TP} is negative as shown in Figures 3b,c at the tropopause. The wave packets over the Asian continent around (120°E, 40°N), (130°E, 40°N), and (135°E, 45°N) are likely mountain waves. As the wind field changes with time, the mountain waves also change their structure.

The arc-shaped waves are robustly simulated in sensitivity experiments with different horizontal resolutions (15 and 20 km), vertical resolutions (50 layers and 210 layers), numerical integration time steps (10 and 20 s), time constants for nonlinear numerical diffusion (1,200 and 2,400 s), or time constants for linear numerical diffusion (600 and 2,400 s). If the number of vertical layers is reduced to 50, the amplitude of the arc-shaped waves in *∂* *w*/*∂* *z* becomes slightly weaker, and the associated peak in {N}_{\text{max}}^{2} also becomes smaller (not shown). The resolution dependency of the peak {N}_{\text{max}}^{2} is consistent with the results of the idealized experiment by Müller and Wirth ([2009]). On the other hand, the weakening of the TIL around the center of the cyclone also becomes gentle. Although the small-scale structures are different, the time evolution of the arc-shaped waves and the negative correlation between *ζ*
_{TP} and {N}_{\text{max}}^{2} does not change much.

Here, temporal modulation of local *N*^{2} by propagating gravity waves or stationary disturbances is discussed in a simple manner because the regions of clear gravity waves coincide with the regions of large {N}_{\text{max}}^{2} (Figures 3c and 6e). If we use standard notations (e.g., Andrews et al. [1987]),

\begin{array}{lcr}\frac{\partial {N}^{2}}{\mathrm{\partial t}}& \equiv & \frac{\partial}{\mathrm{\partial t}}\left(\frac{g}{\theta}\frac{\mathrm{\partial \theta}}{\mathrm{\partial z}}\right)\\ =& \frac{g}{\theta}\left(\frac{\partial}{\mathrm{\partial z}}-\frac{1}{\theta}\frac{\mathrm{\partial \theta}}{\mathrm{\partial z}}\right)\left(\frac{\mathrm{\partial \theta}}{\mathrm{\partial t}}\right),\end{array}

(1)

where *g* is the gravitational acceleration. If the process is adiabatic,

\begin{array}{lcr}\frac{\partial {N}^{2}}{\mathrm{\partial t}}& =& \frac{g}{\theta}\left(\frac{\partial}{\mathrm{\partial z}}-\frac{1}{\theta}\frac{\mathrm{\partial \theta}}{\mathrm{\partial z}}\right)\left(-\mathit{u}\xb7\nabla \theta \right)\\ =& \frac{g}{\theta}\left(-\frac{\partial \mathit{u}}{\mathrm{\partial z}}\xb7\nabla \theta -\mathit{u}\xb7\nabla \frac{\mathrm{\partial \theta}}{\mathrm{\partial z}}+\frac{1}{\theta}\frac{\mathrm{\partial \theta}}{\mathrm{\partial z}}\mathit{u}\xb7\nabla \theta \right),\end{array}

(2)

where *u*≡(*u*,*v*,*w*) denotes three-dimensional winds. If *θ* is assumed to be uniform in the horizontal direction,

\begin{array}{l}\frac{\partial {N}^{2}}{\mathrm{\partial t}}\approx \frac{g}{\theta}\left[-\frac{\mathrm{\partial w}}{\mathrm{\partial z}}\frac{\mathrm{\partial \theta}}{\mathrm{\partial z}}+w\left\{\frac{1}{\theta}{\left(\frac{\mathrm{\partial \theta}}{\mathrm{\partial z}}\right)}^{2}-\frac{{\partial}^{2}\theta}{\partial {z}^{2}}\right\}\right].\end{array}

(3)

#### 3.3.1 Vertically propagating disturbances

If *w*=*A* sin(*m* *z*−*ω* *t*),

\begin{array}{lcr}\frac{\partial {N}^{2}}{\mathrm{\partial t}}& =& A\frac{g}{\theta}\left[-mcos\left(\mathit{\text{mz}}-\mathrm{\omega t}\right)\frac{\mathrm{\partial \theta}}{\mathrm{\partial z}}\right.\\ \phantom{\rule{8.53581pt}{0ex}}\left(\right)close="]">+sin\left(\mathit{\text{mz}}-\mathrm{\omega t}\right)\left\{\frac{1}{\theta}{\left(\frac{\mathrm{\partial \theta}}{\mathrm{\partial z}}\right)}^{2}-\frac{{\partial}^{2}\theta}{\partial {z}^{2}}\right\}& ,\end{array}\n

(4)

where *A* is a constant, *m* is the vertical wave number, and *ω* is the frequency. Assuming that *∂* *θ*/*∂* *z* is a positive constant and *∂* *θ*/*∂* *z*≪*m* *θ*,

\begin{array}{l}\frac{\partial {N}^{2}}{\mathrm{\partial t}}\approx -\mathit{\text{Am}}{N}^{2}cos\left(\mathit{\text{mz}}-\mathrm{\omega t}\right).\end{array}

(5)

This gives

\begin{array}{lcr}{N}^{2}& =& \int \frac{\partial {N}^{2}}{\mathrm{\partial t}}\mathit{\text{dt}}\\ \approx & \frac{\mathit{\text{Am}}{N}^{2}}{\omega}sin\left(\mathit{\text{mz}}-\mathrm{\omega t}\right)+\text{const},\end{array}

(6)

\begin{array}{lcr}\frac{\mathrm{\partial w}}{\mathrm{\partial z}}& =& \mathit{\text{Am}}cos\left(\mathit{\text{mz}}-\mathrm{\omega t}\right),\end{array}

(7)

\begin{array}{lcr}\frac{{\partial}^{2}w}{\partial {z}^{2}}& =& -A{m}^{2}sin\left(\mathit{\text{mz}}-\mathrm{\omega t}\right).\end{array}

(8)

If the phase propagation is downward (*m*/*ω*<0), *N*^{2} and *∂*^{2}
*w*/*∂* *z*^{2} become in phase. For simplicity, the feedback from the changes in *N*^{2} to the dispersion relation of the wave is considered to be free from the variation in *N*^{2} in this paper. Nonlinearity in the TIL (e.g., Miyazaki et al. [2010a]) is beyond the scope of this paper, which focuses on a diagnosis of linear wave propagation.

#### 3.3.2 Stationary disturbances

If *w*=*A* sin(*m* *z*) and the same assumption is made with regard to *θ* and *N*^{2} as above, from Equation 3,

\begin{array}{lcr}{N}^{2}& \approx & -\mathit{\text{Am}}{N}^{2}cos\left(\mathit{\text{mz}}\right)\xb7t+\text{const},\end{array}

(9)

\begin{array}{lcr}\frac{\mathrm{\partial w}}{\mathrm{\partial z}}& =& \mathit{\text{Am}}cos\left(\mathit{\text{mz}}\right).\end{array}

(10)

In this case, −*∂* *w*/*∂* *z* and *N*^{2} are in phase.

Figure 7 shows the vertical cross section of *∂* *w*/*∂* *z* (color) and *N*^{2} (contour) along the line shown in Figure 6b at *t*=27 h. The color coding is the same as that in Figure 6. The green line shows the tropopause. A wave pattern is located at around 0 to 800 km in the horizontal direction and 300 to 200 hPa in the vertical direction, which corresponds to the arc-shaped wave packet in Figure 6.

The dispersion relation of the wave packet around the tropopause at 147°E, 49°N, *t*=27 h is examined because the wave packet is clearer at this point and time. The region has a strong wind shear and a gap in *N*^{2}, showing a nonlinear nature at a later time. From Figures 6b and 7, the wavelengths in the *x*, *y*, and *z* directions are 5.0×10^{2} km, 1.8×10^{2} km, and 2.0–3.0 km, respectively. From Figure 7, *N*^{2} is about 5×10^{−4} s ^{−2}. Using the Coriolis parameter at 50°N, *f*=1.1×10^{−4} s ^{−1}, the dispersion relation for an inertia-gravity wave (e.g., Gill [1982]),

\begin{array}{l}\widehat{\omega}=\omega -\mathit{k}\xb7\mathit{u}=\sqrt{\frac{{N}^{2}({k}^{2}+{l}^{2})+{f}^{2}{m}^{2}}{{k}^{2}+{l}^{2}+{m}^{2}}},\end{array}

(11)

gives an intrinsic frequency \widehat{\omega}=2.9– 4.1×10^{−4} s ^{−1}, or a period of about 2.5– 3.7×10^{2} minutes. Here, *ω* denotes the frequency in a fixed frame of reference, *k* and *l* are the wavenumbers along the *x* and *y* directions, respectively, and *k*≡(*k*,*l*,*m*). We assume *k*>0, *l*<0, and *m*<0 from the phase lines in Figures 6 and 7. Substituting environmental wind speeds of of about (2.8×10, 3.2×10, 1.8×10^{−2}) m s ^{−1}, *ω* becomes −5.4 to −4.0×10^{−4} s ^{−1} or a period of about 1.9 to 2.6×10^{2} min. In the model, the period at a fixed point is about 2.0×10^{2} min with northward phase propagation, or *ω*=−5.2×10^{−3} s ^{−1}, which is close to the value estimated from Equation 11. Thus, this wave satisfies the dispersion relation for an inertia-gravity wave.

From a hodograph analysis of high-pass filtered horizontal winds, it is also confirmed that the wave is an inertia-gravity wave that has a downward phase speed (not shown). Although detailed analysis of gravity waves is beyond the scope of this paper, several discussions will be presented later.

### 3.4 Relationship between TIL and gravity waves

In this subsection, the relationship between the TIL and the gravity waves is further examined to ascertain if the vertical convergence shown in Figure 7 is the dominant factor in the strengthening of the TIL during the development and mature stages of the cyclone.

In Figure 7, *∂* *w*/*∂* *z* is in quadrature with *N*^{2} in the stratosphere; the local maxima of *N*^{2} are located between the local maxima of *∂* *w*/*∂* *z* above and the local minima below. From Equations 6 to 7, this implies that the wave pattern for *N*^{2} is produced by the downward propagating wave disturbance of *w*, which is consistent with the analysis in the previous subsection that indicated that the wave with a downward phase speed is an inertia-gravity wave. These high- *N*^{2} regions are detected as TILs.

It is important to confirm that the amplitudes of the wave patterns in *N*^{2} and *w* are consistent with each other. From Equation 6, the amplitude of *N*^{2} becomes *A* *m* *N*^{2}/*ω*. In Figure 7, the amplitude of *∂* *w*/*∂* *z*, which becomes *Am*, is about 5×10^{−5} s ^{−1}. Substituting *A* *m*=5×10^{−5} s ^{−1}, *N*^{2}=5×10^{−4} s ^{−2}, and *ω*=3.1×10^{−4} s ^{−1} into Equation 6, the amplitude of the wave pattern of *N*^{2} is estimated to be 8.3×10^{−5} s ^{−2}, whereas the actual amplitude in Figure 7 is about 10^{−4} s ^{−2}. These are the same order of magnitude as each other. As the amplitude of the wave pattern for *N*^{2} is proportional to *N*^{2}, the wave pattern is clearly seen only within the TIL (Figure 7).

Figure 8 shows histograms of {N}_{\text{max}}^{2} in the TIL, using the hourly outputs from *t*=24 to 36 h. Horizontal grid points within the analysis regions of 13 snapshots (in total 286,836 grid points) are categorized by the sign of either (a) *ζ*
_{TP}, (b) *∂* *w*/*∂* *z*, or (c) *∂*^{2}
*w*/*∂* *z*^{2}. The histograms of {N}_{\text{max}}^{2} are created for each group. The bin width of the histograms is 5×10^{−5} s ^{−1}. Each histogram is normalized by its sample size.

First, differences due to the sign of *ζ*
_{TP} are examined (Figure 8a). The peak for *ζ*
_{TP}<0 appears at {N}_{\text{max}}^{2}=5.25\times 1{0}^{-4} s ^{−2}, whereas that for *ζ*
_{TP}≥0 appears at {N}_{\text{max}}^{2}=6.25\times 1{0}^{-4} s ^{−2}. This result is consistent with the negative correlation between *ζ*
_{TP} and {N}_{\text{max}}^{2} in Figures 4 to 5.

Next, differences due to the sign of *∂* *w*/*∂* *z* are examined in Figure 8b. Although the two histograms both peak at {N}_{\text{max}}^{2}=5.25\times 1{0}^{-4} s ^{−2}, the distribution for *∂* *w*/*∂* *z*<0 s ^{−1} (short-dashed) appears slightly to the right of that for *∂* *w*/*∂* *z*≥0 s ^{−1} (solid), indicating that {N}_{\text{max}}^{2} increases as *∂* *w*/*∂* *z* decreases. This indicates that stationary disturbances enhance the TIL through vertical convergence. The difference between positive and negative *∂* *w*/*∂* *z* is slightly clearer when the histograms are computed only for the regions of *ζ*
_{TP}<0 (Figure 9). For *ζ*
_{TP}<0, the peak for *∂* *w*/*∂* *z*≥0 is at {N}_{\text{max}}^{2}=5.75\times 1{0}^{-4} s ^{−2}, and that for *∂* *w*/*∂* *z*<0 is at {N}_{\text{max}}^{2}=6.25\times 1{0}^{-4} s ^{−2} (Figure 9a). For *ζ*
_{TP}≥0, the two histograms peak at the same {N}_{\text{max}}^{2} (Figure 9b). Note that the time integration of *∂* *w*/*∂* *z* strengthens *N*^{2} when the stationary disturbances are dominant. Thus, local *∂* *w*/*∂* *z* at a specific location and time may have little correlation with {N}_{\text{max}}^{2}.

Finally, differences due to the sign of *∂*^{2}
*w*/*∂* *z*^{2} are examined in Figure 8c. The two histograms both peak at {N}_{\text{max}}^{2}=5.25\times 1{0}^{-4} s ^{−2}. However, the distribution for *∂*^{2}
*w*/*∂* *z*^{2}≥0 is located to the right of that for *∂*^{2}
*w*/*∂* *z*^{2}<0, indicating that {N}_{\text{max}}^{2} increases as *∂*^{2}
*w*/*∂* *z*^{2} increases. This implies that waves with a downward phase propagation are important for the strengthening of the TIL. The difference between positive and negative *∂*^{2}
*w*/*∂* *z*^{2} is clearer when the histograms are computed only for the regions of *ζ*
_{TP}<0 (Figure 10). For *ζ*
_{TP}<0, the peak for *∂*^{2}
*w*/*∂* *z*^{2}<0 is at {N}_{\text{max}}^{2}=5.75\times 1{0}^{-4} s ^{−2} and that for *∂*^{2}
*w*/*∂* *z*^{2}≥0 is at {N}_{\text{max}}^{2}=6.25\times 1{0}^{-4} s ^{−2} (Figure 10a). For *ζ*
_{TP}≥0, the two histograms peak at the same {N}_{\text{max}}^{2} (Figure 10b). The difference by *∂*^{2}
*w*/*∂* *z*^{2} in Figure 10a is clearer than that by *∂* *w*/*∂* *z* in Figure 9a. Although waves with upward phase propagation may also exist, their contribution seems to be smaller.

It is not a straightforward task to test the statistical significance of spatially correlated samples with appropriate estimates of correlation scales. Thus, we have adopted the moving block bootstrap method ([Künsch 1989]) to test the statistical significance of the differences in mean {N}_{\text{max}}^{2}. Because the optimum block size is unknown, we choose block sizes of 10×10×5, 20×20×10, and 40×40×10 grid points (or, equivalently 200 km × 200 km × 5 h, 400 km × 400 km × 10 h, and 800 km × 800 km × 10 h) in the *x*, *y*, and *t* directions. We obtained 10,000 bootstrap observations each. As a result, the differences in the means in Figure 8a,c are significant to levels higher than 99% except for Figure 8c with the largest block size (see the values at the top left of each figure). The difference in the means in Figure 8b is not as significant as that in other pairs of data except the smallest block size. Statistical tests for Figures 9 and 10 are more difficult to construct, and will not be shown in this paper.