1. Introduction
Static stability and the related buoyancy frequency are important concepts in ocean and atmospheric dynamics. The buoyancy frequency N was first studied, separately, by Vaisala (1925) and Brunt (1927). With the parcel theory, through considering a perturbation of the parcel, the equation for the perturbation displacement was derived. The solution of the equation is determined completely by the base state, especially the vertical structure, of the environment surrounding the parcel (e.g., Peixoto and Oort 1992). Specifically, the buoyancy frequency squared N2, which is denoted hereafter as Z for convenience, measures the static stability of the fluid. Doswell and Markowski (2004) pointed out that the buoyancy is not a relative quantity, and is the static part of an unbalanced vertical pressure gradient force.
The buoyancy frequency was generally treated as a constant in textbooks; for example, for average tropospheric conditions, the value of the N was estimated as 1.2 × 10−2 s−1 (e.g., Holton 1992). In the recent decades, with the availability of the probing and sounding data and the reanalysis data from the models, the variability and changes of the buoyancy frequency and static stability have been analyzed. For the ocean waters, many researchers have investigated the determination and profile of the buoyancy and the stability (e.g., Jackett and Mcdougall 1995; Potetyunko et al. 2003; VanHaren and Millot 2006; King et al. 2012).
For the atmosphere, compared to the lower values in the troposphere, the static stability Z in the stratosphere has higher values, estimated as 5.0 × 10−4 s−2. The tropopause serves as a sharp transition, and the lower stratosphere exhibits a maximum in static stability that is above the stratospheric background value (Grise et al. 2010). Many studies have also focused on the atmospheric buoyancy frequency and static stability in the stratosphere and upper troposphere, especially around the tropopause (e.g., Wirth 2003; Chen et al. 2005; Kunz et al. 2009; Grise et al. 2010; Erler and Wirth 2011; Duran and Molinari 2019). For the atmosphere in the troposphere and the planetary boundary layer, although the stability has lower values, it is associated with the synoptic weather. The corresponding variability and changes have also been emphasized (e.g., Breistein and Parry 1954; Gates 1961; Held 1982; Juckes 2000; Angell et al. 1969). Lee and Mak (1994) examined the variability in the large-scale static stability, and analyzed the climatic mean fields for temporal scales from synoptic, intraseasonal, and seasonal to the interannual scale. They found that the climatological mean field varies over a factor of 2 in the extratropics, and the subseasonal variability reaches 40% of the corresponding seasonal mean values in the two storm-track regions.
In addition to the atmospheric instability, the water vapor contained in the atmosphere is also important to the formation of precipitation. Previous studies have examined the effects of the moisture on the frequency and stability (e.g., Durran and Klemp 1982; Sherwood 2000; Schultz et al. 2000; Liu and Gao 2003). The buoyancy of the stratified atmosphere may cause the gravity waves, and the propagations of the waves may influence the synoptic weather and may bring, in certain circumstances, heavy rainfall (e.g., Ackerman 1956; Li 1978; Chao 1980; Chen 1982; Skopovi and Akylas 2007; Soldatenko 2014; McHugh 2015). For the boundary layer, the static stability may have a connection with the dynamic stability (e.g., Moum et al. 1992; Kwon and Frank 2005), and the Richardson number is generally used to compare the static and dynamic stabilities (e.g., Peixoto and Oort 1992). For the lower stratosphere, the static and dynamic stabilities may also be connected (e.g., Zhang et al. 2019; Kaluza et al. 2019; Kunkel et al. 2019).
In the present study, we focus on the static stability of dry air. A reanalysis dataset is used as an example to calculate it. Considering that the data are from the model and the vertical resolution is relatively lower compared with the observed data (e.g., Grise et al. 2010), as a fundamental step in the study we first compare the results of the static stability of different temporal scales, in both the sign and magnitude, with the atmospheric state ranging from the “instantaneous” one to the more “averaged” one. The purpose is to identify if the results with the resolution can be fundamentally reasonable. When using the output of the four-times-daily data, we may find that at each instantaneous moment the static stability Z can possess negative values in some areas, suggesting that the atmosphere is unstable and leading to convection. By using the averaged model output (e.g., the monthly or even the daily data), the calculated static stability Z may tend to be positive over the entire field.
The fundamental work also includes the examination of whether the horizontal distributions of the static stability Z may display meaningful patterns. The results from the calculations show that the static stability can truly exhibit spatial patterns that seem reasonable, in which Z varies geographically with the latitude and surface conditions (e.g., whether over ocean, coastal, land, or mountain areas). The overall effect of these geographical factors can finally be reflected from the influences of the physical quantities that appear in the formula expression of Z.
The major task of this study is to analyze the relative importance of the physical influencing quantities in the variation of the stability. It was generally regarded that atmospheric static stability is induced by the atmospheric inversion, characterized by the vertical difference of temperature. Through derivations, the conventional formula for Z can finally be expressed as a nonlinear function of the two quantities, the air temperature and the vertical difference of the temperature. Then we have the following questions. Although the vertical difference of temperature is important to the stability, is the variation of the temperature itself important? Can the change of temperature dominate or have a significant contribution to the variation of Z? Is there an easy method to compare the contributions from the two different influencing quantities?
In our previous studies, for the several issues in which the relations are in general nonlinear, we have utilized a simple statistical method to estimate and compare the contributions from the different influencing factors (e.g., Lu et al. 2010, 2014, 2015; Tu and Lu 2020; Lu and Tu 2020). What we consider are the change rates of the variables and the scales of their variations. For the different influencing factors, we normalize them and use the normalized linear regression to fit the relation. The results of the dominance rely on how we sample the data. In this study, we apply this method to analyze the dominance in both the temporal variations and the spatial variation, with the samples being collected respectively from a seasonal variation and an interannual variation of the Z, as well as a spatial variation that include all the grid points in the globe.
The datasets of the National Centers for Environmental Prediction–National Center for Atmospheric Research (NCEP–NCAR) reanalysis (Kalnay et al. 1996) are used, as an example, for the calculations in the present study. They contain 17 pressure levels, with a horizontal resolution of 2.5° × 2.5° in latitude and longitude. The variable required from the datasets is only the temperature, and we use it to calculate the vertical difference of temperature. The data used include the four-times-daily data of 2017 and the daily data from 1948 to 2017.
Methods are introduced in section 2, including the derivation of the expression showing that the static stability Z can be expressed as a nonlinear function of the two influencing factors, the air temperature and the vertical difference of temperature, and the approach for estimating the relative importance of the two influencing quantities. In section 3, we compare the horizontal distributions of Z calculated with the instantaneous data, the daily data, and the monthly mean data. The results of the dominance for the samples that are collected from both the temporal variations and the spatial variation are presented in section 4. A summary and discussion are given in section 5.
2. Methods and considerations
a. The derivation of the physical expression and the influencing quantities
The parcel theory is used to derive the vertical motion equation and to obtain the formula of the static stability and buoyancy frequency. The stratified ambient atmosphere is assumed vertically in a hydrostatic state. When an air parcel develops a perturbation, with a small displacement δz from its initial position, the imbalance between the gravity and the buoyancy restoring force may lead to the acceleration of the parcel. The vertical motion equation can finally be written as (e.g., Holton 1992) d2(δz)/dt2 + (g∂lnθ/∂z) × δz = 0, where g is the gravitational acceleration, z is the height, and θ is the potential temperature.
Figure 1 presents the distributions of Z, calculated from (4), with respect to the X and Y. For different layers, since the parameter k in the equation has different values, the plots may have different patterns. For the specific issue, with the samples obtained from a specific temporal or spatial variation, the corresponding X and Y may have its preferred domains. For the sake of comparison, the three plots in Fig. 1 display the same portion of the distribution (i.e., the same domains of the X and Y for the different layers). Nevertheless the different patterns for the three plots presented all show that, for the domains in the plots, Z increases with X but decreases with Y. The domains of the X and Y are illustrated in Table 1, for some selected temporal and spatial variations.
Distributions of Z (unit: 10−4 s−2), which varies as a function of X (unit: 10−4 K−1) and Y (unit: K). The three plots are for the (a) 10–20-, (b) 600–700-, and (c) 850–925-hPa layers.
Citation: Journal of Climate 34, 12; 10.1175/JCLI-D-20-0615.1
The variation ranges, including the maxima and minima, of the quantities, i.e., the air temperature T, and the related X, which is 1/T; Y, which is ΔT; and Z, which is calculated from the nonlinear relation (4). The three variations include the interannual variation of January for the 600–700-hPa layer over the area 30°–32.5°N, 117.5°–120°E, which is around the city of Nanjing; the seasonal variation of the 12 months over the 70 years, for the same layer and over the same area; and the spatial variation that includes all the grid points in the globe, for the same layer and for January 2017.
b. The approach for estimating the relative importance of the influencing quantities
Using Eq. (4), while we can provide the temporal–spatial variations of the Z, we may also investigate whether the variation of Z is more controlled by the variation of X or by the variation of Y. A simple method has been used in our previous studies, for the several different issues, to assess the relative importance of the two influencing factors (e.g., Lu et al. 2010, 2015; Tu and Lu 2020; Lu and Tu 2020).
Equation (4) shows the nonlinear relation of Z with X and Y, and we may write it as a general form Z = Z(X, Y). With the data collected for the three quantities, we can use a linear regression Z = AX + BY + c to fit the relation. The feasibility of the linear fitting needs to be verified, through the statistical tests for the significance. The meaning of the first two coefficients, through expressing them as A = ∂Z/∂X and B = ∂Z/∂Y, is that they are the change rates of Z with respect to X and Y.
The standard deviations σX and σY are used to reflect the year-to-year perturbation scales of X and Y. Constructed with the change rates and the perturbation scales, the two measures SX ≡ |∂Z/∂X|σX and SY ≡ |∂Z/∂Y|σY can be used to estimate the contributions of X and Y to the variation of Z. With the coefficients fitted from the regression, the measures can be expressed as SX = |A|σX and SY = |B|σY.
Note that in Eq. (5), because of the normalizations, X, Y, and Z can all have positive or negative values. The sign and value of Z depend on the signs and values of X and Y, as well as the signs and values of the coefficients A and B. For assessing the dominance using Eq. (5), since the normalizations, both X and Y have a magnitude of 1, although they can be positive or negative. The magnitudes of the terms AX and BY can thus be measured with the absolute values of A and B. This explains why we can determine whether the variation of Z is dominated by X or Y through examining whether |A| or |B| is greater.
The values of the coefficients A and B depend on the input data for the linear regression. We can use the samples obtained from different variations, both the temporal and the spatial variations. In the present study, we examine the samples from an interannual variation, a seasonal variation, and a spatial variation that is for all the grid points in the globe.
3. Distributions of the static stability with data of different temporal resolutions
Figure 2 shows the horizontal distributions of the static stability Z in the January and July of 2017. We plot the distribution of each layer, and here focus on the layer of 850–925 hPa. The plots in the three rows are for the different temporal resolutions. To provide a more realistic (which can better reflect the instantaneous) atmospheric situation, the 4 times daily data are used. Figures 2a and 2d are the results calculated by the data within the time interval of 6 h around 1200 UTC. In these two plots for the relatively more instantaneous atmospheric states, we can observe the areas over the globe where the value of Z is negative, suggesting that the atmosphere over these areas is unstable and convection may occur. In Fig. 2d, over southern Africa, the Sahara Desert, and Saudi Arabia and the areas north of it, there are several areas with the Z being negative. Note that results are for noon time in summer, and these areas are characterized by desert or dry land. The remarkable instability of the atmosphere over these areas may be related to the intertropical convergence zone (ITCZ) and the mesoscale convective systems.
Distributions of Z (unit: 10−4 s−2) of 2017 at the 850–925-hPa layer, calculated with the four-times-daily data for the periods around 1200 (a) 1 Jan and (d) 1 Jul, the daily data on (b) 1 Jan and (e) 1 Jul, and the monthly data for (c) January and (f) July.
Citation: Journal of Climate 34, 12; 10.1175/JCLI-D-20-0615.1
Figures 2b and 2e are the results obtained from the daily data, which are the average of the above data with the 6-h resolution. The belt region with negative values over the Arctic in Fig. 2a still exists in Fig. 2b. This suggests that the instability over the region may persist during the whole day. In other words, the instability there has less diurnal variation. Differently, other regions marked in yellow in Fig. 2a (e.g., those over Africa) no longer exist in Fig. 2b, which means that the instantaneous unstable state cannot maintain during the whole day, and there may be a diurnal variation. Such difference is more obvious for July (Figs. 2d,e). It is possible that the atmospheric instability over Africa is strong in the noon time, but not in the other times of the day. Convective activity may have diurnal variations, which are mostly linked to the mesoscale convective systems. In Fig. 2b, which is for winter and at the lower level, corresponding to the instability of the atmosphere over the Arctic, the atmosphere over the high-latitude land is very stable. This may be related to the “warm Arctic and cold continents” phenomenon during the winter (e.g., Cohen et al. 2018).
Figures 2c and 2f are the results of Z calculated by using the monthly data. We also plot the distributions that average the daily results of the 31 days with each being calculated from the daily data (figures not shown). Comparisons show that these plots, whether for January or July, are overall almost the same. Because of the 31-day average, we cannot find areas over the entire field where Z has negative values. The plots that use the monthly data are in general similar, in spatial pattern, to the plots that use the daily data, but the values of Z in Figs. 2c and 2f are smaller than the values in Figs. 2b and 2e. It can therefore be concluded that we may observe unstable convective atmosphere in the relatively “instantaneous” state, but the more “averaged” states may tend to have stable atmosphere. The results of the static stability are dependent on the temporal resolutions of the data.
Further intercomparisons between the left panel and the right panel in Fig. 2 indicate that there are high positive-value regions in the winter hemisphere. In the boreal winter (left panel), there are high positive values over the Euro-Asian continent and the North American continent. Careful comparisons show that the distribution pattern of the high positive values follows well the coastal lines of the continents. This may be due to the influence of the land surface on the temperature of the atmosphere. We also notice that large positive values may appear over some of the ocean regions (e.g., the ocean areas immediately west of both North America and South America). This may be caused by the influence of the sea surface temperature on the temperature of the atmosphere.
Figure 3 presents the meridional–vertical sections of the zonally averaged Z. Because of the zonal average, the sections obtained from the daily data (Figs. 3a,b) and the monthly data (Figs. 3c,d) are very similar, and there is no negative value even in the plots obtained from the daily data. Also, because of the zonal average, the effect of the land–sea distribution is weakened, and what is presented in the sections of the 2 months is significantly affected by the meridional movement of the solar radiation.
Meridional–vertical section of the zonal mean Z (unit: 10−4 s−2) in 2017, calculated with the daily data on (a) 1 Jan and (b) 1 Jul, as well as the monthly data from (c) January and (d) July.
Citation: Journal of Climate 34, 12; 10.1175/JCLI-D-20-0615.1
As illustrated in all the plots, the values of Z over the zonal belt from the tropics to the middle latitudes in troposphere are relatively small, with the minimum center at 300 hPa. This coincides with the characteristics that convection is strong and stability is low in the tropospheric atmosphere of this belt, which may be related to the vertical activities of the atmosphere over the ITCZ. The values of Z in the stratosphere over the belt from the tropics to the middle latitudes are large, and the maximum center is around 75 hPa. The tropopause is clearly indicated at the level around 150 hPa where the vertical gradient of Z is the largest over this region.
This zonally averaged meridional–vertical cross section of the static stability Z is fairly stable across the different time scales. The similarity between the plots obtained from the daily and the monthly data suggests that the day-to-day variation of the section is small. The similarity between the plots of January (Figs. 3a,c) and the plots of July (Figs. 3b,d) reveals that the seasonal variation of this zonally averaged stability is also small.
In the troposphere, the maximum center appears in the lower atmosphere. In January, the center is near the surface over the Arctic Ocean. In July, the center is over the Antarctic, extending from the surface to 500 hPa, with much larger values. These may be related to the near-surface cold pools over the polar caps in the respective winter hemisphere, as suggested in AMAP (2015) for the polar dome. In upper troposphere, the values of the Z are large over middle and high latitudes. The maximum center in the stratosphere and the maximum center in the lower troposphere can be connected. Whether they have interactions needs to be further explored. Overall, the horizontal distributions and the vertical structures of the static stability calculated with the reanalysis, which is relatively low in vertical resolution, are consistent with those revealed by Grise et al. (2010) through using the observed data, which are in high vertical resolution.
The vertical structure of the static stability, with Z generally being small in the troposphere but large in the stratosphere, is related to the profile of the lapse rate of temperature or the vertical temperature gradient. For calculations using the data, the vertical temperature gradient contained in the equation is finally replaced by the vertical difference of temperature ΔT, which uses the lower level minus the adjacent upper level. The equation shows that static stability Z decreases with the ΔT. In the troposphere, normally, as the vertical profile shows, temperature decreases with the height, so ΔT is positive. If the positive ΔT is small, Z may acquire relatively small positive values, meaning that the atmosphere can be weakly stable. If the positive ΔT is sufficiently large so that the lapse rate of temperature can be greater than the dry adiabatic lapse rate, Z may acquire negative values, representing an unstable atmosphere, which may lead to convection. However, when there is a temperature inversion in the troposphere, ΔT becomes negative. From the equation, Z can obtain large positive values, implying that the atmosphere is very stable. In the stratosphere, similar to the inversion in troposphere, temperature increases with height, and thus ΔT is also negative. With Z gaining large positive values, the atmosphere is also very stable. Beyond the influence from the vertical temperature gradient, in this study we aim to reveal that the temperature itself may also be important to the variations of the static stability Z.
4. The relative importance for samples from different temporal and spatial variations
With the original data of temperature and the calculated results from Eq. (3), we prepare the data of the X, Y, and Z. Normalizations are performed for them, and then linear regression is established. According to Eq. (4), we finally obtain the fitted relation Z = AX + BY.
The values of the coefficients A and B depend on the data sampled. Here we focus on the samples from three variations. One is an interannual variation, in which we examine the data of each month from all the 70 years, and here only the results of January are presented. The second is a seasonal variation, which includes the data from all the 12 months of the 70 years, so we have 12 × 70 samples. The third is the spatial variation, for each of the 70 years, that includes all the grid points from the globe, and thus we have 144 × 73 samples. The spatial variation is examined for each month, and here we present the results of January and July.
Statistical tests are performed to ensure the reliability of the linearization with the fitting, for the specific relation (4). Figure 4 presents the coefficients of the correlation between the static stability Z calculated with the linear relation (5) obtained from the fitting and the stability Z calculated with the original nonlinear expression (4). Three examples are illustrated, with the data from different temporal and spatial variations. Figure 4a is for an interannual variation, over the 70 years, for January in the layer of 600–700 hPa. Figure 4b shows a seasonal variation, which includes all the 12 months of all years in the layer of 600–700 hPa. Figure 4c is for the spatial variation of January, which considers all the grid points over the globe, and the correlation is for all the vertical levels and all the 70 years. The three plots indicate that for the temporal and spatial variations that have sufficient samples, the correlation coefficient of the static stability between the linearly fitted and the original nonlinearly formulated can all be greater than 0.99.
The coefficient of the correlation between the static stability Z calculated from the linear relation fitted with the regression and the stability calculated with the original nonlinear expression. The data used are from three variations, including (a) the interannual variation for January of all the years (70 samples) in the 600–700-hPa layer, (b) the seasonal variation of the 12 months of all years (12 × 70 samples) in the 600–700-hPa layer, and (c) the spatial variation of January, with all the grid points over the globe (144 × 73 samples), for all the vertical levels and all the 70 years.
Citation: Journal of Climate 34, 12; 10.1175/JCLI-D-20-0615.1
Tests are also conducted for other layers and months, and the results are the same. Hence, the linearization is perfect, and the method is robust. The reason might be that in Eq. (4), while linearly varying with Y, Z is a parabolic function of X. For the samples of a given variation, the data for X are located mainly in one of the two sides in the parabolic curve, and thus the relation of the Z with X can be well approximated as a linear one. The relative importance of X and Y in the variation of Z is assessed through comparing the absolute values of A and B.
For the interannual variation, the seasonal variation, and the spatial variation, the temperature T and the vertical difference of temperature ΔT, and thus X and Y (and Z, calculated from X and Y) may have different variation ranges. Table 1 presents the maxima and minima of these quantities, with an example for each of the interannual variation, seasonal variation, and the spatial variation. Compared with the seasonal variation, the interannual variation has smaller ranges for both the temperature and the vertical difference of temperature, while the spatial variation has larger ranges for the two influencing quantities. Correspondingly, X, Y, and Z, which are obtained directly from the temperature, also display their ranges for the three variations. This table is used to indicate that the dominance analysis method can be applied to the variations in which the input quantities may be in different varying ranges. We understand that the quantities used in Eq. (5), the linear fitting, are all the normalized ones. Also, the three quantities are nonlinearly linked.
a. Sampling from an interannual variation
The relative importance in the interannual variation of Z over the 70 years is examined for each of the 12 months and for all the vertical layers. Figure 5 presents, as examples, the results of January for the three layers, which can be the representatives of all the layers.
Distributions of (top) coefficient A, (middle) coefficient B, and (bottom) |A| − |B| for the (a)–(c) 10–20-, (d)–(f) 600–700-, and (g)–(i) 850–925-hPa layers. The normalized regressions are performed, for each grid point, with the samples from the January of all the 70 years (70 samples).
Citation: Journal of Climate 34, 12; 10.1175/JCLI-D-20-0615.1
In the layer of 10–20 hPa, coefficient A is positive everywhere around the globe (Fig. 5a), indicating that for the interannual variations of this layer, a larger X, or a lower temperature, in a year may lead to a larger Z. The coefficient B is negative across the entire field (Fig. 5b). This means that a lower value of Y (i.e., with a relatively warmer air temperature at the upper level and a colder air temperature at the lower level in a year) may lead to a larger Z. At the time of this month, the coefficients A and B both are the largest in magnitude around the Antarctic.
Figure 5c shows the comparison with the absolute value of A minus the absolute value of B. Over the eastern part of the Antarctic area, the effect from X can be greater than that from Y, and thus the air temperature in the area dominates the interannual variation of the static stability Z. The comparison implies that over this area, although the air in the stratosphere may have negative lapse rate of temperature, which can favor the stability of the atmosphere, the static stability is dominated by the year-to-year perturbation of the temperature itself. Over the remaining region of the globe, the effect from X is less than that from Y. But their differences are small. So, overall, although the lapse rate of temperature (i.e., the vertical difference of temperature) dominates the variation of the static stability, the effect from the temperature itself is also important. In addition, we notice that, in Fig. 5c, the negative difference over the low and midlatitude oceans in the Southern Hemisphere is larger in magnitude than the difference over the lands within this latitudinal belt. This seems mainly due to the contrast of A between the ocean and land (Fig. 5a).
In the lower layer of 850–925 hPa, different from in Fig. 5a, it is shown in Fig. 5g that the coefficient A can be positive, or negative, over nearly half of the global area. Positive values appear over the continents of Eurasia and North America, the Arctic Ocean, and part of the North Pacific and North Atlantic, as well as the 30°–60° belt in the Southern Hemisphere. Negative values can be found over the oceans, the Antarctic, Greenland, Africa, and areas with high topography, such as the Tibetan Plateau and the North American Cordillera. It is interesting that the boundaries of the areas that have positive values are consistent with the coastal lines of the lands (e.g., around the south coasts of Africa and Australia), as well as the west coast of South America (Fig. 5g). Since this is in the lower layer, and A reflects the effect of the temperature, the consistency may be due to the thermal difference of the land and ocean.
Over the eastern part of the Antarctic area, coefficient A has large negative values (Fig. 5g), and coefficient B also has large negative values (Fig. 5h). These lead to the small negative values in the difference field (Fig. 5i). In addition, over the Northern Hemisphere, there are small negative values in the high-latitude regions of the Eurasia and the North America (Fig. 5i). We thus conclude that over these regions in the lower troposphere, while the interannual variation of the static stability is dominated by the lapse rate, or the vertical difference, of temperature, the temperature itself may also have considerable influence.
In the layer of 600–700 hPa, coefficient A has negative values in almost the entire field (Fig. 5d), and the areas with positive values are mostly in Northern Hemisphere. The signs and the spatial patterns of the coefficient A have significant differences among the layers, especially between the troposphere and stratosphere. The negative values of coefficient B are quite large over the lands, and small over the oceans (Fig. 5e). The values of the coefficient B, which reflects the effect of the vertical difference of temperature, are all negative across the entire field in all the vertical layers. In Eq. (3), static stability decreases with the vertical difference of temperature, with the temperature of the lower level minus the temperature of the upper level. The vertical difference of temperature may locally be affected by the surface (e.g., land/ocean) condition and the convection. It may also be influenced by other factors, such as the horizontal atmospheric circulation.
b. Sampling from a seasonal variation
What we examine in section 4a is the interannual variations. Relatively, the interannual variations of the temperature itself are small. The difference of the temperatures between the two adjacent levels may have more randomness in their interannual variations, and thus it may have larger (compared with temperature) variations. Partially because of this, in Fig. 5, coefficient A is smaller than coefficient B in magnitude almost everywhere over the globe.
In this section, we explore the much longer time series, which contain all the 12 months in the 70 years (i.e., 840 samples). In these time series, there are both seasonal variations (or cycles) and interannual variations. Compared with the situation mentioned above for the interannual variations in section 4a, the vertical difference of the temperatures may still have more randomness in their interannual variations, or the interannual variations are large. However, the variations of the temperature itself in the present 840 samples have become sufficiently large because of the annual cycles contained in the time series. It can thus be expected that, comparably, the coefficient A in the present situation may become larger in magnitude.
Figure 6 shows the results for the 840 samples, which reflect more the seasonal variations. In the layer of 10–20 hPa, it is shown that the two coefficients both vary with latitude. Coefficient A has large positive values over both of the two polar areas (Fig. 6a). Coefficient B has large negative values over the high-latitude areas except for Antarctica and Siberia (Fig. 6b). Compared with Fig. 5, because of the increase of the coefficient A in magnitude, the differences of the absolute values of A and B in Fig. 6c have overall become smaller in magnitude. In Fig. 6c, there is a wider area with positive values over the Antarctic region. Also, the negative values over the 30°–60°S belt are not as large in magnitude as in Fig. 5c.
As in Fig. 5, but with the samples from the 12 months of the 70 years (12 × 70 samples).
Citation: Journal of Climate 34, 12; 10.1175/JCLI-D-20-0615.1
Similar situations appear in the 850–925-hPa layer (Figs. 6g–i). Compared with Fig. 5g, the negative values in Fig. 6g have become larger over the North American Cordillera, Greenland, and especially the Tibetan Plateau and the Antarctic region. Over these regions, corresponding to the increase in magnitude of the negative value of coefficient A, coefficient B has become decreased in magnitude. As a result, in Fig. 6i, the differences of the absolute values of A and B have smaller negative or positive values over these regions.
In the 600–700-hPa layer, we notice that the Tibetan Plateau is a special area. Coefficient A has large negative value there (Fig. 6d). This may reflect the temperature or thermal difference of the plateau between winter and summer. The coefficient B has a much larger negative value over the area (Fig. 6e), which may reflect the difference in the stable and convective condition of the atmosphere over the plateau between winter and summer. The difference of the absolute values of A and B in Fig. 6f is thus slightly negative over the Tibetan Plateau. These will also be further studied.
c. Sampling from a spatial variation
We may also examine the difference among the grid points over the globe, and analyze the dominance in the spatial variation of the static stability Z. The dominance analysis is performed for each year and for each layer. Figure 7 shows the coefficients A and B of January and July of all the years at the three layers. It is shown that in the tropospheric layers of 600–700 hPa (Figs. 7c,d) and 850–925 hPa (Figs. 7e,f), the spatial variation of Z is almost totally decided by the spatial difference of Y, the vertical difference of temperature (i.e., the vertical gradient of temperature). The influence from the spatial variation of the temperature is very small. This is the case for both January and July and for all the years.
The coefficients A (red) and B (blue) for (top) January and (bottom) July of each year and for the (a),(b) 10–20-, (c),(d) 600–700-, and (e),(f) 850–925-hPa layers. The normalized regressions are performed with the samples from all the grid points over the globe (144 × 73 samples). For the 10–20-hPa layer in (a) and (b), |B| is also plotted (green).
Citation: Journal of Climate 34, 12; 10.1175/JCLI-D-20-0615.1
However, in the stratospheric layer of 10–20 hPa (Figs. 7a,b), in addition to the spatial variation of the vertical gradient of temperature, the variation of temperature is also important to the spatial variation of Z. It is also shown that the coefficients A and B both have quasi-biennial oscillations over the 70 years. Chen et al. (2005) analyzed the quasi-biennial oscillation of the static stability in stratosphere and pointed out that it might be caused by the change in the temperature field, triggered by the quasi-biennial oscillation of the zonal wind in stratosphere.
Over the 70 years, the interannual variations of the coefficients A and B of January (Fig. 7a) are fairly stable. However, for July (Fig. 7b), the two coefficients both have a decreasing tendency in magnitude. Moreover, in the recent 40 years, coefficient A is greater than coefficient B, in absolute value, and this is true for every year (Fig. 7b). To understand the behaviors of the two coefficients among the stratospheric and tropospheric levels in Fig. 7 and to better discern the gradual changes in the vertical direction, their vertical profiles are further examined.
Figure 8 presents the profiles for January and July of 2017. In the lower-tropospheric levels below 700 hPa, the atmosphere can largely be affected by Earth’s surface. The sensible and latent heat fluxes from the surface may have substantial effects on the temperature as well as the lapse rate of temperature in the boundary layer. The result is that the static stability is dominated by the vertical difference of temperature, and the variation of the temperature itself has very little contribution. In the middle levels between 700 and 300 hPa, the static stability may be influenced by the dynamic and thermodynamic processes in the free atmosphere. While the vertical difference of temperature still dominates the static stability, the spatial variation of the air temperature may also have certain contribution. From 300 hPa through the tropopause to 50 hPa, while the negative value of the B does not change much, coefficient A becomes positive and increases rapidly with the height. In the upper stratosphere from 50 to 15 hPa, there is a very sharp increase in the value of A. At 15 hPa, the value of the A becomes even greater than the absolute value of B, suggesting that at this height the spatial variation of the temperature can dominate the variation of the static stability. Comparably, the atmospheric convection, or the vertical difference of temperature, is weak. Previous studies revealed that the atmosphere in the stratosphere, which has less direct influence from Earth’s surface, may be affected by the solar radiation and the chemistry–climate interaction (e.g., Birner 2010; Grise et al. 2010).
The profiles of the coefficients A (black) and B (gray). The normalized regressions are performed with the samples of (a) January and (b) July 2017 from all grid points over the globe (144 × 73 samples).
Citation: Journal of Climate 34, 12; 10.1175/JCLI-D-20-0615.1
5. Summary and discussion
Static stability Z and buoyancy frequency N are fundamental concepts in ocean and atmospheric dynamics. The static stability Z can be negative, indicating that the atmosphere is unstable, which may lead to convection. When Z is positive, meaning that the atmosphere is stable, with a perturbation, the air parcel forced by the gravity and the buoyancy restoring force can oscillate vertically around its initial position. From the derived wave equation, the frequency of the oscillation (i.e., the buoyancy frequency N) is the square root of the static stability, thus Z = N2. Because of the buoyancy in the stable stratified atmosphere, the horizontal propagations of the vertical oscillation of the air parcel can form gravity waves, which may influence the weather in the downstream regions. Static stability and buoyancy frequency are presently provided as output data in the climate models (e.g., Neale et al. 2012).
The major task of this study is to derive the expression that shows that the static stability Z can be influenced by the vertical difference of temperature (i.e., the vertical temperature gradient) as well as the temperature itself, and then analyze the relative importance of the two influencing quantities in the temporal and spatial variations of the stability. Since the reanalysis data we used as example to calculate the static stability are from the model and the vertical resolution is relatively low, the spatial distributions of the static stability calculated with the “instantaneous” data and the “averaged” data are presented and compared to illustrate the reasonability of their patterns; that is, the static stability can vary geographically with the latitude, ocean/land, and topographical conditions.
The distributions of the static stability Z calculated with the 4 times daily data, which can be regarded as an “instantaneous” state, show that there are some areas where Z can have negative values, and thus the atmosphere is unstable and convection may occur there. Over the remaining majority of regions in the globe, Z has positive values, meaning that the atmosphere is stable. As mentioned above, the perturbations of the air parcel in the stable stratified atmosphere may lead to gravity waves through the propagation of the vertical oscillation. In the plots of Z calculated with the daily data, we can still find some areas where the atmosphere is unstable, although the areas are smaller in size. When calculated with monthly data, which reflects a more “averaged” state of the atmosphere, the values of the static stability Z are all positive across the entire field. The horizontal distribution patterns of the static stability for all the three temporal scales suggest that although it varies greatly from place to place, the static stability might be influenced by a variety of geographic factors combined, including the latitude and height, whether it is over ocean or land, and whether it is over a coastal or mountain region. The overall effect of these geographic influences can finally be represented by the physical quantities that appear in the formula of the Z.
The vertical structure of the static stability is related to the lapse rate of temperature or the vertical temperature gradient. With ΔT being the difference of the lower level from the adjacent upper level, the static stability Z decreases with ΔT. For the troposphere, there are three situations. One is that ΔT is positive and sufficiently large, with the lapse rate of temperature being greater than the dry adiabatic lapse rate, and then Z may gain negative values, which means that the atmosphere is unstable and convection may occur. The second is that ΔT is positive and small; then Z may obtain small positive values, meaning that the atmosphere can be weakly stable. The third is that ΔT becomes negative, with a temperature inversion in the troposphere; then Z can receive large positive values, and thus the atmosphere can be very stable. In the stratosphere, similar to the inversion in troposphere, temperature increases with height, and thus ΔT is negative. With Z attaining large positive values, the atmosphere is also very stable.
Derivations show that while it links to the vertical difference of temperature, which is generally emphasized, the static stability also relates to the temperature itself. The conventional expression of the static stability Z can thus be converted into a nonlinear function of the quantity X, which reflects the influence from the temperature, and the quantity Y, which measures the vertical difference of temperature. Whether the variation of temperature, or X, is important to the variation of Z needs to be assessed, and should be compared with the effect of the variation in the vertical difference of temperature. A method is required for the comparison.
For the nonlinear issue, to make it convenient to estimate the contributions of the two influencing quantities and assess their relative importance, we use a linear regression to fit the relation. To compare the changes of the different quantities, before the fitting, all quantities are normalized with the data. Tests show that the linear fitting is perfect for the different samples, which may come from both the temporal and spatial variations. Because of the normalizations, the absolute values of the two coefficients obtained from the linear fitting can be used as the measures for estimating the relative contributions of the two influencing quantities.
The results of the dominance depend on the data examined. In this study, data are sampled from two temporal variations and one spatial variation. The results of the dominance for the interannual and seasonal variations show that their spatial distributions are overall very similar. In lower troposphere, it is true that the vertical difference of temperature is very important, and it dominates the variation of the static stability over the entire field. In the high latitudes of the two hemispheres, the effect of temperature is also important. In the stratosphere, the change of temperature can even dominate the interannual variation of Z over the Antarctic region. In the remaining regions, although the vertical difference of temperature dominates the variation, the change of temperature is also important.
For the spatial variations consisting of all the grid points in the globe, the vertical difference of temperature dominates the variation of static stability in lower levels, and the contribution from temperature is small. This is the case for every year. In the stratosphere, temperature has positive contributions to the spatial variation of static stability in every year, corresponding to the negative contributions from the vertical difference of temperature. Both of them vary greatly from year to year. Overall, they can balance with each other. The result for July is interesting. The two contributions both show declining tendency over the 70 years. In each of the recent 40 years, temperature dominates the spatial variations of the static stability.
In sum, the purpose of this study is to provide a novel understanding of the temporal–spatial variations of the static stability and the physical quantities that dominate the variations. The static stability is typically expressed with the potential temperature. Considering temperature as a fundamental climate quantity, here we express the static stability in terms of the temperature. Our derivations show that in the obtained formula, while there is the vertical temperature gradient, there also appears the temperature itself. For the static stability of a single moment, vertical temperature gradient is important; it is the mechanism responsible for the sign and value of the static stability. However, when we examine the temporal–spatial variations of the static stability, the effect of the temperature may become important. It is possible that during a variation of static stability, the vertical temperature gradient remains constant or does not change much, while relatively the temperature varies greatly. In this study, we treat temperature and the vertical difference of temperature, which represents the gradient, as two influencing quantities, and explore which of them dominates the variation of the static stability. A quantitative method designed in our previous studies is utilized, and the results obtained are reasonable. As mentioned above, we truly find the circumstances in which temperature dominates the variation of the static stability. These include the interannual and seasonal variations over the Antarctic region in the stratosphere, as well as the spatial variation in the stratosphere during the recent 40 years. For the variations of static stability in the troposphere, temperature may also have contributions.
The static stability examined in this study is for dry air. In our future work, we will continue to examine the moist air. Through introducing a moisture quantity, we may assess the dominance among the humidity quantity and the temperature-related quantities. More reanalysis datasets (e.g., JRA-55, MERRA-2, and ERA5), along with available observed data, will be used to illustrate whether the vertical difference of temperature may contain errors and whether the results of static stability from different datasets may display uncertainties. Dominance can also be explored between the static stability and the dynamic stability, which is generally linked to the motions and the wind shears, when the atmosphere is related to different conditions (e.g., the mesoscale system, synoptic weather, and large-scale motion). Case studies can be conducted for some key regions. For example, over the Tibetan Plateau in the troposphere, the thermal effect may influence both the temperature and the convection. Over high-latitude regions, the temperature can exhibit very large annual ranges. In the free atmosphere, temperature and convection may both be significantly influenced by the atmospheric circulation. In the stratosphere, although the negative lapse rate of temperature can help the stability gain large values, the variations of the static stability may sometimes be dominated by the change of the temperature, which might be affected by the radiation and chemistry processes.
Acknowledgments
This study was supported by the National Natural Science Foundation of China (Grant 41991281), the National Key Research and Development Program of China (Grant 2018YFC1507704), and the Priority Academic Program Development of Jiangsu Higher Education Institutions (PAPD). The reviewers and the editor are thanked for their helpful comments and suggestions. The NCEP/NCAR reanalysis data were provided by the NOAA/OAR/ESRL PSL, Boulder, Colorado, USA, from their website at https://psl.noaa.gov/.
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