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Before starting, we note that the variable of MLD works throughout Eq. (2). To explore the variation of MLD in the Niño-3.4 region, this paper performs a 40-year analysis.
Results based on the data derived from GODAS reveal that the annual mean of MLD is found to vary from 30 to 110 m (Fig. 1a), accompanied by a decreasing gradient from west to east while being shallower on the equator than off the equator. In addition, the MLD fluctuates with time as well (Fig. 1b). Here, it is necessary to pay attention to the impact of MLD variations, as its spatial and temporal differences have been ignored. The MLD was defined at 50 m in the Niño-3.4 region according to previous studies (Ren and Wang, 2020; Chen and Jin, 2022). Next, we will take the SST as an example to compare the differences in adopting distinct MLD selection criteria.
Figure 1. (a) The annual mean MLD in the Niño-3.4 region during 1981–2020 (units: m). The contour interval is 5 m for MLD. (b) The area-averaged anomalous MLD in the Niño-3.4 region during 1981–2020 (units: m). Three super ENSO events of 1982–83, 1997–98, and 2015–16 are shaded, where red ones represent the El Niño events and blue shades are La Niña events, respectively. A 3-month running mean has been applied in both plots.
Figure 2 displays the time evolution of the area-averaged SST anomaly (SSTA), which was calculated with the variable MLD and fixed at 50 m in the Niño-3.4 region compared to the SSTA at 5 m during 1981–2020, respectively. It is obvious to see that
$ T'_{\_50} $ has a certain deviation from$T'_{\_5} $ , as this pattern can be found in three super ENSO events as well. Meanwhile, the root-mean-square error (RMSE) between$ T'_{\_{\text{MLD}}} $ and$ T'_{\_5} $ is much smaller than the one between$ T'_{\_50} $ and$ T'_{\_5} $ . Hence, considering the variation of MLD in the diagnostic process is more capable and accurate in characterizing the real situation of the surface. Note that, the analysis of the results of ocean heat balance is not sensitive to the selection criteria for the location of the bottom of the mixed layer (Huang et al., 2010). The selection criterion from GODAS is based on a buoyancy difference of 0.03 kg m–3 with the sea surface. It has been found that either temperature differences or density differences are alternatives used to reveal the variation of MLD (Kara et al., 2000; Thomson and Fine, 2003; Huang et al., 2010).Figure 2. (a) Comparison between
$T'_{\_5}$ (averaged SSTA in the Niño-3.4 region at 5 m, blue solid line) and$T'_{\_50}$ (area averaged SSTA from sea surface to the fixed MLD at 50 m in the Niño-3.4 region, red dotted line). (b) Comparison between$T'_{\_5}$ and$T'_{{\text{\_MLD}}}$ (area averaged SSTA from sea surface to the varying MLD bottom in the Niño-3.4 region, red dotted line). Shaded bars represent three super ENSO events of 1982–83, 1997–98, and 2015–16, where red ones denote the El Niño events and blue shades are La Niña events, respectively. A 3-month running mean has been applied to both plots. -
The previous subsection discussed the spatial and temporal differences of MLD over the past 40 years and the advantages of considering its variability. After that, the annual mean of the mixed layer heat budgets in the Niño-3.4 region of each term in Eq. (2) is discussed.
It is apparent that the strong heating by
${Q_q}$ (Fig. 3e) acts as the main positive contributor across the Niño-3.4 region with maximum amplitude over 2°C month–1 near 120°W. There is an increasing gradient of${Q_q}$ from west to east owing to the influence of surface radiation. That is because a cold tongue is established in the east and a warm pool in the west over the equatorial Pacific under the climatic background. Due to the effects of the Walker Circulation, the eastward sea surface receives more solar shortwave radiation than the west, accompanied by less outward longwave radiation due to the cooler sea surface in the east. The positive contributions of annual mean surface heat flux caused by radiation (${Q_q}$ ) and zonal advection (${Q_u}$ ) (Fig. 3a) along the equator are balanced by several cooling processes from meridional advection (${Q_v}$ ) (Fig. 3b), vertical diffusion ($ Q_{zz} $ ) (Fig. 3d), and vertical entrainment ($ Q_{w} $ ) (Fig. 3c) to a substantial extent. Typically,${Q_u}$ plays a negative role within 2° of equator while the maximum cooling centers of${Q_v}$ exist near 2°N with magnitudes of over 1°C month–1 and amplitudes of over 0.5°C month–1 near 2°S, close to the eastern Pacific. In contrast, cooling centers from${Q_w}$ and${Q_{zz}}$ are primarily concentrated near the equator (Figs. 3d, e). Although both of them are dynamic processes in the vertical direction,${Q_{zz}}$ has a greater cooling effect at a rate of 1°C month–1 approaching 120°W than${Q_w}$ . At the same time,${Q_{zz}}$ goes along with a broader meridional extension than${Q_w}$ as well. As for the vertical entrainment process (Fig. 3c), it is a narrow strip constrained within 3° of the equator, mainly resulting from the upwelling caused by Ekman transport.Figure 3. Annual mean of the mixed layer heat budgets for (a)
${Q_u}$ , (b)${Q_v}$ , (c)${Q_w}$ , (d)${Q_{zz}}$ , and (e)${Q_q}$ in the Niño-3.4 region during 1981–2020 (units: °C month–1). Positive areas are surrounded by solid lines and negative areas are circled by dotted lines. Irregular contour intervals are 0, ±0.2, ±0.5, ±1, ±1.5, ±2, ±2.5, and ±3.5°C month–1. A 3-month running mean has been applied to all plots. -
This subsection focuses on the seasonal cycle from the right-hand terms of the mixed layer heat budget equation at the equator (0°). As shown in Fig. 4, the forcing terms contributing to the variation of temperature tendency present noticeable seasonal cycles in the central to eastern Pacific.
${Q_u}$ produces a cooling effect from July to March and makes a positive contribution from April to June near east of 150°W with the maximum positive and negative amplitude appearing around May and September (Fig. 4a). The heating by zonal advection generally occupies the central Pacific all year round. According to the previous statistical analysis (Wang and McPhaden, 1999, 2000, 2001), the southward current in the central Pacific is salient throughout most of the year, while the northward flow near the eastern Pacific tends to peak from May to December. Thus, the seasonal variation of${Q_v}$ only weakly contributes to the temperature tendency (Fig. 4b). It is easy to capture a prominent semiannual cycle in${Q_q}$ due to the regulation from solar shortwave radiation with the maximum amplitude occurring in the boreal spring and the second-largest magnitude taking place in September close to 120°W (Fig. 4d). The heating effect from${Q_q}$ is quite intensely inclined eastward, which is the result of a shallower MLD and a relative decrease in cloudiness inherent to the eastern Pacific. Evidently,${Q_q}$ makes the most critical positive contribution to the mixed layer temperature tendency at a rate of 0°C–2.5°C month–1, noting that this is widely balanced by a cooling effect of terms of${Q_w}$ and${Q_{zz}}$ (Fig. 4c). Seasonal cycles for both of the vertical processes are negative throughout the year followed by two maximum magnitudes exceeding 2°C month–1 that appear during boreal spring and autumn. The cooling sources from${Q_w} + {Q_{zz}}$ are more pronounced further east in the equatorial Pacific all year round, where the MLD distribution is relatively shallow with a larger SST vertical gradient as well.Figure 4. Seasonal cycles of the mixed layer heat budgets for (a)
${Q_u}$ , (b)${Q_v}$ , (c)${Q_w} + {Q_{zz}}$ , and (d)${Q_q}$ in the equatorial Pacific (0°) during 1981–2020 (units: °C month–1). Irregular contour intervals are 0, ±0.2, ±0.5, ±1, ±1.5, ±2, ±2.5, and ±3.5°C/month. A 3-month running mean has been applied to the plots. -
Based on the RO theory (Jin, 1997), two approaches to investigate the SST growth rate in the Niño-3.4 region were outlined in Eqs. (5) and (7). Growth rates estimated by linear regression utilizing the anomalous mixed layer temperature tendency and external forcing terms are performed by considering the variation in MLD. Generally speaking, both
$ T'_t\_R $ and$ {F'}({\text{varying MLD}})\_R $ exhibit similar characteristics (Figs. 5a, b), i.e., the positive growth rate peaks in September and the most negative one occurs in the early spring, which is reasonably consistent with previous studies (Li, 1997; Stein et al., 2010; Kim and An, 2021; Chen and Jin, 2022). However, there is a bias between growth rates calculated by forcing terms with fixed and varying MLD. The seasonal cycle of$ {F'}({\text{fixed MLD)}}\_R $ tends to be the strongest during the boreal autumn still, but the weakest growth rate shifts to May with indistinguishable differences in the four former months (Fig. 5c). This indicates that it is of great importance for considering the variation of MLD as a factor in diagnosing and analyzing the SST growth rate, a consideration that has been neglected by previous researchers (Boucharel et al., 2015; Ren and Wang, 2020; Chen and Jin, 2022).Figure 5. Curves with dots depict the seasonal cycle of the SST growth rate in the Niño-3.4 region diagnosed from (a) the anomalous mixed layer temperature tendency
$ T'_t\_R $ , (b) the anomalous dynamic and thermodynamic processes with a variable MLD$ {F'}({\text{varying MLD}})\_R $ , and (c) the anomalous dynamic and thermodynamic forcing terms with a fixed MLD$ {F'}({\text{fixed MLD}})\_R $ (units: month–1). Blue shading indicates the 95% confidence interval in (a) and the 80% confidence interval in (b) and (c). A 3-month running mean has been applied to the plots.Subsequently, as the forcing term in Eq. (4) is composed of five processes, the contribution of each term to the growth rate can be obtained (Fig. 6a). Taking
$ Q'_u $ as an example, the corresponding contribution of$ Q'_u\_R $ can be measured through linear regression by replacing$ {F'} $ in Eq. (5). As shown in Fig. 6a, the contribution calculated by$ Q'_v $ ($ Q'_q $ ) is always positive (negative) throughout the year. The variation of$ Q'_{zz} \_R $ is opposite to the growth rate indicating that$ Q'_{zz} $ essentially makes a negative contribution and that the magnitude of$ Q'_w\_R $ is the smallest among the five terms. Meanwhile, it is not hard to tell that$ Q'_u \_R $ is the closest to the growth rate in terms of its amplitude and variation trend.Figure 6. (a) The SST growth rates diagnosed from the anomalous forcing term
$ {F'} $ (Growth Rate, blue curve with dots) with its decomposition into$ Q'_u $ ($ Q'_u\_R $ , red curve),$ Q'_v $ ($ Q'_v\_R $ , green curve),$ Q'_w $ ($ Q_w'\_R $ , purple curve),$ Q'_{zz} $ ($ Q'_{zz}\_R $ , grey curve), and$ Q'_q $ ($ Q'_q \_R $ , orange curve). (b)$ Q'_w $ ($ Q'_w \_R $ , red curve) with its decomposition, EK ($ {\text{EK}}\_R $ , green curve), TH ($ {\text{TH\_}}R $ , grey curve), and nonlinear processes ($ {\text{Nonlinear}}\_R $ , orange curve). (c) Decomposition of$ Q'_q $ , longwave radiation ($ {\text{Longwave rad}}\_R $ , red curve), shortwave radiation ($ {\text{Shortwave rad}}\_R $ , green curve), sensible heat ($ {\text{Sensible heat}}\_R $ , purple curve), latent heat ($ {\text{Latent heat}}\_R $ , grey curve), and penetrative shortwave radiation ($ Q'_{pen} \_R $ , orange curve). (d)$ Q'_u $ ($ Q'_u\_R $ , red curve) with its decomposition, ZA ($ {\text{ZA}}\_R $ , green curve),$ - \overline {{u_a}} {{\partial T'_a }}/{{\partial x}} $ ($ - \overline u {T'}\_R $ , grey curve), and nonlinear term ($ {\text{Nonlinear}}\_R $ , orange curve). A 3-month running mean has been applied in each diagnosing process.As can be readily seen, the development of
$ Q'_v\_R $ and$ Q'_{zz} \_R $ are quite similar, with both of their processes characterized by initial growth and subsequent decay. On the contrary,$ Q'_u\_R $ and$ Q'_q\_R $ resemble a seasonal trend of first rising and then decreasing. From December to July, the contribution of$ Q'_q $ is evidently greater than that of$ Q'_u $ during the negative SST growth rate. In the period characterized by the positive development of growth rates,$ Q'_v $ and$ Q'_u $ are roughly comparable between August and November. Quantitatively, the primary and secondary contributions to the seasonal cycle of growth rate in each calendar month are outlined in Table 1.Jan Feb Mar Apr May Jun Jul Aug Sep Oct Nov Dec Primary $ Q_q' $ $ Q_q' $ $ Q_q' $ $ Q_q' $ $ Q_q' $ $ Q_q' $ $ Q_q' $ $ Q_v' $ $ Q_u' $ $ Q_u' $ $ Q_v' $ $ Q_q' $ Secondary $ Q_u' $ $ Q_u' $ $ Q_u' $ $ Q_u' $ $ Q_u' $ $ Q_u' $ $ Q_{zz}' $ $ Q_u' $ $ Q_v' $ $ Q_v' $ $ Q_u' $ $ Q_{zz}' $ Table 1. The primary and secondary contributions of forcing terms in each calendar month for the seasonal cycle of SST growth rate.
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We have learned that the contribution of
$ Q'_w $ displays the smallest amplitude among the five feedback processes; however, it contains two important vertical dynamic processes: the thermocline and Ekman feedback. It is possible that contributions from these processes largely offset each other, resulting in the small magnitude of$ Q'_w\_R $ . The influence of both processes upon growth rate is dissected in the following discussion. Decomposition from$ Q'_w $ consists of three components (Fig. 6b), namely the thermocline feedback (TH), Ekman feedback (EK), and nonlinear processes. It is easy to tell that amplitudes of all these terms are fairly small and that the contributions from both EK and the nonlinear term impact the development of the growth rate only slightly between February to May. Besides, these results rule out the speculation of these two vertical dynamic terms offsetting each other as previously conjectured.In quantifying the contribution in Eq. (7), it is clear that
$ Q'_q $ occupies a prominent position in the phase where the growth rate is negative. To analyze the influence of this thermodynamic process, the decomposition of$ Q'_q $ is examined. As shown in Fig. 6c,$ Q'_q $ includes five components, namely shortwave radiation, longwave radiation, latent heat, sensible heat, and shortwave radiation penetrating through the bottom of the mixed layer. It can be observed that shortwave radiation accounts for the largest share (Fig. 6c) and shows a characteristic similar to that of$ Q'_q $ , highlighting the significant contribution of solar radiation to the growth rate. There is a remarkable semiannual cycle of shortwave radiation flux (Fig. 7a), with two heating centers corresponding to spring and autumn, which is closely related to the seasonal shift of ITCZ. It is found that$ {\text{Shortwave rad}}\_R $ mainly drives the growth rate during the spring and summer, with its intensity rapidly decaying soon thereafter. This occurs even though more solar radiation is received during autumn in the Niño-3.4 region, and can be understood through the balance that results from the strong cooling attributed to other thermodynamic processes, resulting in a smaller intensity at that time (Fig. 7). Therefore, the contribution of$ Q'_q $ to the growth rate is highlighted in the phase between spring and summer.Figure 7. Heat fluxes in the equatorial Pacific (0°) for (a) downward shortwave radiation, (b) downward longwave radiation, (c) sensible heat, (d) latent heat, and (e) penetrative shortwave radiation during 1981–2020 (units: W m–2). Contour intervals (C.I.) are (a) 10, (b) 2, (c) 10, and (d) 1 W m–2. A 3-month running mean has been applied to the plots.
As mentioned earlier,
$ Q'_u\_R $ is marked to be the optimal metric, either in trend or strength, to approach the growth rate compared to the remaining four items. Therefore, it is important to figure out the basic mechanism by which$ Q'_u $ operates on the growth rate. Through its decomposition,$ Q'_u $ consists chiefly of three components,$ - u'_a({{\partial \overline {{T_a}} }}/{{\partial x}}) $ , which is the ZA,$ - \overline {{u_a}} ({{\partial T'_a}}/{{\partial x}}) $ , and nonlinear processes, and the relative contributions corresponding to each item in$ Q'_u $ are placed in Fig. 6d. It can be seen that$ - \overline u {T'}\_R $ oscillates around zero with the smallest amplitude throughout the year. The contribution obtained from the nonlinear term has an opposite trend to the evolution of$ Q'_u\_R $ , with a negative contribution to the development of the growth rate from January to June, diminishing to practically zero in the latter part of the year. In contrast, ZA accounts for the largest proportion of$ Q'_u $ , which conveys the meaning that ZA plays a dominant role in the seasonal cycle of growth rate. Therefore, the interpretation of the ZA turns out to be key for unraveling the development of the SST growth rate.As indicated by the expression of ZA,
$ \overline {{T_a}} $ , which manifests the variation of the climatic background, determines the strength of this feedback. Figure 8 shows the Hovmöller diagram of$ {{\partial \overline {{T_a}} }}/{{\partial x}} $ ; it can be seen that this parameter is consistently negative for the whole year in the Niño-3.4 region with the maximum and minimum occurring in the boreal spring and near September, respectively. Many studies have revealed that the seasonal variation of SST considerably depends on the meridional migration of ITCZ (Philander, 1983; Xie and Philander, 1994; Tziperman et al., 1997; Xie et al., 2018). As the most characterized seasonal evolution in the tropical Pacific, the ITCZ exerts a large influence on the air-sea coupled system instability. There is a clear correspondence between the seasonal distribution of the ITCZ and its main rainbands as well as the associated warmer SST (Xie et al., 2018; Fang and Xie, 2020). The seasonal variation in Fig. 9a illustrates that the southernmost approach of the ITCZ is near 2°S in February while it marches northward to its farthest extent from the equator in boreal autumn. It is worth noting that the correlation between the seasonal evolution of ITCZ and SST growth rate is markedly significant, both of which present a minimum in boreal spring and a maximum occurring around September, suggesting that the SST growth rate variation is markedly controlled by the seasonal migration of ITCZ.Figure 8. The Hovmöller diagram of
$ {{\partial \overline {{T_a}} }}/{{\partial x}} $ (units: °C deg–1) averaged between 5°N–5°S from 1981 to 2020. The contour interval is 0.1°C deg–1 for$ {{\partial \overline {{T_a}} }}/{{\partial x}} $ . A 3-month running mean has been applied to the plot.Figure 9. (a) Seasonal immigration of ITCZ in the central to eastern Pacific (170°–120°W) during 1981–2020. (b) Seasonal evolution of
$ - {{\partial \overline {{T_a}} }}/{{\partial x}} $ in the Niño-3.4 region during 1981–2020 (units: °C deg–1). A 3-month running mean has been applied to both of the plots.A brief explanation of mechanisms that potentially affect the modulation of
$ {{\partial \overline {{T_a}} }}/{{\partial x}} $ by the ITCZ follows. As the most salient seasonal movement in the tropical region, the ITCZ is at its southernmost extent in the spring and is accompanied by weak trade winds and westward currents, resulting in the accumulation of more warm water volume to the east tending to maximize around the eastern Pacific. As its location marches gradually northward, the land placed north of the equator becomes a strong heat source related to geographic asymmetries in the tropical Pacific, which leads to strengthening the trans-equatorial flow and southeastward wind resulting in additional warm water transport to the west. Soon afterward, the ITCZ stretches over 6°N in the autumn with a perfectly set up cold tongue near the central to eastern Pacific through several effects including the intense southeasterly trade winds, oceanic evaporation, and latent heat release caused by the wind-driven surface currents and upwelling. Although the received solar shortwave radiation is enhanced along the eastern tropical region during the autumnal equinox, Liu et al. (2005) indicated that the contributions of those cooling processes are far greater than the heating effects from the radiation. It follows that when these factors are coupled together they readily bring the coolest SST during the annual cycle. With the movement of the subsolar point, the ITCZ shifts equatorward causing the eastern Pacific warm again. Accordingly, the tropical Pacific background state repeatedly circulates and is governed by the seasonal migration of ITCZ. This process is of great importance for the SST variation and ZA which plays a dominant role in regulating the seasonal cycle of growth rate. As displayed in Fig. 9b, the seasonal development of ZA has a proximate trend with either the ITCZ or growth rate, all of which tend to be at a minimum during spring before peaking in autumn. However, due to some nonlinear effects, the timing for the minimum of ZA and growth rate do not exactly match, but in general, it can still explain the variation pattern for the seasonal cycle of SST growth rate. -
As mentioned by Huang et al. (2010), the mixed layer temperature heat budgets can be described as
where F consists of five terms, zonal advection [
$ {Q_u} = - {u_a}({{\partial {T_a}}}/{{\partial x}}) $ ], meridional advection [$ {Q_v} = - {v_a} ({{\partial {T_a}}}/ {{\partial y}}) $ ], vertical entrainment [$ {Q_w} = - w({{\partial {T_a} - \partial {T_{{\text{MLD}}}}}})/{{\partial z}} $ ], vertical diffusion [$ {Q_{zz}} = - ({{{\mathrm{Ec}}}}/{{{\text{MLD}}}})({{\partial {T_a} - \partial {T_{{\text{MLD}}}}}})/{{\partial z}} $ ], and adjusted surface heat flux caused by radiation ($ {Q_q} $ ). Variables with the subscript a denote the vertical average from the sea surface to the mixed layer bottom, and the subscript MLD is the mixed layer depth. The subscript MLD indicates a certain value of the variable at the bottom of the mixed layer.$ T $ ,$ u $ , and$ v $ represent the temperature and horizontal ocean currents. Since the vertical processes of$ {Q_w} $ and$ {Q_{zz}} $ are difficult to capture and treated as residual terms at some point, these can be characterized through parameterization. Here,$ w $ in$ {Q_w} $ is derived from:Other than that,
$ {\mathrm{Ec}} $ is calculated by:where Ri in Eq. (A4) is the Richardson number described as:
note that Ri is confined with the minimum value of –0.1, to ensure a rational diagnosis.
$ \lambda $ in Eq. (A4) and$ \delta $ Eq. (A5) are given by:and
respectively.
The last term
$ {Q_q} $ is defined asin which
$ {Q_t} $ is the total downward heat flux,$ {Q_s} $ represents the shortwave radiation, and$ \rho $ and$ {c_p} $ are the water density and heat capacity, respectively.The constant parameters from Eq. (A4) to (A8) are summarized in Table A1. Simple physical interpretations and specific algorithms of each dynamic and thermodynamic process are displayed in Table A2 as well.
Parameters Values $ {E_z} $ 1 × 10–5 $ {{\text{m}}^{\text{2}}}{\text{ }}{{\text{s}}^{{{ - 1}}}} $ $ \alpha $ 5 $ g $ 9.8 $ {\text{m }}{{\text{s}}^{{{ - 2}}}} $ $ \varepsilon $ 8.75 × 10–6 $ \rho $ 103 kg m–3 $ {c_p} $ 4.2 × 103 J kg–1 °C $ {\lambda _0} $ 1 × 10−4 $ {{\text{m}}^{\text{2}}}{\text{ }}{{\text{s}}^{{{ - 1}}}} $ $ {\lambda _1} $ 3.5 × 10–3 $ {{\text{m}}^{\text{2}}}{\text{ }}{{\text{s}}^{{{ - 1}}}} $ Table A1. Constant parameters and corresponding values from Eqs. (A4) to (A8).
Term Physical interpretation Specific algorithm $ {Q_u} $ zonal advection $ - {u_a}({{\partial {T_a}}}/{{\partial x}}) $ $ {Q_v} $ meridional advection $ - {v_a}({{\partial {T_a}}}/{{\partial y}}) $ $ {Q_w} $ vertical entrainment $ - w({{\partial {T_a} - \partial {T_{{\text{MLD}}}}}})/{{\partial z}} $ $ {Q_{zz}} $ vertical diffusion $ - ({Ec}/{{\text{MLD}}})({\partial {T_a} - \partial {T_{{\text{MLD}}}}})/{{\partial z}} $ $ {Q_q} $ adjusted heat flux by radiation $ {Q_t} - {Q_s}(0.58{e^{\frac{{ - {\text{MLD}}}}{{0.35}}}} + 0.42{e^{\frac{{ - {\text{MLD}}}}{{23}}}})/\rho {c_p}{\text{MLD}} $ $ {T_t} $ temperature tendency $ {{\partial {T_a}}}/{{\partial t}} $ TH thermocline feedback $ - \overline w ({{\partial T_a' - \partial T_{{\text{MLD}}}'}})/{{\partial z}} $ EK Ekman feedback $ - {w'}({{\partial{ \overline {T_a'}} - \partial \overline {T_{{\text{MLD}}}'} }})/{{\partial z}} $ ZA zonal advective feedback $ - u_a' ({{\partial \overline {{T_a}} }}/{{\partial x}} )$ Table A2. Simple physical interpretations and specific algorithms of each dynamic and thermodynamic process.
Eq. (A2) is written by removing the climatology in each term as follows:
where the Eddy term has been neglected.
Here,
and
In addition,
and
Jan | Feb | Mar | Apr | May | Jun | Jul | Aug | Sep | Oct | Nov | Dec | |
Primary | $ Q_q' $ | $ Q_q' $ | $ Q_q' $ | $ Q_q' $ | $ Q_q' $ | $ Q_q' $ | $ Q_q' $ | $ Q_v' $ | $ Q_u' $ | $ Q_u' $ | $ Q_v' $ | $ Q_q' $ |
Secondary | $ Q_u' $ | $ Q_u' $ | $ Q_u' $ | $ Q_u' $ | $ Q_u' $ | $ Q_u' $ | $ Q_{zz}' $ | $ Q_u' $ | $ Q_v' $ | $ Q_v' $ | $ Q_u' $ | $ Q_{zz}' $ |