Why do the KOE-SSTA events with larger mature-to-decaying transition rates have a greater possibility of yielding a more significant SPB? What is the relationship between the transition rate of SSTA events and the error growth rate associated with the SPB? To address these issues, we explore which physical processes are responsible for the error growth associated with the SPB and compare their differences between the KOE-SSTA events in the two categories.
(Duan and Wu, 2015) explored the physical mechanisms responsible for the SPB through mixed-layer heat budget analysis based on the equation governing the mixed-layer temperature, which is a good proxy for SST. The equation can be expressed as: \begin{eqnarray} \dfrac{\partial T}{\partial t}=\dfrac{Q}{\rho c_pz}-\left(u\dfrac{\partial T}{\partial x}+v\dfrac{\partial T}{\partial y}\right)- \Delta Tw-\Delta T \dfrac{\partial z}{\partial t}-\Delta Tr+R . \ \ (4)\end{eqnarray} On the right-hand side of Eq. (4), Q in the first term represents the net sea surface heat flux; ρ,cp and z are respectively the density of sea water, the specific heat capacity and the mixed-layer depth, which is defined as the layer depth where the sea temperature is 0.5°C less than the SST. The second term, -[u(∂ T/∂ x)+v(∂ T/∂ y)], is the horizontal advection by zonal velocity u and meridional velocity v. In the third term, -∆ Tw, ∆ T=(T-T(-z))/z is the entrainment due to the vertical velocity w. The vertical advection induced by the Ekman pumping is one component of this entrainment term because the Ekman pumping is the vertical velocity induced by the wind-stress curl and w implicitly includes the vertical velocity due to the wind stress. On seasonal to annual timescales, the vertical velocity field is usually considered as naturally filtered, and is then approximately equal to the vertical Ekman advection, i.e., w≈ w E (de Boisséson et al., 2010). The analysis of this study mainly focuses on seasonal timescales. Therefore, this entrainment term in this study is dominated by the vertical advection induced by Ekman pumping. The fourth and fifth terms are respectively the entrainment due to the tendency of the mixed layer depth and the entrainment due to "advection of the mixed layer depth", in which r=-[u(∂ z)/(∂ x)+v(∂ z)/(∂ y)]. The last term includes the turbulent mixing and heat diffusion. In the heat budget analysis of KOE-SSTA events (not shown here), it is found that the SSTA tendency of KOE-SSTA events is largely dominated by the net sea surface heat flux, the vertical advection and the horizontal advection. The contributions of the entrainment terms (i.e., -∆ T(∂ z/∂ t) and -∆ Tr) and the turbulent mixing and heat diffusion term R are negligible, which thus can be neglected in our following analysis. Therefore, the equation governing the evolution of KOE-SSTA prediction errors can be expressed as: \begin{eqnarray} \dfrac{\partial T'}{\partial t}&=&\dfrac{\partial(\overline{T}+T^*+T')}{\partial t}-\dfrac{\partial(\overline{T}+T^*)}{\partial t}\nonumber\\ &=&\dfrac{Q'}{\rho c_p\overline{h}}+(\overline{U}_{\rm adv}+U_{\rm adv}^*+U'_{\rm adv})+(\overline{V}_{\rm adv}+V_{\rm adv}^*+V'_{\rm adv})\nonumber\\ &&+(\overline{W}_{\rm adv}+W_{\rm adv}^*+W'_{\rm adv}) , \ \ (5)\end{eqnarray} in which,
\begin{eqnarray} \overline{U}_{\rm adv}&=&-\overline{u}\dfrac{\partial T'}{\partial x}-u'\dfrac{\partial \bar{T}}{\partial x} ,\quad U_{\rm adv}^*=-u^*\dfrac{\partial T'}{\partial x}-u'\dfrac{\partial T^*}{\partial x} ,\quad U'_{\rm adv}=-u'\dfrac{\partial T'}{\partial x} ,\nonumber\\ \overline{V}_{\rm adv}&=&-\bar{v}\dfrac{\partial T'}{\partial y}-v'\dfrac{\partial\overline{T}}{\partial y} ,\quad V_{\rm adv}^*=-v^*\dfrac{\partial T'}{\partial y}-v'\dfrac{\partial T^*}{\partial y} ,\quad V'_{\rm adv}=-v'\dfrac{\partial T'}{\partial y} ,\nonumber\\ \overline{W}_{\rm adv}&=&-\overline{w}(\Delta T)'-w'\overline{\Delta T} ,\quad W_{\rm adv}^*=-w^*(\Delta T)'-w'(\Delta T)^* ,\quad W'_{\rm adv}==-w'(\Delta T)' . \ \ (6)\end{eqnarray}
In Eqs. (5)-(6), the climatological mean state, the anomaly and the error are respectively denoted by an overbar, asterisk and prime. Q' represents the sea surface heat flux error and is the sum of the latent heat flux error Q' LH, sensible heat flux error Q' SH, shortwave radiation flux error Q' SWH, and longwave radiation flux error Q' LWH. \(\overline{h}\) is the climatological monthly mean mixed-layer depth. As in observations (Wang et al., 2012), the simulated \(\overline{h}\) in the KOE region is deeper than 150 m in boreal winter and shallower than 30 m in boreal summer. The terms in Eq. (6) indicate the effects of oceanic temperature advection on the SSTA error growth. (Duan and Wu, 2015) revealed that the latent heat flux errors (Q' LH) and the vertical oceanic temperature advection associated with the climatological mean state (\(\overline{W}_\rm adv\)), which are both largely forced by the sea surface wind stress errors, dominate the SSTA error growth associated with the SPB. Clearly, the effects of both Q' LH and \(\overline{W}_\rm adv\) on the error growth are directly influenced by the climatological annual cycle and prediction errors, but not the SSTA events to be predicted. In Figs. 5a and b, we show the ensemble means of Q' LH and \(\overline{W}_\rm adv\) averaged over the KOE region in the ASO season for the warm and cold events in Category-1 (black bars) and Category-2 (gray bars), respectively. They both have few differences between the two categories (differences shown in Fig. 5e), indicating that the larger error growth rates of the SSTA events in Category-1 are not due to the physical processes of Q' LH and \(\overline{W}_\rm adv\). Furthermore, the terms Q' SH, Q' SWH, Q' LWH, \(\overline{U}_\rm adv\), U' adv, \(\overline{V}_\rm adv\), V' adv and W' adv are not directly influenced by the SSTA events to be predicted and make little contribution to the different error growth rates between Category-1 and Category-2 (not shown here).
Among all terms in Eqs. (5)-(6), only the physical processes of U adv*=-u*(∂ T')/∂ x-u'(∂ T*)/∂ x, V adv*=-v*(∂ T')/∂ y-u'(∂ T*)/∂ y and W adv*=-w*(∆ T)'-w'(∆ T)* are directly related with the SSTA events to be predicted, and they respectively describe the effect of the prediction errors of the zonal, meridional and vertical oceanic temperature advection associated with the warm or cold events on the SSTA error growth. The u*, v*, w* and T* are respectively the anomalies of the zonal, meridional, vertical current velocities and SSTA in the North Pacific; and u', v', w' and T' represent their related prediction errors. A larger absolute value of U adv*, V adv* or W adv* causes a larger growth tendency (i.e., growth rate) of prediction errors (i.e., (∂ T')/∂ t) and, as a result, leads to faster error growth. To investigate which of these terms contribute most to the larger error growth rates in the ASO season for the SSTA events in Category-1, the regional-mean U adv*, V adv* or W adv* in the ASO season for the warm and cold events in Category-1 (black bars) and Category-2 (gray bars) are illustrated in Figs. 5c and d. It is shown that only the W adv*, i.e., the prediction errors of the vertical oceanic temperature advection associated with the SSTA events to be predicted, exhibits a significant difference between the two categories (as shown in Fig. 5e), implying that the difference in W adv* is the major factor contributing the most to the large difference in error growth rates between the Category-1 and Category-2 events. In addition, a positive (negative) value of the processes on the right-hand side of Eq. (4) indicates the effect of favoring the error growth for warm (cold) events. Therefore, as shown in Figs. 5c and d, a positive (negative) value of W adv* for the warm (cold) events in Category-1 causes error growth, while a negative (positive) value for the warm (cold) events in Category-2 suppresses error growth, which therefore leads to much larger error growth for the SSTA events in Category-1 than those in Category-2.
W adv*=-w*(∆ T)'-w'(∆ T)* is composed of the oceanic temperature advection errors by anomalous vertical currents of KOE-SSTA events [i.e., \(A=-w*(\Delta T)'=-w^*(T'-T'_h)/\overline{h}\), T' is the SST error, and T'h is the temperature error just below the mixed layer base] and the anomalous oceanic temperature advection by vertical current errors [i.e., \(B=-w'(\Delta T)^*=-w'(T^*-T_h^*)/\overline{h}\), T* is the SST anomaly, and Th* is the temperature anomaly just below the mixed layer base]. In Fig. 5f, we plot the differences in term A and term B between the two categories for both warm and cold events in the ASO season. It is shown that the dynamical process indicated by term A plays a more important role in the contribution from W adv* to the large difference in SSTA error growth between Category-1 and Category-2. Term A, the oceanic temperature advection errors by anomalous vertical currents of KOE-SSTA events, is dominated by the anomalous vertical currents (i.e., the anomalous upwelling or downwelling) and the difference between the SST error and the temperature error just below the mixed layer base [i.e., \((\Delta T)'=(T'-T'_h)/\overline{h}\)]. The anomalous upwelling or downwelling can be caused by the Ekman pumping, which is generated by the wind stress curl anomaly. In fact, Ekman pumping can be related to the wind stress by W E= curlz(τ/ρ f), where τ is the vector wind stress, ρ is the density of sea water and f is the Coriolis parameter (Stewart, 2008, Chapter 9). Obviously, the positive (negative) wind stress curl can cause the upwelling (downwelling). Here, we show the anomalous Ekman pumping and sea surface wind stress anomalies in the ASO season for the warm and cold events in the two categories in Fig. 6, where the positive (negative) shaded values indicate the anomalous upwelling (downwelling). The \((\Delta T)^*=(T^*-T_h^*)/\overline{h}\) and \((\Delta T)'=(T'-T'_h)/\overline{h}\) in the ASO season for both categories are also shown in Fig. 6.
For warm events, the anomalous upwelling induced by the cyclonic wind stress anomalies over the KOE region for Category-1 is much more significant than for Category-2 (Figs. 6a and b). The positive \((\Delta T)^*=(T^*-T_h^*)/\overline{h}\), which also shows a larger value for Category-1 than Category-2 (blue bars and yellow bars in Fig. 6c), implies that the anomalous upwelling brings the cold water to upper ocean layers and leads to the cooling of KOE-SSTA. Therefore, the more significant anomalous upwelling for Category-1 causes much more cooling of KOE-SSTA, which explains why the warm events in Category-1 transfer more rapidly from the mature to decaying phase than those in Category-2. Moreover, Fig. 6c shows that the \((\Delta T)'=(T'-T)/\overline{h}\) is negative with a large absolute value for Category-1 (green bars). Therefore, for the warm events in Category-1, the significant upwelling (i.e., a positive value of vertical velocity w* in term A) and a negative value of \((\Delta T)'=(T'-T'_h)/\overline{h}\) lead to a positive term A (i.e., positive oceanic temperature advection errors by anomalous vertical currents of the SSTA events), and in turn cause a positive W adv*, which favors the rapid growth of warm prediction errors according to Eq. (5). Compared with Category-1, the error growth induced by W adv* is much weaker due to the negligible vertical advection for Category-2 (Fig. 6b). As shown in Fig. 5c, the large difference in W adv* enhances the difference in error growth rates between the SSTA events in Category-1 and Category-2, and therefore favors a greater possibility of a more significant SPB for the SSTA events in Category-1. The mechanisms are similar for the cold events (see Figs. 6d-f).
Overall, the mature-to-decaying transition of SSTA and the growth of prediction errors in the ASO season are both related with the anomalous upwelling or downwelling in the transition phase of the SSTA events to be predicted. The anomalous upwelling or downwelling in the ASO season for the SSTA events in Category-1 is much more significant than in Category-2, which not only leads to the largest mature-to-decaying transition rate of SSTA but also results in the fastest error growth for the SSTA events in Category-1. Therefore, this explains why the SSTA events transferring more rapidly from the mature to decaying phase tend to yield a greater possibility of a more significant SPB.