Effectiveness of the Bjerknes stability index in representing ocean dynamics

Abstract

The El Niño-Southern Oscillation (ENSO) is a naturally occurring coupled phenomenon originating in the tropical Pacific Ocean that relies on ocean–atmosphere feedbacks. The Bjerknes stability index (BJ index), derived from the mixed-layer heat budget, aims to quantify the ENSO feedback process in order to explore the linear stability properties of ENSO. More recently, the BJ index has been used for model intercomparisons, particularly for the CMIP3 and CMIP5 models. This study investigates the effectiveness of the BJ index in representing the key ENSO ocean feedbacks—namely the thermocline, zonal advective, and Ekman feedbacks—by evaluating the amplitudes and phases of the BJ index terms against the corresponding heat budget terms from which they were derived. The output from Australian Community Climate and Earth System Simulator Ocean Model (a global ocean/sea ice flux-forced model) is used to calculate the heat budget in the equatorial Pacific. Through the model evaluation process, the robustness of the BJ index terms are tested. We find that the BJ index overestimates the relative importance of the thermocline feedback to the zonal advective feedback when compared with the corresponding terms from the heat budget equation. The assumption of linearity between variables in the BJ index formulation is the primary reason for these differences. Our results imply that a model intercomparison relying on the BJ index to explain ENSO behavior is not necessarily an accurate quantification of dynamical differences between models that are inherently nonlinear. For these reasons, the BJ index may not fully explain underpinning changes in ENSO under global warming scenarios.

Appendix

1.1 Derivation of the BJ index

The BJ index is derived from the linearized anomalous temperature tendency equation, namely

$$\frac{\partial T^{\prime}}{\partial t} = -\overline{u}\frac{\partial T^{\prime}}{\partial x} -\overline{v}\frac{\partial T^{\prime}}{\partial y} -\overline{w}\frac{\partial T^{\prime}}{\partial z} - u^{\prime}\frac{\partial \overline{T}}{\partial x} - v^{\prime}\frac{\partial \overline{T}}{\partial y} - w^{\prime}\frac{\partial \overline{T}}{\partial z} + {Q_{q}^{\prime}},$$

(19)

where the overline notation denotes climatological fields (i.e. averaged over the full time period) and the prime denotes anomalous fields that have the seasonal cycle removed. In what follows we drop the prime notation. The terms in Eq. (19) are averaged vertically, from the ocean surface to the MLD, and horizontally in the central-eastern equatorial Pacific 5°S–5°N, 175°E–80°W), in which the majority of ENSO variability occurs, yielding

$$\begin{aligned} \left\langle\frac{\partial{T}}{\partial t}\right\rangle_E = -\left(\frac{\langle\overline{u}\rangle_E}{L_x} + \frac{\langle-2y\overline{v}\rangle_E}{L_y^2} + \frac{{\langle {\overline{w}}\rangle_E}}{H_{BJ}}\right){\langle T\rangle_E} + {\langle Q\rangle_E}\\ - {\left\langle\frac{\partial \overline{T}}{\partial x}\right\rangle_E}{\langle u\rangle_E} - {\left\langle\frac{\partial \overline{T}}{\partial z}\right\rangle_E}{\langle H(\overline{w}){w}\rangle_E} + {\left\langle\frac{\overline{w}}{H_{BJ}}\right\rangle_E}\langle H(\overline{w})T_H\rangle_{E}, \end{aligned}$$

(20)

where L _x and L _y are the longitudinal and latitudinal extents of the central-eastern box, respectively, and the factor −2y/L _y assumes that the tropical SST anomalies are Gaussian with an e-folding decay scale of L _y . The term H _BJ is the MLD, T _H is the temperature at the grid point just below the mixed layer, and $H(\overline{w})$ ensures that only the vertical motion into the mixed layer affects the mixed layer heat budget. Note that in deriving Eq. (20), the small term $-v^{\prime} \partial T/\partial y$ has been omitted, consistent with Jin et al. (2006). A series of balance equations from section 3 [Eqs. (9–11), (13), (15), and (17)] are applied to approximate the terms in Eq. (20). These balance equations yield coefficients that estimate the strength of the air–sea coupling μ, the sensitivity of oceanic responses to surface winds β _h , β _u , β _w , and the magnitude of advection by mean currents and thermodynamic damping. Collectively, they enable the temperature tendency in Eq. (20) to be separated into growth and frequency components, expressed in the form of the recharge oscillator model (Jin 1997a), namely,

$$\left\langle\frac{\partial{T}}{\partial t}\right\rangle_E = R{\langle T\rangle_E} + F{\langle h\rangle_{W}}.$$

(21)

Here the coefficients R and F are explicit functions of the basic state, rather than simply coefficients estimated via regression to observations or model data. R is the growth term that underpins the BJ index, and is given by

$$R = -\left(\frac{\langle\overline{u}\rangle_E}{L_x} + \frac{\langle-2y\overline{v}\rangle_E}{L_y^2} + \frac{{\langle {\overline{w}}\rangle_E}}{H_{BJ}}\right) - \alpha + a_h\beta_h\mu{\left\langle\frac{\overline{w}}{H_{BJ}}\right\rangle_E} + \beta_u\mu{\left\langle-\frac{\partial \overline{T}}{\partial x}\right\rangle_E} + \beta_w\mu{\left\langle-\frac{\partial \overline{T}}{\partial z}\right\rangle_{E}},$$

(22)

and the frequency term F is given by

$$F = \beta_{uh}{\left\langle-\frac{\partial \overline{T}}{\partial x}\right\rangle_E} + a_h{\left\langle\frac{\overline{w}}{H_{BJ}}\right\rangle_{E}}.$$

(23)