Sensitivity of El Niño diversity prediction to parameters in an intermediate coupled model

Abstract

There are large uncertainties that exist in El Niño–Southern Oscillation (ENSO) predictions. Errors in model parameters are one of the factors limiting ENSO prediction skill. In this study, we look for the model parameters that can induce the largest least-square changes in the sea surface temperature predictions for four El Niño events that were optimized to fit observations in the tropical Pacific: 1982 and 1987 eastern Pacific (EP) events and 1990 and 1994 central Pacific (CP) events. The ENSO model is an intermediate coupled model used at the Institute of Oceanology, Chinese Academy of Sciences (IOCAS ICM). The sensitivity of the prediction skill for each type of El Niño to the model parameters is analyzed using an optimization parameter sensitivity analysis (OPSA) method. Perturbation experiments allow us to identify three important model parameters that play a key role in ENSO dynamics from nine selected parameters. These three, α_Tea, α_τ, and α_HF, are related to the thermocline feedback, wind stress, and heat flux, respectively. α_Tea and α_τ are important for both types of El Niño events, while α_HF is another important parameter that impacts CP events. Our results show that ENSO predictions can be improved by accurate estimates of the sensitive parameters identified here. In particular, it is reasonable that accurate estimates of α_HF make more sense for improving CP El Niño prediction than for improving EP El Niño prediction. Increasing α_Tea or α_τ or decreasing α_HF tends to increase the amplitudes of these warm events. The two types of El Niño events have different sensitivities to the parameter α_HF, which is attributed to the role played by the vertical mixing process in the eastern equatorial Pacific. These parameter sensitivity analyses provide an improved understanding of ENSO predictability, and highlight the special importance of α_HF for enhancing the prediction skill for the CP El Niño.

Appendices

Appendix A. Governing equations of IOCAS ICM

The ICM consists of an intermediate complexity dynamic ocean model, an SSTA model, and a statistical atmospheric model. Here, the dynamic ocean model is further composed of a linear component and a nonlinear component. In the present study, we would only give the formulation of the SSTA model. For details of the dynamic ocean model, we refer readers to Keenlyside (2001), Gao et al. (2017), and Zhang et al. (2020).

The SSTA model equation can be written as

$$\begin{gathered} \frac{{\partial T^{\prime}}}{\partial t} = \mathop { - u^{\prime}\frac{{\partial \overline{T}}}{\partial x}}\limits_{(1)} \mathop { - \left( {\overline{u} + u^{\prime}} \right)\frac{{\partial T^{\prime}}}{\partial x}}\limits_{(2)} \mathop { - v^{\prime}\frac{{\partial \overline{T}}}{\partial y}}\limits_{(3)} \mathop { - \left( {\overline{v} + v^{\prime}} \right)\frac{{\partial T^{\prime}}}{\partial y}}\limits_{(4)} \hfill \\ \begin{array}{*{20}c} {} & {} \\ \end{array} \mathop { - \left[ {\left( {\overline{w} + w^{\prime}} \right)M\left( { - \overline{w} - w^{\prime}} \right) - \overline{w}M\left( { - \overline{w}} \right)} \right]\frac{{\alpha_{Tec} \overline{T}_{e} - \overline{T}}}{H}}\limits_{(5)} \hfill \\ \begin{array}{*{20}c} {} & {} \\ \end{array} \mathop { - \left( {\overline{w} + w^{\prime}} \right)M\left( { - \overline{w} - w^{\prime}} \right)\frac{{\alpha_{Tea} T^{\prime}_{e} - T^{\prime}}}{H}}\limits_{(6)} \mathop { - \alpha_{hf} T^{\prime}}\limits_{\begin{subarray}{l} \\ (7) \end{subarray} } \hfill \\ \begin{array}{*{20}c} {} & {} \\ \end{array} \mathop { + \frac{{\kappa_{h} }}{H}\nabla_{h} \cdot \left( {H\nabla_{h} T^{\prime}} \right)}\limits_{(8)} \mathop { + \frac{{2\kappa_{v} }}{{H\left( {H + H_{2} } \right)}}\left( {\alpha_{Tea} T^{\prime}_{e} - T^{\prime}} \right)}\limits_{(9)} \hfill \\ \end{gathered}$$

(A1)

where overbars indicate climatological values, and primes indicate anomalies with respect to the climatological mean state. Here, $T$ and $T_{e}$ are SST and temperature of the subsurface water entrained into the mixed layer; $t$ is the time; $u$, $v$, and $w$ are the zonal ($x$), meridional ($y$), and vertical velocity, respectively;$H$ is the horizontally varying depth of the mixed layer; $H_{2}$ is a constant (125 m); $M\left( x \right)$ is the Heaviside step function; $\alpha_{hf}$ is the thermal damping coefficient; $\kappa_{h}$ and $\kappa_{v}$ are the coefficients for horizontal and vertical diffusivity of temperature, respectively; $\alpha_{Tea}$ and $\alpha_{Tec}$ are scaling coefficients of entraining anomalous and climatological subsurface temperatures, respectively. The function $M\left( x \right)$ accounts for the fact that SSTAs are affected by vertical advection only when subsurface water is entrained into the mixed layer, but SSTAs can always be influenced by subsurface temperature variability through vertical mixing.

Appendix B. Formulation of the OPSA method

In general, a numerical model can be formally expressed by:

$${\mathbf{X}}\left( t \right) = {\mathbf{M}}_{t} \left( {\mathbf{P}} \right)\left( {{\mathbf{X}}_{0} } \right)$$

(B1)

where ${\mathbf{X}}\left( t \right)$ is the model state vector at time $t$, ${\mathbf{M}}_{t} \left( {\mathbf{P}} \right)$ represents a nonlinear propagator with a parameter vector ${\mathbf{P}} = \left( {P_{1} ,P_{2} ,...,P_{m} } \right)$, and ${\mathbf{X}}_{0}$ is the initial state vector. Here, we will not consider the uncertainty in the initial conditions and mainly focus on the uncertainty in the model parameters. The parameter uncertainties are represented by perturbations in all model parameters, which are denoted as ${\mathbf{p}} = \left( {p_{1} ,p_{2} ,...,p_{m} } \right)$.

A parameter should have a limited range, meaning that it falls within some intervals. Therefore, the parameter perturbation magnitude needs to be constrained by a specific condition, i.e., ${\mathbf{p}} \in C_{\varepsilon }$, where $\varepsilon$ is the constraint value. $C_{\varepsilon }$ should be determined with a priori knowledge about the region of the parameter space in which reasonable physical values reside. To evaluate the effect of parameter perturbation $p_{i}$, it is impractical to take different parameter perturbations and repeatedly integrate the model. Alternatively, we can use an objective procedure to rank parameters in terms of the norm of the changes in the model state they induce after some integration time. Formally, we search the whole feasible region $C_{\varepsilon }$ for the maximum of the objective function $J_{i}$ by solving the following constraint optimization problem:

$$\begin{gathered} J_{i,\varepsilon } \left( {p_{1,\varepsilon } ,...,p_{m,\varepsilon } } \right) = \mathop {\max }\limits_{{\left( {p_{1} ,...,p_{m} } \right) \in C_{\varepsilon } }} J_{i} \left( {p_{1} ,...,p_{m} } \right) \hfill \\ = \mathop {\max }\limits_{{\left( {p_{1} ,...,p_{m} } \right) \in C_{\varepsilon } }} \left[ {J_{total} \left( {p_{1} ,...,p_{m} } \right) - J_{ - i} \left( {p_{1} ,...,p_{i - 1} ,p_{i + 1} ,...,p_{m} } \right)} \right] \hfill \\ \end{gathered}$$

(B2)

Here, $J_{total}$ represents the magnitude of state perturbations in the model output (prediction or simulation) caused by the perturbations in all parameters and is calculated by.

$$J_{total} \left( {p_{1} ,...,p_{m} } \right) = \left\| {{\mathbf{M}}\left( {P_{1} + p_{1} ,...,P_{m} + p_{m} } \right)\left( {{\mathbf{X}}_{0} } \right) - {\mathbf{M}}\left( {P_{1} ,...,P_{m} } \right)\left( {{\mathbf{X}}_{0} } \right)} \right\|$$

(B3)

where $\left\| \cdot \right\|$ indicates the L2 norm that measures the perturbation magnitude in the whole model domain $\Omega$. $J_{ - i}$ represents the magnitude of the state perturbations induced by all parameter perturbations except for the $i$ th perturbation $p_{i}$, $1 \le i \le m$(i.e., setting $p_{i} = 0$), and can be written as:

$$\begin{gathered} J_{ - i} \left( {p_{1} ,...,p_{i - 1} ,p_{i + 1} ,...,p_{m} } \right) \hfill \\ = \left\| {{\mathbf{M}}\left( {P_{1} + p_{1} ,...,P_{i - 1} + p_{i - 1} ,P_{i} ,P_{i + 1} + p_{i + 1} ,...,P_{m} + p_{m} } \right)\left( {{\mathbf{X}}_{0} } \right) - {\mathbf{M}}\left( {P_{1} ,...,P_{m} } \right)\left( {{\mathbf{X}}_{0} } \right)} \right\| \hfill \\ \end{gathered}$$

(B4)

Then, the objective function $J_{i} = J_{total} - J_{ - i}$ is actually the magnitude of the state perturbations induced by the parameter perturbation $p_{i}$ itself and its interaction with perturbations in other parameters.

Appendix C. Effect of parameters on model improvement

We take 100 random perturbations that are uniformly distributed within the constraint interval for each parameter. Subsequently, these perturbations are superimposed on the reference values of the corresponding model parameters, and the ICM is run for a one-year period. The magnitude of the SSTA prediction error over the tropical Pacific induced by perturbations in all model parameters can be measured by the norm.

$$E_{1} = \sqrt {\int_{Jan}^{Dec} {\iint_{\Omega } {\left[ {T^{\prime}\left( {{\mathbf{P}} + {\mathbf{p}}} \right) - T^{\prime}\left( {\mathbf{P}} \right)} \right]^{2} dxdydt}} }$$

(C1)

where $T^{\prime}$ represents the SSTA, and ${\mathbf{P}}$ and ${\mathbf{p}}$ denote the reference parameter and the parameter perturbation, respectively, and $\Omega$ is the whole model domain. If the sensitive parameters have an improved effect, the prediction error can be estimated as:

$$E_{2} = \sqrt {\int_{Jan}^{Dec} {\iint_{\Omega } {\left[ {T^{\prime}\left( {{\mathbf{P}} + {\mathbf{p}} \cdot {{\varvec{\Lambda}}}} \right) - T^{\prime}\left( {\mathbf{P}} \right)} \right]^{2} dxdydt}} }$$

(C2)

where ${{\varvec{\Lambda}}}$ is a 9 diagonal matrix (9 model parameters), in which each diagonal element is either 0 or 1. If the $i$ th parameter is a sensitive one, the $i$ th diagonal element of ${{\varvec{\Lambda}}}$ is set to 0, or else it is set to 1 and thereby the insensitive parameters are fully weighted. The prediction error induced by improving the insensitive parameters can be estimated in a similar way. The degree of model improvement is measured by the following relative difference:

$$improvement = \frac{{E_{1} - E_{2} }}{{E_{1} }}.$$

(C3)

For each event, the mean value of model improvements of the 100-member ensemble is shown in this study.