The fastest growing initial error in prediction of the Kuroshio Extension state transition processes and its growth

Abstract

In this study, the predictability of the Kuroshio Extension (KE) transition processes is explored from an error growth perspective. The fastest growing initial errors (FGIEs) are obtained through the conditional nonlinear optimal perturbation approach within a reduced-gravity shallow-water model forced by steady winds, which provides a fairly realistic simulation of the KE low-frequency variability of intrinsic origin. The large amplitudes of the FGIEs for both the transitions from a typical low-energy state to a typical high-energy state (LH) and the opposite transition (HL), are found mainly in the Kuroshio large meander region south of Japan and in the KE region. The FGIE grows more rapidly for the HL process than for the LH process, implying that the HL transition process may be more difficult to predict. The evolution processes of the FGIEs and the related mechanisms are revealed by investigating the evolution of the potential vorticity anomalies caused by the FGIEs. The dominant physical processes governing the FGIE growth are found to be different for the LH and HL processes. For the LH process, the evolution is mainly governed by linear advection processes and interfacial friction, while for the HL process, in addition to these two processes, the nonlinear advection process also plays a vital role in the evolution. This indicates that nonlinear intrinsic oceanic processes affect considerably the error growth, especially in the HL transition process, suggesting that the intrinsic processes should be carefully considered when exploring the predictability and forecast of the KE low-frequency variability.

Appendix: Derivation of the evolution equation of the PV anomaly

To obtain the evolution equation of the PV anomaly, we first derive the governing equation for the PV based on the shallow-water model. Taking the curl of Eqs. (1a) and (1b), we obtain

$$\begin{aligned} \frac{\partial \zeta }{\partial t} + \zeta \nabla \cdot \varvec{V} + \varvec{V} \cdot \nabla \zeta + f\nabla \cdot \varvec{V} + v\frac{\partial f}{\partial y} & = curl\left( {A_{H} \nabla^{2} u,A_{H} \nabla^{2} v} \right) \\ & \quad + curl\left( { - \gamma u\sqrt {u^{2} + v^{2} } , - \gamma v\sqrt {u^{2} + v^{2} } } \right) + curl\left( {\frac{\tau }{\rho h},0} \right), \\ \end{aligned}$$

(12)

where $\varvec{V} = \left( {u,v} \right)$. For simplicity, we denote ${\text{AH}} = curl\left( {A_{H} \nabla^{2} u,A_{H} \nabla^{2} v} \right)$, ${\text{FR}} = curl\left( { - \gamma u\sqrt {u^{2} + v^{2} } , - \gamma v\sqrt {u^{2} + v^{2} } } \right)$, ${\text{WS}} = curl\left( {\frac{\tau }{\rho h},0} \right)$ and rearrange Eq. (12) as follows,

$$\frac{{D\left( {\zeta + f} \right)}}{dt} + \left( {\zeta + f} \right)\nabla \cdot \varvec{V} = {\text{AH}} + {\text{FR}} + {\text{WS}},$$

(13)

where $D/dt$ denotes material derivative.

According to Eq. (1c), we have

$$\nabla \cdot \varvec{V} = - \frac{1}{h}\frac{Dh}{dt}.$$

(14)

Substituting Eq. (14) into Eq. (13), we can obtain the following PV evolution equation

$$\frac{\partial q}{\partial t} = - \varvec{ V} \cdot \nabla q + \frac{\text{AH}}{h} + \frac{\text{FR}}{h} + \frac{\text{WS}}{h}.$$

(15)

For the reference state, the evolution equation becomes,

$$\frac{{\partial q_{r} }}{\partial t} = - \varvec{ V}_{r} \cdot \nabla q_{r} + \frac{{{\text{AH}}_{r} }}{{h_{r} }} + \frac{{{\text{FR}}_{r} }}{{h_{r} }} + \frac{{{\text{WS}}_{r} }}{{h_{r} }},$$

(16)

where the subscript $r$ denotes the reference state. Accordingly, for the state obtained after perturbing the initial reference state with the FGIE, the PV equation is,

$$\begin{aligned} \frac{\partial \left( {q_{r} + q^{\prime}} \right)}{\partial t} & = - \left( {\varvec{V}_{r} + \varvec{V}^{\prime}} \right) \cdot \nabla \left( {q_{r} + q^{\prime}} \right) \\&\quad+ \frac{{{\text{AH}}_{p} }}{{h_{r} + h^{\prime}}} + \frac{{{\text{FR}}_{p} }}{{h_{r} + h^{\prime}}} + \frac{{{\text{WS}}_{p} }}{{h_{r} + h^{\prime}}}, \end{aligned}$$

(17)

where the prime denotes the anomaly caused by the FGIE and the subscript $p$ represents the term calculated after superimposing the FGIE on the reference state.

Subtracting Eq. (16) from Eq. (17), we obtain the equation governing the evolution of the PV anomaly as follows

$$\frac{{\partial q^{\prime}}}{\partial t} = - \varvec{V}_{r} \cdot \nabla q^{\prime} - \varvec{V}^{\prime} \cdot \nabla q_{r} - \varvec{V}^{\prime} \cdot \nabla q^{\prime} + \frac{{{\text{AH}}_{p} }}{{h_{r} + h^{\prime}}} - \frac{{{\text{AH}}_{r} }}{{h_{r} }} + \frac{{{\text{FR}}_{p} }}{{h_{r} + h^{\prime}}} - \frac{{{\text{FR}}_{r} }}{{h_{r} }} + \frac{{{\text{WS}}_{p} }}{{h_{r} + h^{\prime}}} - \frac{{{\text{WS}}_{r} }}{{h_{r} }}.$$

(18)

Hence, the expression of each term in the right-hand side of Eq. (11) is

$${\text{LADV}}1 = - \varvec{V}_{r} \cdot \nabla q^{\prime},$$

(19)

$${\text{LADV}}2 = - \varvec{V}^{\prime} \cdot \nabla q_{r} ,$$

(20)

$${\text{NADV}} = - \varvec{V}^{\prime} \cdot \nabla q^{\prime},$$

(21)

$${\text{VISC}} = \frac{{{\text{AH}}_{p} }}{{h_{r} + h^{\prime}}} - \frac{{{\text{AH}}_{r} }}{{h_{r} }},$$

(22)

$${\text{FRIC}} = \frac{{{\text{FR}}_{p} }}{{h_{r} + h^{\prime}}} - \frac{{{\text{FR}}_{r} }}{{h_{r} }},$$

(23)

$${\text{WIND}} = \frac{{{\text{WS}}_{p} }}{{h_{r} + h^{\prime}}} - \frac{{{\text{WS}}_{r} }}{{h_{r} }}.$$

(24)