Instability processes of mesoscale features in the Kuroshio Extension reproduced through assimilation of altimeter data into a quasi-geostrophic model using the variational method

Abstract

The Kuroshio, one of the most energetic western boundary currents in the world, shows variations in its mesoscale features and recirculation gyres, providing an excellent test case of interactions between the mesoscale field and Kuroshio Extension (KE) states. A three-layer quasi-geostrophic model was used to reconstruct flow fields continuous in time and the horizontal plane from the TOPEX/POSEIDON altimeter data based on the variational method. Compared with the solutions obtained by the nudging method, the present results proved that the variational solution was closer to the real field. In the assimilation period, 1993–1997, the baroclinic instability index (BII) was defined to be the phase shift from the uppermost layer to the lowest layer with mesoscale features. In the first half of the assimilation period, the KE took the transition from the elongated to contracted states, in which BII decreased gradually, as a consequence of the KE state shift. In the second half period, BII increased in the downstream region just west of the Shatsky Rise, in which baroclinic instability contributed to the final stage of the contracted state, and was followed by rapidly weakened instability as a trigger of the opposite transition from the contracted to elongated states. The wind-driven recirculation gyre played an active role on the KE transition in the first half period, although not in the second half.

Appendix

The Lagrange function, L, is defined as the sum of cost function, J, and the product of the model equations and the respective Lagrange multipliers. To realize this optimization, J is set as

$$ \begin{aligned} J = & \, \sigma_{k} [\Uppsi^{\prime}_{1} (m,n,10k - 4) \, - \Uppsi^{\prime}_{\text{D}} (10n - 10 + 10k - 4)]^{2} + \, \sigma_{j} B_{j} [[\Uppsi^{\prime}_{j} (m,n,1) \\ & \quad - \{ \Uppsi^{\prime}_{j} (m - 1,n - 1,11) + \Uppsi^{\prime}_{j} (m - 1,n,1)\} /2]^{2} + [\Uppsi^{\prime}_{j} (m,n,11) \\ & \quad - \{ \Uppsi^{\prime}_{j} (m - 1,n - 1,21) + \Uppsi^{\prime}_{j} (m - 1,n,11) + \Uppsi^{\prime}_{j} (m - 1,n + 1,1)\} /3]^{2} \\ & \quad + [\Uppsi^{\prime}_{j} (m,n,21) - \{ \Uppsi^{\prime}_{j} (m - 1,n - 1,31) + \Uppsi^{\prime}_{j} (m - 1,n,21) + \Uppsi^{\prime}_{j} (m - 1,n + 1,11)\} /3]^{2} \\ & \quad + [\Uppsi^{\prime}_{j} (m,n,31) - \{ \Uppsi^{\prime}_{j} (m - 1,n,31) + \Uppsi^{\prime}_{j} (m - 1,n + 1,21)\} /2]^{2} ] \\ \end{aligned} $$

(A.1)

($\sigma$ _k is a summation for three 10-day periods with k = 1, 2 and 3, and 3 _j is a summation for three layers with j = 1, 2, and 3) where the second to fifth lines are excluded for m = 1, m denotes the set number, and n denotes the sequential numbers of the 30-day periods. The third indices for $\Uppsi^{\prime}_{j}$ are days in each 30-day period, and the index for $\Uppsi^{\prime}_{\text{D}}$ is a day in the entire assimilation period. The coefficient B ₁, B ₂ and B ₃ are chosen as 1, 2.5 and 5.0, respectively. It is noted that n is 2 for the first 30-day period, and n is 221 for the last 30-day period, where (A.1) is valid for n = 3–220. For n = 2, the second-to-fifth lines in (A.1) are revised to be

$$ \begin{aligned} & + \sigma_{j} B_{j} [[\Uppsi^{\prime}_{j} (m,2,11) - \, \{ \Uppsi^{\prime}_{j} (m - 1,2,11) + \Uppsi^{\prime}_{j} (m - 1,3,1)\} /2]^{2} \\ & + [\Uppsi^{\prime}_{j} (m,2,21) \, - \, \{ \Uppsi^{\prime}_{j} (m - 1,2,21) + \Uppsi^{\prime}_{j} (m - 1,3,11)\} /2]^{2} \\ & + [\Uppsi^{\prime}_{j} (m,2,31) \, - \, \{ \Uppsi^{\prime}_{j} (m - 1,2,31) + \Uppsi^{\prime}_{j} (m - 1,3,21)\} /2]^{2} ] \\ \end{aligned} $$

(A.2)

For n = 221, the second-to-fifth lines in (A.1) are revised to be

$$ \begin{aligned} & + \, \sigma_{j} B_{j} [[\Uppsi^{\prime}_{j} (m,221,1) \, - \, \{ \Uppsi^{\prime}_{j} (m - 1,220,11) + \Uppsi^{\prime}_{j} (m - 1,221,1)\} /2]^{2} \\ & + [\Uppsi^{\prime}_{j} (m,221,11) \, - \, \{ \Uppsi^{\prime}_{j} (m - 1,220,21) + \Uppsi^{\prime}_{j} (m - 1,221,11)\} /2]^{2} \\ & + [\Uppsi^{\prime}_{j} (m,221,21) \, - \, \{ \Uppsi^{\prime}_{j} (m - 1,220,31) + \Uppsi^{\prime}_{j} (m - 1,221,21)\} /2]^{2} ] \\ \end{aligned} $$

(A.3)

The variational method is subject to vanishing partial derivatives of L with respect to the control variables. The partial derivatives are obtained through the adjoint equations for the Lagrangian multipliers related to $\Uppsi^{\prime}_{j}$ and $\omega^{\prime}_{j}$. The variational method has the following steps:

• (a) The first-guess field is chosen for the control variables, which are the initial conditions,

• (b) The governing equations for $\Uppsi^{\prime}_{j}$ and $\omega^{\prime}_{j}$ are solved forward in time,

• (c) The adjoint equations for the Lagrangian multipliers are solved backward in time by taking the cost function,

• (d) The gradients of the cost function with respect to the control variables are calculated, and

• (e) The control variables are corrected using a well selected descent algorithm.

These steps are iterated 60 times, and one examines whether the data-model misfit is reduced.

Each one-period solution of the individual variational solutions with constraint data for three cycles is not identical to the solutions in the periods separated by 10 days. Once we define the entire variational solution as the first 10 days of the variational solutions of 220 periods, the entire solution is not continuous in time. In order to obtain a continuous solution, the gaps of the consecutive variational solutions have to be optimized by including the second-to-fifth lines of (A.1) in J. Thus, the set of adjoint equations for 220 periods is iterated 30 times so that the gaps become very small.

The performance is summarized here. The first-guess field is taken from the nudging solution, which has a variability amplitude about 80 % of the data converted from the T/P sea surface data. Mismatches are clear in all three layers at the times of nudging every 10 days. First in the variational assimilation, the individual variational solution was obtained for all 30-day periods, and the cost function J was confirmed to decrease during 60 iterations. Then, these individual solutions were optimized by including the mismatches between two neighboring 30-day periods into the cost function. This optimization was also iteratively carried out 30 times. The spatial and temporal mean of the mismatches reduced significantly for about 10 iterations (Fig. 5), proving that the variational solutions are much more appropriate for flow field analyses than the nudging solutions are.