Intraseasonal variability of sea level and circulation in the Gulf of Thailand: the role of the Madden–Julian Oscillation

Abstract

Intraseasonal variability of the tropical Indo-Pacific ocean is strongly related to the Madden–Julian Oscillation (MJO). Shallow seas in this region, such as the Gulf of Thailand, act as amplifiers of the direct ocean response to surface wind forcing by efficient setup of sea level. Intraseasonal ocean variability in the Gulf of Thailand region is examined using statistical analysis of local tide gauge observations and surface winds. The tide gauges detect variability on intraseasonal time scales that is related to the MJO through its effect on local wind. The relationship between the MJO and the surface wind is strongly seasonal, being most vigorous during the monsoon, and direction-dependent. The observations are then supplemented with simulations of sea level and circulation from a fully nonlinear barotropic numerical ocean model (Princeton Ocean Model). The numerical model reproduces well the intraseasonal sea level variability in the Gulf of Thailand and its seasonal modulations. The model is then used to map the wind-driven response of sea level and circulation in the entire Gulf of Thailand. Finally, the predictability of the setup and setdown signal is discussed by relating it to the, potentially predictable, MJO index.

Appendix: Spectral analysis

Consider two scalar random processes {x _t } and {y _t } where $t=1,2,\ldots$ is a time index. The power spectra of {x _t } and {y _t } are denoted f _xx (ω) and f _yy (ω), respectively, and their cross-spectrum is denoted f _xy (ω), where ω is the frequency. The power spectra and cross-spectrum are calculated from

$$ f_{xy}(\omega) = \langle I_{xy}(\omega) \rangle $$

(2)

where I _xy (ω) = d _x (ω)d ^* _y (ω) is the periodogram and d _x (ω) is the discrete Fourier transform of {x _t } and is in general complex valued. The complex conjugate is denoted by ^* and a weighted running average, in the frequency dimension, is denoted by 〈〉. The weighted running average is required because without performing such smoothing the periodogram does not provide a consistent estimator of the power spectra (e.g., Shumway and Stoffer 2000). An N-point Bartlett window, with N typically 10–20 % of the length of the original time series, is used in this study.

The squared coherence between {x _t } and {y _t } is defined by Priestley (1981)

$$ \kappa^2_{xy}(\omega) = \frac{|f_{xy}(\omega)|^2}{f_{xx}(\omega)f_{yy}(\omega)}, $$

(3)

The coherence is confined to lie between 0 and 1 and can roughly be thought of as a “frequency-dependent correlation coefficient” (Eq. 3 is similar in form to the equation for squared correlation if the cross-spectrum is replaced by the cross-covariance and the power spectra are replaced by the auto-covariances). The associated phase spectrum ϕ _xy (ω) is given by

$$ \phi_{xy}(\omega) = \arctan\left( \frac{\hbox{Im}(f_{xy}(\omega))}{\hbox{Re}(f_{xy}(\omega))} \right) $$

(4)

where Re(z) and Im(z) represent the real and imaginary parts of z respectively. Given a statistically significant value of κ _xy at some frequency ω₀ then ϕ(ω₀) represents the phase offset, in radians, between the coherent signals of {x _t } and {y _t } at the frequency ω₀.

Given the bivariate random process $\{{\user2{x}}_t\}$ where ${\user2{x}}_t$ is given by [x ₁ x ₂] _t the coherence between {y _t } and both components of $\{{\user2{x}}_t\}$ can be calculated using the squared multiple coherence (Priestley 1981):

$$ \kappa^2_{{\user2{x}}y}(\omega) = \frac{{{\user2{f}}_{{\user2{x}}y}(\omega) {\user2{f}}_{\user2{xx}}}^{-1} {\user2{f}}_{{\user2{x}}y}^{*}(\omega)}{f_{yy}(\omega)}, $$

(5)

where

$$ {\user2{f}}_{{\user2{x}}y}(\omega) = \left[\begin{array}{ll} f_{x_1y}(\omega) f_{x_2y}(\omega) \end{array}\right]\quad\quad {{\user2{f}}_{\user2{xx}}}(\omega) = \left[\begin{array}{ll} f_{x_1x_1}(\omega) & f_{x_1x_2}^{*}(\omega) \\ f_{x_1x_2}(\omega) & f_{x_2x_2}(\omega) \end{array}\right] $$

(6)

and * now corresponds to the conjugate transpose. Equation 5 generalises for multivariate $\{{\user2{x}}_t\}$. The equivalent metric in the time domain is the coefficient of determination for a multiple linear regression model.

Seasonally stratified power spectra are calculated as follows. Consider the time series x _s , defined as a 181-day subset of {x _t } centred on t = s:

$$ x_s = \{ x_t | t = s-90,\ldots,s-1,s,s+1,\ldots,s+90 \}. $$

(7)

All points in time occurring on February 29 are removed so that each year has exactly 365 days. The evolutionary power spectrum, as a function of both ω and s, is given by

$$ f_{xy}(\omega,s) = \langle I_{xy}(\omega,s) \rangle $$

(8)

where the periodogram is given by I _xy (ω, s) = d _x (ω, s) d ^* _y (ω, s) and d _x (ω, s) is the discrete Fourier transform of x _s . The seasonal power spectra f ^S _xy (ω, n) is given by the averaging over all values of s which fall on the same day of each year n

$$ f^{\rm S}_{xy}(\omega,n) = \frac{\sum_{i=1}^{N} f_{xy}(\omega,365(i-1) + n) }{N}\quad\hbox{for}\;n=1,2,\ldots, 365 $$

(9)

where N is the number of years in the series. In summary, power spectra are first calculated over 181-day subsets of {x _t } with each successive block shifted by one day to yield the evolutionary spectrum. The seasonal power spectrum is then calculated by averaging across years for the same mid-date of the 181-day subsets (i.e., each January 1, January 2, etc).