Convectively coupled equatorial waves within the MJO during CINDY/DYNAMO: slow Kelvin waves as building blocks

Abstract

This study examines the relationship between the MJO and convectively coupled equatorial waves (CCEWs) during the CINDY2011/DYNAMO field campaign using satellite-borne infrared radiation data, in order to better understand the interaction between convection and the large-scale circulation. The spatio-temporal wavelet transform (STWT) enables us to document the convective signals within the MJO envelope in terms of CCEWs in great detail, through localization of space–time spectra at any given location and time. Three MJO events that occurred in October, November, and December 2011 are examined. It is, in general, difficult to find universal relationships between the MJO and CCEWs, implying that MJOs are diverse in terms of the types of disturbances that make up its convective envelope. However, it is found in all MJO events that the major convective body of the MJO is made up mainly by slow convectively coupled Kelvin waves. These Kelvin waves have relatively fast phase speeds of 10–13 m s⁻¹ outside of, and slow phase speeds of ~8–9 m s⁻¹ within the MJO. Sometimes even slower eastward propagating signals with 3–5 m s⁻¹ phase speed show up within the MJO, which, as well as the slow Kelvin waves, appear to comprise major building blocks of the MJO. It is also suggested that these eastward propagating waves often occur coincident with n = 1 WIG waves, which is consistent with the schematic model from Nakazawa in 1988. Some practical aspects that facilitate use of the STWT are also elaborated upon and discussed.

Appendix: Energy conversion from STWT space to Fourier space

Presenting the calculated STWT spectra in its native space (as a function of scale and speed tuning parameter) makes the STWT difficult to compare with popular FT approaches and hinders interpretation. In order to fill the gap in representation, we developed a method to convert the STWT spectra in terms of zonal wavenumber and frequency. By considering how a sinusoidal wave can be represented in the STWT and FT, $a$ and $c$ can be associated with $k$ and $\omega$ as follows (Kikuchi and Wang 2010):

$$\begin{aligned} a & =\frac{1}{2}{(\frac{{k_M}{\omega _M}}{k\omega} )^{1/2}} \\ c & =\frac{{k_M}\omega}{k{\omega _M}} \\ \end{aligned}$$

(13)

where ${k_M}={k_0}+{\left( {k_{0}^{2}+2} \right)^{1/2}},\quad {\omega _M}={\omega _0}+{\left( {\omega _{0}^{2}+2} \right)^{1/2}}.$

In light of the energy conservation (3), the discretized STWT spectrum ${\mathop {\tilde{W}}\nolimits^{} _{i,n}}$ in Fourier space at zonal wavenumber ${k_i}$ and frequency ${f_n}$ can be estimated using the computed STWT spectrum ${W_{j,q}}$ at ${a_j}$ and ${c_q}$ as

$$\left| {\tilde{W}_{{i,n}} } \right| = \frac{1}{{C_{{\psi _{{c \ne 0}} }} }}\sum\nolimits_{{j = 1}}^{J} {\sum\nolimits_{{q = 1}}^{Q} {\frac{{\left( {log2} \right)^{2} }}{{a_{j}^{2} }}} } \left| {W_{{j,q}} } \right|^{2} \frac{\alpha _{{i,n;j,q}}} {\Delta k\Delta \omega}$$

(14)

where ${\alpha _{i,n;j,q}}$ is the coefficient that measures the ratio of the area of the segment $(j,q)$ enclosed by the curves $P_{1}^{'},~P_{2}^{'},~P_{3}^{'},~P_{4}^{'}$ to the total area of the segment $(j,q)$ (Fig. 11), and $\Delta k$ and $\Delta f$ are the desired zonal wavenumber and frequency resolutions of the ${\mathop {\tilde{W}}\nolimits^{} _{i,n}}$ spectrum in the Fourier space.

Fig. 11

Schematic illustrating how the STWT spectra are represented in terms of the Fourier space. a Spectrum at zonal wavenumber k and frequency f represents the energy contained in a rectangular defined by ${P_1}\left( {{{k - \Delta k} \mathord{\left/ {\vphantom {{k - \Delta k} 2}} \right. \kern-0pt} 2},\;{{f - \Delta f} \mathord{\left/ {\vphantom {{f - \Delta f} 2}} \right. \kern-0pt} 2}} \right),\quad {P_2}\left( {{{k+\Delta k} \mathord{\left/ {\vphantom {{k+\Delta k} 2}} \right. \kern-0pt} 2},\;{{f - \Delta f} \mathord{\left/ {\vphantom {{f - \Delta f} 2}} \right. \kern-0pt} 2}} \right),\;{P_3}\left( {{{k+\Delta k} \mathord{\left/ {\vphantom {{k+\Delta k} 2}} \right. \kern-0pt} 2},\;{{f+\Delta f} \mathord{\left/ {\vphantom {{f+\Delta f} 2}} \right. \kern-0pt} 2}} \right),\;{P_4}\left( {{{k - \Delta k} \mathord{\left/ {\vphantom {{k - \Delta k} 2}} \right. \kern-0pt} 2},\;{{f+\Delta f} \mathord{\left/ {\vphantom {{f+\Delta f} 2}} \right. \kern-0pt} 2}} \right)$ and b the corresponding energy represented in the wavelet domain. The curves $P_{1}^{\prime },\;P_{2}^{\prime },\;P_{3}^{\prime },\;P_{4}^{\prime },$ are obtained by means of Eq. (13)