1 Assessing the equatorial long-wave approximation: asymptotics and observational data analysis (Supplementary Material) 3 by H. Reed Ogrosky and Samuel N. Stechmann 4 1. Kelvin and MRG waves 5 6 The analysis of section 6 is repeated here for the φ and φ 1 components of equation (5.) from the main text corresponding to the Kelvin, MRG, and EIG waves. 7 a. Definition of wave variables 8 1) KELVIN WAVE 9 The φ component of (5.) is a single PDE governing a Kelvin wave, t r + x r =. (S1) 1 ) MRG AND EIG WAVES 11 The φ 1 component of (5.) is a system of two coupled PDEs governing an MRG and EIG wave, 1 t r 1 + x r 1 v =, t v + r 1 =. (Sa) (Sb) 13 Each of the variables in (S) can be expressed as a superposition of plane-wave ansatzes ˆr 1 (k,ω)e i(kx ωt), ˆv (k,ω)e i(kx ωt) ; (S3) 1
14 substituting (S3) into (S) gives i(k ω) 1 1 iω ˆr 1 ˆv =. (S4) 15 We are interested in finding the eigenmodes of (S4); its characteristic equation is ω kω 1 =. (S5) 16 17 There are two solutions, ω j, for j {MRG,EIG } to (S5). Each eigenvalue ω j for j {MRG,EIG } is associated with an eigenvector of the form ( ê j = i ) T, 1. (S6) k ω j 18 19 The resulting eigenvectors are shown in Fig. S1 after normalization. The degree to which the spatial structure of each of these two waves is seen in reanalysis data can be calculated by the projection technique described in section 6. Using this approach, the 1 Fourier coefficients of the MRG and EIG wave structures are defined as MRG(k) = ê MRG (k) ˆr 1 (k), ÊIG (k) = ê EIG (k) ˆv (k) ˆr 1 (k) ˆv (k), (S7) 3 respectively, where crosses denote the conjugate transpose. This spectral data may then be trans- formed back into physical space through an inverse Fourier transform. 4 b. Long-wave theory with wave variables 5 1) KELVIN WAVE 6 The Kelvin wave is unaffected by the long-wave approximation.
7 ) MRG AND EIG WAVES 8 Projecting (5.7), i.e. with δ 1, onto φ 1 results in two coupled PDEs, t r 1 + x r 1 v =, δ t v + r 1 = ; (S8a) (S8b) 9 In the limit of small δ, the system (S8) can be expressed as an eigenvalue problem (written here in terms of ˆv = δ ˆv ), 3 iδ(k ω) 1 1 iδω ˆr 1 ˆv =. (S9) 31 We are interested in finding the eigenmodes of (S9); its characteristic equation is δ ω δ kω 1 =. (S1) 3 33 34 There are again two solutions, ω j, for j {MRG,EIG } to (S1), but in the limit δ both of these roots are singular. Each eigenvalue ω j is associated with an eigenvector of the form ( ê j = i ) T, δ. (S11) k ω j In the long-wave limit, approximate eigenvalues may be found by expanding in powers of δ, ω MRG = δ 1 + O(1), ω EIG = δ 1 + O(1), (S1a) (S1b) 35 36 After a phase shift so that the r 1 component is positive and real, the normalized long-wave eigen- vectors are, to leading order in δ, given by MRG : ê MRG = ( 1, i ) T, EIG : ê EIG = ( 1, i ) T. (S13a) (S13b) 37 Note that the structure of these long-wave eigenvectors is independent of k. 3
38 c. Observational data analysis 39 1) KELVIN WAVE 4 41 4 43 A Hovmoller plot of the Kelvin wave structure isolated in reanalysis data is shown in Fig. S(a) for the one year period 1 July 9 through 3 June 1. Note the abundance of information propagating rapidly to the east, reminiscent of Kelvin waves. The power spectrum of the Kelvin wave structure is shown in the main text in Fig. 18(a). 44 ) MRG AND EIG WAVES 45 46 47 48 49 5 51 5 53 54 Fig. S(b-c) shows the non-long-wave MRG and EIG wave structures isolated in reanalysis data. The MRG wave exhibits the strongest activity between 18 and 9W, similar to the location of highest activity in the meridional winds v (see Fig. 4b in the main text). This can be anticipated in light of the MRG wave being comprised almost entirely of v at moderate to high wavenumbers, as in Fig. S1(a). Note also the large discrepancy between the MRG and EIG structures in amplitude: the EIG wave contains very little power at all frequencies and wavenumbers, while the MRG wave contains significant power at moderate wavenumbers. Both waves contain very little low-wavenumber information. These figures demonstrate that reanalysis data projects weakly onto the MRG and EIG waves over spatial and temporal scales where the long-wave approximation holds. 4
55 56 LIST OF FIGURES Fig. S1. Eigenvector components of the (a) MRG and (b) EIG wave structures........ 6 57 58 Fig. S. Hovmoller plot of (a) Kelvin, (b) MRG, and (c) EIG anomalies from a seasonal cycle. Time period shown is 1 July 9 through 3 June 1.............. 7 5
.5 1 (a) Components of ê MRG Re(r 1 ) Im(v ).5 1 (b) Components of ê EIG -.5 -.5-1 -1 1 k -1-1 1 k FIG. S1. Eigenvector components of the (a) MRG and (b) EIG wave structures. 6
E I G M RG K Jun May Apr Mar Feb Jan Dec Nov Oct Sep Aug 59 6 Jul 9E 18 9W (a).5.5 9E 18 9W.5 (b).5 (c).5 9E 18 9W.5 F IG. S. Hovmoller plot of (a) Kelvin, (b) MRG, and (c) EIG anomalies from a seasonal cycle. Time period shown is 1 July 9 through 3 June 1. 7