Hayashi Space–Time Spectral Analysis

Hayashi Space–Time Spectral Analysis—frequently paired with its modern tropical diagnostic extension, the Wheeler–Kiladis Wavenumber–Frequency Spectrum—is the canonical two-dimensional Fourier cross-spectral method used in dynamical meteorology, oceanography, and climate diagnostics to decompose a longitude–time geophysical field Ψ(λ, t) into eastward-propagating (P_+k,+ω), westward-propagating (P_−k,+ω), and standing wave variance components as a function of zonal wavenumber k and temporal frequency ω. Formulated by Japanese atmospheric physicist Yoshikazu Hayashi at the Geophysical Fluid Dynamics Laboratory (GFDL) in a sequence of foundational papers (1971, 1973, 1977, 1979, and 1982) building on earlier space–time power formulations by Kao (1968) and Deland (1964), the method resolves the directional ambiguity of one-dimensional spatial Fourier series by computing the temporal auto-spectra, cospectrum, and quadrature spectrum between the real cosine harmonic coefficient C_k(t) and sine harmonic coefficient S_k(t) around a latitude circle. In 1999, Matthew Wheeler and George N. Kiladis (Journal of the Atmospheric Sciences) combined Hayashi's directional space–time power decomposition with an equatorial symmetric/antisymmetric latitudinal partition and red-noise background spectrum division on satellite-observed outgoing longwave radiation (OLR). The resulting Wheeler–Kiladis diagram revealed that the power spectrum of convectively coupled tropical cloudiness in the atmosphere of Earth aligns along the theoretical Taroh Matsuno (1966) shallow-water equatorial β-plane dispersion curves—isolating Equatorial Kelvin waves, n = 1 Equatorial Rossby waves, n = 0 Mixed Rossby–Gravity (Yanai) waves, n = 0, 1, 2 Eastward and Westward Inertio-Gravity waves, and the intraseasonal Madden–Julian Oscillation (MJO).

The Directional Ambiguity of Spatial Fourier Harmonics

Consider a real-valued meteorological or oceanographic field Ψ(λ, ϕ, t)—such as 200-hPa zonal wind u, 500-hPa geopotential height Z, sea surface temperature, or satellite outgoing longwave radiation (OLR)—sampled around a closed latitude circle ϕ₀ as a periodic function of longitude λ ∈ [0, 2π) and time t. Expanding Ψ(λ, t) at each instant t into a spatial Fourier series across integer zonal wavenumbers k = 0, 1, 2, … (where k counts the number of complete wavelengths encircling the globe at latitude ϕ₀, corresponding to dimensional zonal wavenumber k_dim = k / (a cos ϕ₀)): Ψ(λ, t) = C₀(t) + ∑_k=1^∞ [ C_k(t) cos(k λ) + S_k(t) sin(k λ) ] where the time-dependent cosine coefficient C_k(t) and sine coefficient S_k(t) are given by the orthogonal spatial projections:

• C_k(t) = (1 / π) ∫₀^2π Ψ(λ, t) cos(k λ) dλ
• S_k(t) = (1 / π) ∫₀^2π Ψ(λ, t) sin(k λ) dλ

As noted by Deland (1964) and Hayashi (1971), computing the temporal power spectra P_ω(C_k) and P_ω(S_k) separately cannot distinguish whether a wave of zonal wavenumber k and positive temporal frequency ω > 0 is propagating eastward, propagating westward, or oscillating in place as a standing wave (A cos(kλ) cos(ωt) = (1/2)A cos(kλ − ωt) + (1/2)A cos(kλ + ωt)).

Cosine–Sine Quadrature Spectrum and the Hayashi Decomposition Formula

To resolve the direction of zonal phase propagation via the fast Fourier transform (FFT), Hayashi (1971) expanded Ψ(λ, t) into traveling progressive waves moving eastward (kλ − ωt) and westward (kλ + ωt): Ψ(λ, t) = ∑_k=1^∞ ∑_ω>0 [ R_+k,+ω cos(kλ − ωt + θ_+k,+ω) + R_−k,+ω cos(kλ + ωt + θ_−k,+ω) ] Applying trigonometric addition identities shows that for an eastward-traveling wave cos(kλ − ωt) = cos(kλ)cos(ωt) + sin(kλ)sin(ωt), the sine coefficient S_k(t) lags C_k(t) by 90° (+π/2 phase quadrature), whereas for a westward-traveling wave cos(kλ + ωt) = cos(kλ)cos(ωt) − sin(kλ)sin(ωt), S_k(t) leads C_k(t) by 90° (−π/2 phase quadrature).

Let P_ω(C_k) and P_ω(S_k) denote the one-sided temporal power spectral densities of C_k(t) and S_k(t) at frequency ω > 0, and let K_ω(C_k, S_k) − i Q_ω(C_k, S_k) denote their complex cross-spectrum, where K_ω(C_k, S_k) is the in-phase cospectrum and Q_ω(C_k, S_k) is the 90°-out-of-phase quadrature spectrum. Hayashi (1971) proved that the total eastward-propagating variance P_+k,ω and westward-propagating variance P_−k,ω are given by the algebraic identities:

• Eastward-Propagating Power (c_x = +ω / k > 0):

P_+k,ω = (1 / 4) [ P_ω(C_k) + P_ω(S_k) ] + (1 / 2) Q_ω(C_k, S_k)

• Westward-Propagating Power (c_x = −ω / k < 0):

P_−k,ω = (1 / 4) [ P_ω(C_k) + P_ω(S_k) ] − (1 / 2) Q_ω(C_k, S_k)

• Net Zonal Propagation Power Difference:

P_+k,ω − P_−k,ω = Q_ω(C_k, S_k)

Partitioning Into Standing vs. Traveling Wave Variance

When an eastward wave R_+k,ω and a westward wave R_−k,ω of the same wavenumber k and frequency ω are phase-locked (coherent)—for example, due to continental land–sea heating contrasts or stationary orographic forcing alongside Charney–DeVore multiple equilibria—they interfere to form a standing wave oscillation. In his 1977 and 1979 refinements, Hayashi showed that a standing wave generates a non-zero difference between the cosine and sine auto-spectra (P_ω(C_k) − P_ω(S_k) ≠ 0) and/or a non-zero cospectrum K_ω(C_k, S_k) ≠ 0:

1. Standing Wave Power Component (P_stand(k, ω)):

P_stand(k, ω) = √[ ( (1 / 4) [ P_ω(C_k) − P_ω(S_k) ] )² + ( (1 / 2) K_ω(C_k, S_k) )² ]

1. Coherence-Adjusted Traveling Wave Power (P_trav,±(k, ω)): If eastward and westward waves are statistically independent (incoherent), their phase difference is random and P_stand(k, ω) vanishes on ensemble averaging, so that the 2D FFT modulus-squared spectrum |Ψ̂(±k, ω)|² equals Hayashi's P_±k,ω directly.

Step 1: Latitudinal Decomposition Into Equatorial Symmetric and Antisymmetric Components

While Hayashi (1971, 1974) originally applied space–time cross-spectral analysis to midlatitude baroclinic eddies and stratospheric planetary waves (quantified vertically by the Plumb Eliassen–Palm flux), Matthew Wheeler and George N. Kiladis (1999) recognized that equatorially trapped waves on the equatorial β-plane (where the Coriolis frequency is f = β y with β = 2Ω / a ≈ 2.29 × 10⁻¹¹ m⁻¹ s⁻¹ and the Rossby number increases near the equator) possess eigenstates characterized by Hermite polynomials D_n(y / L_e) of meridional mode index n = 0, 1, 2, … that are either symmetric or antisymmetric about the equator (ϕ = 0°).

Before computing the space–time FFT across longitude and time, Wheeler and Kiladis decomposed the tropical field Ψ(λ, ϕ, t) within the equatorial waveguide (15°S ≤ ϕ ≤ 15°N) at conjugate latitudes (±ϕ) into two orthogonal sub-fields:

1. Equatorially Symmetric Component (Ψ_sym):

Ψ_sym(λ, ϕ, t) = (1 / 2) [ Ψ(λ, +ϕ, t) + Ψ(λ, −ϕ, t) ] For divergence, geopotential, or OLR, the symmetric spectrum isolates modes with even Hermite structure in pressure/OLR (and odd meridional wind v): the Equatorial Kelvin wave (n = −1), the n = 1 Equatorial Rossby wave, the n = 1 Inertio-Gravity waves (EIG1, WIG1), and the symmetric component of the Madden–Julian Oscillation (MJO).

1. Equatorially Antisymmetric Component (Ψ_anti):

Ψ_anti(λ, ϕ, t) = (1 / 2) [ Ψ(λ, +ϕ, t) − Ψ(λ, −ϕ, t) ] For OLR or geopotential, the antisymmetric spectrum isolates modes with odd Hermite structure in pressure/OLR (and symmetric cross-equatorial meridional wind v peaking on the equator): the n = 0 Mixed Rossby–Gravity (Yanai) wave, the n = 0 Eastward Inertio-Gravity wave (EIG0), the n = 2 Equatorial Rossby wave, and the n = 2 Inertio-Gravity waves (WIG2, EIG2).

Step 2: Overlapping Time Windowing and Latitudinal Power Averaging

Following the Welch–Bartlett spectral estimation procedure:

• Multi-decadal daily or twice-daily satellite OLR or reanalysis records are detrended, tapered with a Hanning window across overlapping 96-day time segments, and transformed via a two-dimensional fast Fourier transform over longitude λ and time t at each latitude ϕ ∈ [0°, 15°].
• The resulting raw symmetric power P_sym(k, ω; ϕ) and antisymmetric power P_anti(k, ω; ϕ) are summed between 15°S and 15°N and ensemble-averaged over all 96-day windows.

Step 3: Red-Noise Background Spectrum Construction and Signal-to-Noise Ratio Division

When plotted in raw power units (W² m⁻⁴ per wavenumber per frequency), the space–time spectrum of tropical convection is dominated by a smooth "red-noise" continuum—power increases steeply toward low frequencies (ω → 0) and small planetary wavenumbers (k → 0), obscuring individual wave dispersion ridges.

Wheeler and Kiladis constructed an empirical smoothed background spectrum B(k, ω) by averaging [ P_sym(k, ω) + P_anti(k, ω) ] / 2 and applying repeated 1–2–1 smoothing passes in frequency ω and zonal wavenumber k. Dividing the raw symmetric and antisymmetric spectra by this shared red-noise background:

• R_sym(k, ω) = P_sym(k, ω) / B(k, ω)
• R_anti(k, ω) = P_anti(k, ω) / B(k, ω)

transforms the red-noise mound into a high-contrast signal-to-noise ratio (SNR) map (where contour thresholds of R ≥ 1.1 to 1.2 exceed the 95%–99% χ² significance level).

Alignment With Taroh Matsuno's (1966) Shallow-Water Equatorial Wave Theory

When Wheeler and Kiladis (1999) plotted the background-divided ratios R_sym(k, ω) and R_anti(k, ω) of satellite OLR (1979–1996), the significant spectral ridges aligned onto the analytical dispersion curves derived by Taroh Matsuno (1966) for linearized shallow-water waves on an equatorial β-plane: ( ω / c )² − k_dim² − ( β k_dim / ω ) = (2n + 1) ( β / c ) (for n = 0, 1, 2, …) together with the non-dispersive Equatorial Kelvin wave branch (ω = + c k_dim, where v ≡ 0). Here c = √(g h_e) is the gravity-wave speed governed by the equivalent depth h_e, and the equatorial Rossby radius is L_e = √(c / β).

Why Convectively Coupled Waves Exhibit Shallow Equivalent Depths (h_e ≈ 12–50 m)

A major physical insight revealed by the Wheeler–Kiladis diagram concerns the magnitude of the observed equivalent depth h_e:

• For a dry, adiabatic, first-baroclinic vertical mode spanning a tropical troposphere of depth H_T ≈ 15–16 km (vertical wavelength λ_z ≈ 30 km, static stability N² ≈ 1.2 × 10⁻⁴ s⁻²), normal-mode separation predicts h_e,dry ≈ 200 m and a fast phase speed of c_dry = √(g h_e,dry) ≈ 45–50 m s⁻¹ (observed in dry stratospheric Kelvin waves).
• In contrast, the convectively coupled tropospheric OLR peaks fall between h_e = 12 m and h_e = 50 m (centered near h_e ≈ 25 m), corresponding to a slowed phase speed of c ≈ 11 to 22 m s⁻¹.
• This order-of-magnitude reduction in equivalent depth occurs because latent heat release inside deep cumulonimbus convection partially cancels adiabatic cooling of ascending air parcels—reducing the effective moist static stability (N_eff² ≪ N_dry²) via mechanisms analyzed in Charney–Eliassen CISK, stratiform-instability second-baroclinic coupling (Mapes, 2000), and Emanuel's WISHE mechanism.

The Madden–Julian Oscillation (MJO) as a Distinct Non-Matsuno Spectral Peak

The symmetric Wheeler–Kiladis spectrum also highlights a prominent spectral peak at eastward zonal wavenumbers k = +1 to +3 and intraseasonal periods of 30 to 60 days (frequency ω ≈ 0.017 to 0.033 cycles per day, eastward phase speed ~5 m s⁻¹): the Madden–Julian Oscillation (MJO). Unlike the Kelvin, Equatorial Rossby, Mixed Rossby–Gravity, and Inertio-Gravity peaks, the MJO spectral island does not lie along any constant-h_e Matsuno shallow-water dispersion curve, demonstrating that the MJO is not a linear shallow-water Kelvin wave, but a multi-scale moisture mode coupled to radiative-convective feedbacks, surface latent-heat fluxes, and horizontal moisture advection across the Indo-Pacific warm pool (complementing tropical circulation frameworks such as the Lindzen–Hou Hadley circulation and the Kuo barotropic instability theorem).

The Hayashi–Wheeler–Kiladis space–time spectral diagnostic is a standard benchmark in the World Climate Research Programme's Coupled Model Intercomparison Project (CMIP5 / CMIP6) and operational weather prediction centers:

• Diagnosing Cumulus Parameterization Fidelity: General circulation models (GCMs) whose deep convective parameterizations trigger too easily with weak moisture sensitivity produce a flat wavenumber–frequency spectrum lacking a distinct MJO peak and displaying overly fast Kelvin waves (h_e > 50–90 m), whereas models incorporating prognostic convective entrainment and stratiform rain evaporation reproduce the observed h_e ≈ 12–50 m dispersion ridges.
• Real-Time Wave Filtering for Subseasonal Forecasting: By applying an inverse 2D space–time FFT restricted to the spectral bounding box of a single mode (the Wheeler–Kiladis spectral filter for Kelvin, ER, MRG, or MJO bands), forecasters reconstruct real-time longitude–time Hovmöller diagrams of equatorial waves to predict tropical cyclogenesis and monsoon active/break cycles 1 to 4 weeks in advance.

Why is the quadrature spectrum needed in Hayashi's space–time spectral analysis?

Expanding a meteorological field around a latitude circle into spatial cosine and sine harmonics (C_k(t) cos(kλ) + S_k(t) sin(kλ)) and computing only their separate temporal power spectra P_ω(C_k) and P_ω(S_k) discards the relative temporal phase between the cosine and sine waves, so one cannot tell whether the wave is traveling eastward or westward. In an eastward-traveling wave cos(kλ − ωt), S_k(t) lags C_k(t) by 90° (+π/2), producing a positive quadrature spectrum (Q_ω(C_k, S_k) > 0); in a westward-traveling wave cos(kλ + ωt), S_k(t) leads C_k(t) by 90° (−π/2), producing a negative quadrature spectrum (Q_ω < 0). Hayashi's formula uses ±(1/2)Q_ω(C_k, S_k) to separate eastward from westward wave power.

Why do Wheeler–Kiladis diagrams divide the raw spectrum by a smoothed red-noise background?

The raw wavenumber–frequency power spectrum of tropical cloudiness (OLR) and rainfall is dominated by a steep red-noise background continuum in which variance increases by orders of magnitude toward low frequencies (ω → 0) and planetary scales (k = 1–3). In a raw power plot, higher-frequency equatorially trapped waves appear only as faint shoulders on the red-noise slope. Dividing the raw symmetric and antisymmetric spectra by a smoothed background B(k, ω) converts the plot into a dimensionless signal-to-noise ratio, causing the narrow dispersion ridges of Matsuno's (1966) equatorial wave modes to stand out clearly above the 95% statistical significance threshold.

Why is the equivalent depth (h_e ≈ 12–50 m) of convectively coupled equatorial waves much smaller than the dry tropospheric value (~200 m)?

In a dry atmosphere without condensation, a vertical wave spanning the 15-km tropical troposphere has an equivalent depth of h_e ≈ 200 m and a gravity-wave speed of c = √(g h_e) ≈ 45–50 m s⁻¹. When an equatorial wave is coupled to deep moist convection and stratiform anvil clouds, latent heat release warms ascending parcels and nearly cancels adiabatic cooling, while stratiform rain evaporation cools the lower troposphere (creating a shorter, second-baroclinic vertical wavelength). Together, this reduction in effective static stability and shorter vertical wavelength reduces the effective equivalent depth to h_e ≈ 12 to 50 m, slowing the convectively coupled wave phase speed to 11–22 m s⁻¹.

1. Matsuno, T. (1966). "Quasi-geostrophic motions in the equatorial area." Journal of the Meteorological Society of Japan, Series II, 44(1), 25–43. https://doi.org/10.2151/jmsj1965.44.1_25
2. Hayashi, Y. (1971). "A generalized method of resolving disturbances into progressive and retrogressive waves by space Fourier and time cross-spectral analysis." Journal of the Meteorological Society of Japan, Series II, 49(2), 125–128. https://doi.org/10.2151/jmsj1965.49.2_125
3. Hayashi, Y. (1979). "A generalized method of resolving transient disturbances into standing and traveling waves by space–time spectral analysis." Journal of the Atmospheric Sciences, 36(6), 1017–1029. https://doi.org/10.1175/1520-0469(1979)036<1017:AGMORT>2.0.CO;2036<1017:AGMORT>2.0.CO;2)
4. Hayashi, Y. (1982). "Space–time spectral analysis and its applications to atmospheric waves." Journal of the Meteorological Society of Japan, Series II, 60(1), 156–171. https://doi.org/10.2151/jmsj1965.60.1_156
5. Wheeler, M., & Kiladis, G. N. (1999). "Convectively coupled equatorial waves: Analysis of clouds and temperature in the wavenumber–frequency domain." Journal of the Atmospheric Sciences, 56(3), 374–399. https://doi.org/10.1175/1520-0469(1999)056<0374:CCEWAO>2.0.CO;2056<0374:CCEWAO>2.0.CO;2)