Rotary Spectral Analysis

Advanced time-frequency analysis of rotating flows using rotary spectra and continuous wavelet transforms by Claudio Iturra

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Velocity Data Input

Enter your u and v velocity time series with temporal resolution

Input Configuration

⏱️ Temporal Parameters

Analysis Options

Rotary Analysis Theory

🌀 Rotary Components

  • • Clockwise (CW): Negative rotary component
  • • Counterclockwise (CCW): Positive rotary component
  • • Complex velocity: w = u + iv
  • • Rotary decomposition: w = w⁺ + w⁻

📊 Spectral Analysis

  • • Power spectral density: Energy vs frequency
  • • Rotary ratio: CW/CCW energy ratio
  • • Frequency resolution: Δf = 1/(N×Δt)
  • • Nyquist frequency: fₙ = 1/(2×Δt)

🌊 Wavelet Transform

  • • Time-frequency: Localized spectral analysis
  • • Cone of influence: Edge effects boundary
  • • Scale-frequency: s = f₀/f relationship
  • • Wavelet coherence: Phase relationships

🎯 Applications

Oceanography: Inertial oscillations, tidal analysis
Meteorology: Atmospheric waves, cyclone tracking
Geophysics: Seismic wave polarization
Engineering: Rotating machinery vibration

Rotary Spectral Analysis Results

Comprehensive time-frequency analysis of rotating flows

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Enter your velocity data and perform rotary spectral analysis

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Rotary Spectral Theory

Mathematical foundations of rotary analysis and wavelet transforms

Rotary Decomposition

Complex velocity representation:

w(t) = u(t) + iv(t)

Rotary components:

w⁺(t) = ½[w(t) + iw*(t)] (CCW)
w⁻(t) = ½[w(t) - iw*(t)] (CW)

Where w*(t) is the complex conjugate of w(t).

Rotary Spectra

Power spectral densities:

S⁺(f) = |W⁺(f)|² (CCW spectrum)
S⁻(f) = |W⁻(f)|² (CW spectrum)

Rotary ratio:

R(f) = S⁻(f) / S⁺(f)

R > 1: CW dominant, R < 1: CCW dominant

Continuous Wavelet Transform

Wavelet transform of rotary components:

W⁺(a,b) = ∫ w⁺(t)ψ*((t-b)/a)dt
W⁻(a,b) = ∫ w⁻(t)ψ*((t-b)/a)dt

Where ψ(t) is the mother wavelet, a is scale, b is translation.

Common wavelets: Morlet, Paul, DOG (Derivative of Gaussian)

Physical Interpretation

Inertial Oscillations

Near-inertial frequency f = 2Ω sin(φ)

Tidal Currents

Elliptical tidal motion decomposition

Atmospheric Waves

Gravity waves and planetary waves

Turbulent Eddies

Coherent vortical structures