Kolmogorov 1941 Turbulence Theory (K41)
Energy Spectrum and the Inertial Range
Within the inertial range of fully developed turbulence (high Reynolds number), viscosity \(\nu\) is irrelevant for the energy cascade. The total turbulent kinetic energy is given by the integral of the energy spectrum:
\[ \int_0^{\infty} E(k)\, dk = \frac{1}{2}\langle u^2 \rangle \]
where:
- \(E(k)\) is the spectral energy density
- \(k\) is the wavenumber
- \(\langle u^2 \rangle\) is the mean square velocity fluctuation
Wavenumber Interpretation
The wavenumber \(k\) represents an inverse length scale:
\[ k \; [L^{-1}] \]
Interpretation:
- Large eddies (large length scale \(\ell\)) \(\rightarrow\) small \(k\)
- Small eddies (small length scale \(\ell\)) \(\rightarrow\) large \(k\)
Spectral Energy Density
The dimensional units of the energy spectrum are
\[ E(k) \quad [L^3 T^{-2}] \]
Kolmogorov Hypothesis
For sufficiently high Reynolds numbers and within the inertial subrange, the spectrum depends only on:
- the energy dissipation rate per unit mass, \(\varepsilon\)
- the wavenumber, \(k\)
Therefore,
\[ E(k) = f(\varepsilon, k) \]
with
\[ \varepsilon \quad [L^2 T^{-3}] \]
Dimensional Analysis (Buckingham \(\pi\) Theorem)
Assume a power-law relationship:
\[ E(k) \propto \varepsilon^a k^b \]
Dimensions
\[ [E] = L^3 T^{-2} \]
\[ [\varepsilon] = L^2 T^{-3} \]
\[ [k] = L^{-1} \]
Time Dimension
\[ -3a = -2 \]
\[ a = \frac{2}{3} \]
Length Dimension
\[ 2a - b = 3 \]
Substitute \(a\):
\[ 2\left(\frac{2}{3}\right) - b = 3 \]
\[ \frac{4}{3} - b = 3 \]
\[ b = -\frac{5}{3} \]
Kolmogorov Energy Spectrum
The inertial-range spectrum therefore becomes
\[ E(k) = C \, \varepsilon^{2/3} k^{-5/3} \]
where \(C\) is the Kolmogorov constant, empirically found to be approximately
\[ C \approx 1.5 \]
Physical Interpretation
- Energy is injected at large scales (small \(k\)).
- It cascades through the inertial range without loss.
- Dissipation occurs only at small scales (large \(k\)).
The \(-5/3\) slope is the defining signature of the turbulent energy cascade.
1 Introduction
Environmental time series such as atmospheric CO₂, ocean pCO₂, temperature, or aerosol concentration often exhibit variability across many temporal scales.
A common explanation is turbulent cascade dynamics, where fluctuations generated at large scales are transferred toward smaller scales through nonlinear interactions.
Such signals often display scale-invariant statistics that can be studied using:
- spectral analysis
- increment statistics
- multifractal scaling
2 Spectral analysis
Power spectral density
For a stationary time series \(X(t)\), the autocorrelation function is
\[ C(\tau) = \langle X(t)X(t+\tau) \rangle \]
Using the Wiener–Khinchin theorem, the spectral energy density is
\[ E(f) = \int_{-\infty}^{\infty} C(\tau)e^{-2\pi i f\tau}d\tau \]
where
\(f\) = frequency
\(E(f)\) = power spectral density.
(The Wiener–Khinchin theorem uses autocorrelation because it contains the complete temporal structure of the signal, whereas the mean misses out on meaningful variations)
3 Power-law spectra
Many turbulent signals follow a power law
\[ E(f) \propto f^{-\beta} \]
where \(\beta\) is the spectral slope.
Typical cases
White noise
\[ \beta = 0 \]
Brownian motion
\[ \beta = 2 \]
Passive scalar turbulence
\[ \beta = \frac{5}{3} \]
4 Self-similar processes
A stochastic process is self-similar if
\[ X(\lambda t) \sim \lambda^{H} X(t) \]
where \(H\) is the Hurst exponent.
5 The Hurst exponent
The Hurst exponent describes long-range dependence.
Interpretation
Random walk
\[ H = 0.5 \]
Persistent behaviour
\[ H > 0.5 \]
Anti-persistent behaviour
\[ H < 0.5 \]
6 Relation between spectral slope and Hurst exponent
For self-similar processes such as fractional Brownian motion
\[ \beta = 2H + 1 \]
Therefore
\[ H = \frac{\beta - 1}{2} \]
Example
If
\[ \beta = 1.67 \]
then
\[ H \approx 0.33 \]
7 Increment statistics
Fluctuations at scale \(\tau\) are defined by increments
\[ \Delta_\tau X = X(t+\tau) - X(t) \]
8 Structure functions
Moments of increments are called structure functions
\[ S_q(\tau) = \langle |\Delta_\tau X|^q \rangle \]
If scaling exists
\[ S_q(\tau) \propto \tau^{\zeta(q)} \]
9 Monofractal scaling
For a self-similar process
\[ \zeta(q) = qH \]
This corresponds to monofractal behaviour.
10 Intermittency
Real turbulent signals exhibit deviations from monofractal scaling
\[ \zeta(q) \neq qH \]
Large fluctuations appear more frequently than predicted by Gaussian statistics.
This phenomenon is called intermittency.
11 Lognormal multifractal model
When nergy dissipation follows a lognormal distribution.
The scaling exponent becomes
\[ \zeta(q) = qH - \frac{\mu}{2}(q^2 - q) \]
where
\(\mu\) = intermittency parameter.
12 Interpretation of parameters
Three parameters summarize the multiscale dynamics
spectral slope
\[ \beta \]
long-range memory
\[ H \]
intermittency strength
\[ \mu \]
13 Summary
Environmental time series with turbulent dynamics typically show
- power-law spectra
- long-range correlations
- intermittent fluctuations
These properties can be quantified through spectral slopes, Hurst exponents, and multifractal scaling analysis.
Exercise:
Interpret these plots?
