1 2D Continuous wavelet transform

More documents

Recommendations

Info

2Dcwttools.tex Last compiled: Monday 22 nd January, 2007 15:33Matlab’s atan function returns an angle that is defined from [−π/2, π/2]. Becausethe argument is the ratio of the imaginary and real components, the function can notdistinguish between phase angles where the sign of the real and imaginary components areswapped. Matlab’s atan2 function accepts as inputs the imaginary and real componentsand can distinguish the full 2π angle of phase. However the result from atan2 is definedon the [−π, π] interval. We must therefore adjust the result to fit onto the [0, 2π] interval.1.3 Analysis and diagnosticsHere we list some diagnostic tools that we can calculate from the coefficients of the wavelettransform [?] (repeat of annual review).1.3.1 Energy diagnosticsThe first is the space-scale energy density (proposed by Moret-Bailly et al 1991)E(l, x) = | ˜f(l, x)| 2(22)l nwhich can be integrated over scale to obtain the local energy density∫ ∞E(x) = C −1ψE(l, x) dl(23)0 l+Instead of integrating over scale, we can integrate over space to obtain the globalwavelet energy spectrum (a function of scale)∫E(l) = E(l, x)d n x (24)R nwhich is related to the Fourier energy spectrum∫E(l) = E(k)| ˆψ(lk)| 2 d n k (25)R nwhich is equivalent to the Fourier energy spectrum of the signal smoothed by thewavelet Fourier energy spectrum (E(k) = | ˆf(k| 2 ).We can recover the total energy of the signal by integrating over scaleE = C −1ψ∫ ∞E(l) dl0 l+(26)1.3.2 Local intermittency factorThe local intermittency factor measures the spatially local deviation from the Fourierenergy spectrum (or the global energy spectrum?? wavelet energy spectrum averagedover position) at a given scale.I(l, x) =| ˜f(l, x)| 2< | ˜f(l, x)| 2 > x(27)8
2Dcwttools.tex Last compiled: Monday 22 nd January, 2007 15:33Signals where I(l, x) = 1 have energy distributed evenly in space. Regions of signalswhere the intermittency factor is much greater than 1 correspond to intermittent features.(presumably these intermittent features will correspond to the coherent structures!)1.3.3 Local Reynolds numberFor a signal that corresponds to a turbulent flow we can construct the Reynolds numberRe as the non-dimensional ratio of the nonlinear advection terms to the linear viscousterms in the Navier-Stokes equations. This gives the level of nonlinearity of the flowand is a measure of the intensity of turbulence (i.e. Re ≫ 1 is generally turbulent,though depending upon the specifics flow, and is indicative of the range of scales ofmotion excited). A global Reynolds number, based upon characteristics of the flow, canbe estimated asRe = UL(28)νwhere U is a characteristic velocity scale (i.e the RMS turbulent velocity), L a characteristiclength scale, and ν the kinematic viscosity of the flow. We can construct theequivalent from the wavelet coefficients. A space-scale Reynolds number (Farge et al1990)ṽ(l, x)lRe(l, x) = (29)νwhere ṽ(l, x) is the characteristic RMS velocity that can be calculated by summingover the CWT of the various components of the velocity. i.e. in 3Dṽ(l, x) = [(3C ψ ) −13∑|ṽ n (l, x)| 2 ] −1/2 (30)n=1One can recover a local (in scale) Reynolds number by integrating over space∫Re(l) = Re(l, x)d n x (31)R n...global flow intermittency1.3.4 Local Rossby numbersHere I propose a measure of the local Rossby number for a rotating turbulent flow. TheRossby number Ro is a non-dimensional parameter based upon the ratio of the nonlinearterms to the Coriolis term in the Navier-Stokes equations with rotation. There are twoglobal Rossby numbers that one can construct. One is based upon the advection termRo =U(32)2ΩLwhere U is a characteristic velocity scale, 2Ω is the background vorticity (the rotationof the system), and L is a characteristic length scale of the flow. Another Rossby numbercan be constructed from the nonlinear term containing the time derivative of the velocity.9
Page 1 and 2: 2Dcwttools.tex Last compiled: Monda
Page 10: 2Dcwttools.tex Last compiled: Monda

1 2D Continuous wavelet transform

You also want an ePaper? Increase the reach of your titles

Delete template?

Save as template?