The Colour Figures Captions Only

The Colour Figures Captions only

Chapter 1:

Fig. 1.1: Effective viscosity as a function of scale, reproduced from (Monin, 1972), adapted from (Richardson, 1926). The text (inserted by Monin) should read “region of free turbulence(!)”.

Fig. 1.2: Spectra from ≈ 1000 orbits of the VIRS instrument (Visible Infrared Scanner) on the TRMM satellite channels 1-5 (at wavelengths of 0.630, 1.60, 3.75, 10.8, 12.0 mm from top to bottom, displaced in the vertical for clarity). The data are for the period January through March 1998 and have nominal resolutions of 2.2 km. The straight regression lines have spectral exponents b = 1.35, 1.29, 1.41, 1.47, 1.49 respectively, close to the value b =1.53 corresponding to the spectrum of passive scalars (= 5/3 minus intermittency corrections, see ch.3). The units are such that k = 1 is the wavenumber corresponding to the size of the planet (20000 km)-1. Channels 1, 2 are reflected solar radiation so that only the 15600 km sections of orbits with maximum solar radiation were used. The high-wavenumber falloff is due to the finite resolution of the instruments. To understand the figure we note that the VIRS bands 1, 2 are essentially reflected sunlight (with very little emission and absorption) so that for thin clouds, the signal comes from variations in the surface albedo (influenced by the topography and other factors), while for thicker clouds it comes from nearer the cloud top via (multiple) geometric and Mie scattering. As the wavelength increases into the thermal IR, the radiances are increasingly due to black body emission and absorption with very little multiple scattering. Whereas at the visible wavelengths we would expect the signal to be influenced by the statistics of cloud liquid water density, for the thermal IR wavelengths it would rather be dominated by the statistics of temperature variations - themselves also close to those of passive scalars. Adapted from (Lovejoy et al., 2008).

Fig. 1.3: Spectra of radiances from the Thematic Microwave Imager (TMI) from the TRMM satellite, ≈ 1000 orbits from January through March 1998. From bottom to top, the data are from channels 1, 3, 5, 6, 8 (vertical polarizations, 2.8, 1.55, 1.41, 0.81, 0.351 cm) with spectral exponents b = 1.68, 1.65, 1.75, 1.65, 1.46 respectively at resolutions 117, 65, 26, 26, 13 km, (hence the high wavenumber cut-offs) each are separated by one order of magnitude for clarity. To understand these thermal microwave results, recall that they have contributions from surface reflectance, water vapour and cloud and rain. Since the particles are smaller than the wavelengths this is the Rayleigh scattering regime and as the wavelength increases from 3.5 mm to 2.8 cm the emissivity/absorbtivity due to cloud and precipitation decreases so that more and more of the signal originates in the lower reaches of clouds and underlying surface. Also, the ratio of scattering to absorption increases with increasing wavelength so that at 2.8 cm multiple scattering can be important in raining regions. The overall result is that the horizontal gradients -which will influence the spectrum – will increasingly reflect large internal liquid water gradients.

Fig. 1.4a: A sample of cloud pictures taken looking upward from the ground near midday, Montreal, Quebec. To get a useful resolution of several thousand pixels on a side, the standard 8 bit imagery of commercial digital cameras is not adequate. In the figure it was necessary to scan black and white negatives (with effectively 13 -14 bit dynamical range); the figure shows typical results in the latter case using large format (60 mm x 60 mm) negatives to resolutions (for low lying clouds) down to 50 cm or so. Reproduced from (Sachs et al., 2002).

Fig. 1.4b: The spectra of the 19 (of 38) highest resolution clouds analyzed in with a spectral slope b≈ 2, see fig. 1.4 a for 12 of the samples. Reproduced from (Sachs et al., 2002).

Fig. 1.5a: Intercomparison of various reanalysis fields for 1 Jan 2006, 0Z, ECMWF interim. This shows the specific humidity (top left), temperature (top right), (zonal, meridional wind) middle left and right, and vertical wind and geopotential height (bottom left and right). All fields are at 700 mb, reproduced from (Lovejoy and Schertzer, 2011).

Fig. 1.5b: Inter-comparisons of the spectra of different atmospheric fields from the ECMWF interim reanalysis. Top (red) is the geopotential (b = 3.35), second from the top (green) is the zonal wind (b = 2.40), 3rd from the top (cyan) is the meridional wind (b = 2.40), 4th from the top (blue) is the temperature (b = 2.40) 5th from the top (orange) is the vertical wind (b = 0.4), at the bottom, (purple) is the specific humidity (b = 1.6). All are at 700 mb and between ±45o latitude, every day in 2006 at GMT. The scale at the far left corresponds to 20000 km in the east-west direction, at the far right to 660 km. Note that for these 2-D spectra, Gaussian white noise would yield b = -1 (i.e. a positive slope = +1). Reproduced from (Lovejoy and Schertzer, 2011).

Fig. 1.6a: Aircraft temperature spectra; red slopes are 1.9, black, 1.7. The bottom three curves are averages of 10 samples and each curve is taken at roughly a one year interval, the top curve is the overall ensemble average. The curves are displaced in the vertical for clarity. Adapted from (Chigirinskaya et al., 1994).

Fig. 1.6b: The same as fig. 1.6a but for the horizontal wind spectrum, slopes of 1.68 are indicated. Adapted from (Chigirinskaya et al., 1994).

Fig. 1.6c: Aircraft spectra of temperature (bottom, blue), humidity (middle, red), log potential temperature (top, gold), reference lines β = 2. These are averages over 24 isobaric aircraft “legs” near 200 mb taken over the Pacific Ocean during the Pacific Winter Storms 2004 experiment, the resolution was 280 m (Nyquist wavenumber = (560 m)-1. Adapted from (Lovejoy et al., 2010).

Fig. 1.7a: Adapted from (Lazarev et al., 1994), slope -2.4 indicated (287 radio sondes, 50 m resolution, dropped from 13.3 km altitude.

Fig. 1.7b Typical vertical-horizontal cross section acquired on August 14 2001. The scale (bottom) is logarithmic: darker is for smaller backscatter (aerosol density surrogate), lighter is for larger backscatter. The black shapes along the bottom are mountains in the British Columbia region. The line at 4.6 km altitude shows the aircraft trajectory (b) Enlarged content of the (700–1600 m) box in (a). Note that small structures become more vertically aligned while large structures are fairly flat. The aspect ratio is 1:96. Reproduced from (Lilley et al. 2004).

Fig. 1.7c: Zoom of previous showing that at the small scales, structures are beginning to show vertical (rather than horizontal) “stratification” (even though the visual impression is magnified by the 1:40 aspect ratio, the change in stratification at smaller and smaller scales is visually obvious). Reproduced from (Lilley et al. 2004).

Fig. 1.7d: The lower curve is the power spectrum for the fluctuations in the lidar backscatter ratio, a surrogate for the aerosol density (B) as a function of horizontal wave number k (in m−1) with a line of best fit with slope bh =1.61. The upper trace is the power spectrum for the fluctuations in B as a function of vertical number k with a line of best fit with slope bv = 2.15. Adapted from (Lilley et al., 2004).

Fig. 1.8a: An example of a drop reconstruction. For clarity only the 10% largest drops are shown, only the relative sizes and positions of the drops are correct, the colours code the size of the drops. The boundaries are defined by the flash lamps used for lighting the drops and by the depth of field of the photographs. Reproduced from (Lovejoy and Schertzer, 2008).

Fig. 1.8b: The angle averaged drop spectra for
5 storms, 18 image triplets Corsin-Obukhov passive scalar theory (rain has statistics like tracer). This shows the 3D isotropic (angle-integrated) spectrum of the 19 stereophotographic drop reconstructions, for r, the particle mass density. Each of the five storms had 3–7 ‘scenes’ (from matched stereographic triplets) with 5000−40 000 drops (see table 1) each taken over a 15–30 min period (orange = f207, yellow = f295, green = f229, blue-green = f142 and cyan = f145; the numbers refer to the different storms). The data were taken from regions roughly 4.4 x 4.4 x 9.2 m3 in extent (slight changes in the geometry were made between storms). The region was broken into 1283 cells (3.4 x 3.4x 7.2 cm2, geometric mean = 4.4 cm); we used the approximation that the extreme low wavenumber (log10k = 0) corresponds to the geometric mean, i.e. 5.6 m, the minimum in the plot corresponds to about 40–70 cm). The single lowest wavenumbers (k = 1) are not shown since the largest scales are nonuniform due to poor lighting and focus on the edges. The reference lines have slopes −5/3, +2, i.e. the theoretical values for the Corrsin–Obukhov (l1/3) law and white noise, respectively. Adapted from (Lovejoy and Schertzer, 2008).

Fig. 1.9a: This shows the scaling of hourly surface temperatures from 4 stations in the northwest US, for 4 years (2005-2008) from the US Climate Reference Network. The data are discussed more fully in section 8.1.2. To reduce noise the data were divided into 112 day long sections and the 48 spectra averaged (the rise at the extreme low frequency is connected with the annual cycle; see fig. 8.3 c for the full 4 year spectra and discussion of detrending). One can see that in spite of the strong diurnal cycle (and harmonics), that the basic scaling extends to about 7 days. The reference lines (with absolute slopes 0.2, 2 are theoretically motivated, see ch. 10.

Fig. 1.9 b: 1-D spectra from the thermal infra red over the Pacific Ocean (MTSAT). Analyzed in time (blue (with diurnal peak), in the east-west, direction, (pink, bottom at right), in the north-south direction (orange). Units are such that the highest wavenumber is (60 km)-1 and highest frequency is (2 hrs)-1 (i.e. the Nyquist wavenumber and frequency of data at 30 km and 1 hour resolutions). In ch. 8 we show that the low frequency/wavenumber curvature is an artefact of the finite geometry of the MTSAT scene coupled with some horizontal and space-time anisotropy. The reference line has slope =1.5. Reproduced from (Pinel, 2012).

Fig. 1.9 c: This shows a composite spectrum of the GRIP (summit) ice core d18O (a temperature proxy, low resolutions, (and the first 91 kyrs at high resolution at left), with the spectrum of the (mean) 75oN 20th Century reanalysis temperature spectrum, at 6 hour resolution, from 1871-2008, at 700 mb (right). The overlap (from 10 – 138 yr scales) is used for calibrating the former (moving them vertically on the log-log plot). All spectra are averaged over logarithmically spaced bins, ten per order of magnitude in frequency. Three regimes are shown corresponding to the weather regime (which apparently extends down to turbulent dissipation scales ≈1 ms i.e. another 7 orders of magnitude to the right), with bw= 2; note that the diurnal variation and subharmonic at 12 hours are visible at the extreme right. The central low frequency weather “plateau” is shown along with the theoretically predicted bwc = 0.2 - 0.4 regime; see ch. 10. Finally, at longer time scales (the left), a new scaling climate regime with exponent bc ≈1.4 continues to about 100 kyrs.

This composite uses a single instrumental and single paleo data source: data from the 138 year long 20th Century reanalysis at 75oN. This is roughly the same latitude as the paleo temperatures from d18O proxy temperature series from the famous GRIP Greenland summit ice core (Members, 1993). More details on these data are given in chs. 8, 10.

Fig. 1.9 d: This is similar to fig. 1.9 c, but showing only the high resolution paleo spectrum (GRIP 90kyr, the average of nine consecutive 10 kyr sections at 5.2 yr resolution, left) the the daily resolution, annually detrended 20CR spectrum at 75oN. These empirical spectra are compared with model spectra: the stochastic (Fractionally Integrated Flux model, daily resolution) and the control run of the IPSL GCM at monthly resolution, both dashed.

Fig. 1.10a: ETOPO5 topography data set at 5 minutes of arc resolution (roughly 10 km). The squares delineate regions that were subject to a special intercomparison of continental versus bathymetric/oceanic regions (the H exponents were found to be a bit different ≈0.4 and 0.7 respectively, see ch. 5). Reproduced from (Gagnon et al., 2006).

Fig. 1.10b: A log-log plot of the spectral power as a function of wavenumber for four Digital Elevation Models. From right to left Lower Saxony (with trees, top), without trees (bottom), US (in grey), GTOPO30 and ETOPO5. A reference line of slope -2.10 is shown for comparison. The small arrows show the frequency at which the spectra are not well estimated due to their limited dynamical range (for this and scale dependent corrections, see (Gagnon et al., 2006).

Fig. 1.11: The Ocean, channels 1-8 offset for clarity, 8 visible channels ocean colour, 210km long swath, 28500X1024 pixels, 7m resolution, (8 visible channels). The extreme high wavenumber is (14 m)-1. Adapted from (Lovejoy et al., 2001).

Fig. 1.12: Spectra of six bands of MODIS radiances over a 512x512 pixel region of Spain (at 250 m resolution; k=1 correspoinds to 128 km): E(k)) as a function of the modulus of the wave vector. In order from top to bottom at the point log10k = 0.7, the curves are: purple= Band 6, black = Band 1, magenta = Band 7, light green = Band 2, cyan = Band 4, dark green = Band 3. Reference lines have slopes -1.3. The band wavelengths are (in nm): channel 1: 620–670, 2: 841–876, 3: 459–479, 4: 545–565, 5: 1230–1250, 6: 1628–1652, 7: 2105–2155. These data are used for determining both vegetation and soil moisture indices. Adapted from Fig. 3a (Lovejoy et al., 2007).

Fig. 1.13: This self-affine simulation illustrates the “phenomenological fallacy” since both the top and bottom look quite different while having the same generators of the scale changing operator G (see ch. 6; G is diagonal with elements 0.8, 1.2), same (anisotropic) statistics at scales differing by a factor of 64 (top and bottom blow-up). The figure shows the proverbial geologists’ lens cap at two resolutions differing by a factor of 64. Seen from afar (top), the structures seem to be composed of left to right ridges, however closer inspection (bottom) shows that in fact this is not the case at the smaller scales. Reproduced from (Lovejoy and Schertzer, 2007).