Quasinormal scale elimination theory of the anisotropic energy spectra of atmospheric and oceanic turbulence

Progress in the rapidly expanding exploration of planetary atmospheric and oceanic environments demands an adequate qualitative and quantitative representation of various processes in anisotropic turbulence. The existing analytical spectral theories are developed for homogeneous isotropic flows. They quickly become very complicated when expanded to anisotropic flows with waves. It is possible, however, to extend one such theory, the quasinormal scale elimination (QNSE), to stably stratified and rotating flows. Here the results of the theory are compared with a large variety of oceanic and atmospheric flows. These comparisons make it possible to clarify the physics of some processes governing the atmospheric and oceanic dynamics, quantify their spectra, and investigate their latitudinal and longitudinal variabilities. Some of the main results of this analysis are that vertical and horizontal spectra of atmospheric and oceanic turbulence can be derived within QNSE analytically; there exists a quantitative affinity between atmospheric and oceanic spectra; on large scales, spectral amplitudes are determined by the extra strains that cause flow anisotropization, rather than the energy or enstrophy fluxes; and planetary circulations appear to be amenable to classification as flows with compactified (compressed) dimensionality.

Figure 1

Ratio $\mathcal{R}\equiv E_T\left(k_1\right)/E_L\left(k_1\right)$ based upon Eqs. (17) and (18). The latitude $\theta ={30}^{\circ }$ N and the energy transfer rate ${\mathrm{\Pi }}_{\varepsilon }=5\times {10}^{-5}{\text{m}}^2{\text{s}}^{-3}$.

Figure 2

Longitudinal KE spectra from [88] for flight segments in the troposphere and stratosphere in the latitudinal band ${25}^{\circ }\mathrm{N}$–${50}^{\circ }\mathrm{N}$. Red lines are the QNSE analytical solutions (17). The spectrum for the meridional wind is shifted one decade to the right. The latitude for the theoretical lines is $\theta ={30}^{\circ }\mathrm{N}$ and the energy flux is ${\mathrm{\Pi }}_{\varepsilon }=5\times {10}^{-5}{\mathrm{m}}^2{\mathrm{s}}^{-3}$ [99]. An estimate in [100] is ${\mathrm{\Pi }}_{\varepsilon }=6\times {10}^{-5}{\mathrm{m}}^2{\mathrm{s}}^{-3}$. Straight black lines show Kolmogorov's $-5/3$ and synoptic $-3$ slopes.

Figure 3

Latitudinal variation of the NG spectra. The latitudes increase from bottom to top and their values are specified in the plot. The data points are from [88]. The spectra for the meridional wind are shifted one decade to the right.

Figure 4

Comparison of the longitudinal (red) and transverse (blue) observational spectra from [81] (solid lines) with (17) and (18) (dashed lines) for $\theta ={60}^{\circ }\mathrm{N}$ and ${\mathrm{\Pi }}_{\varepsilon }=5\times {10}^{-4}{\mathrm{m}}^2{\mathrm{s}}^{-3}$.

Figure 5

Comparison of the longitudinal (red) and transverse (blue) density-weighed observational spectra from [108] (solid lines) with (17) and (18) (dashed lines) for $\theta ={60}^{\circ }\mathrm{N}$ and ${\mathrm{\Pi }}_{\varepsilon }=5.5\times {10}^{-4}{\mathrm{m}}^2{\mathrm{s}}^{-3}$.

Figure 6

Comparison of the theoretical longitudinal (blue) and transverse (red) second-order structure functions with the empirical ones (black dotted lines) computed by Lindborg [111] based upon the MOZAIC data [98]. Here $\theta ={20}^{\circ }$ and ${\mathrm{\Pi }}_{\varepsilon }=8\times {10}^{-5}{\mathrm{m}}^2{\mathrm{s}}^{-3}$.

Figure 7

The KE spectra from the 3-km mpas global simulation for the troposphere and the stratosphere. Solid lines represent the results of simulations in [118] while the gray line is specified by Eq. (26). Vertical solid lines show the effect of the resolution as elaborated by Skamarock et al. [118].

Figure 8

Longitudinal (red) and transverse (blue) spectra obtained along ${137}^{\circ }\mathrm{E}$ in the northwest Pacific for four latitudinal regions. The latitudes are $\theta ={31}^{\circ }\mathrm{N}$, ${18}^{\circ }\mathrm{N}$, ${14}^{\circ }\mathrm{N}$, and $8^{\circ }\mathrm{N}$ starting at the Kuroshio band and moving southward. The corresponding values of ${\mathrm{\Pi }}_{\varepsilon }$ are estimated at $3\times {10}^{-6}$, $3\times {10}^{-6}$, $2\times {10}^{-6}$, and $3\times {10}^{-6}{\mathrm{m}}^2{\mathrm{s}}^{-3}$. The data are from [133]. The dashed red and blue lines are Eqs. (17) and (18), respectively. The top plate also shows a line with a slope $k_1^{-3}$.

Figure 9

(a) Longitudinal (blue solid line) and transverse (red solid line) spectra obtained from the 13 years of shipboard ADCP measurements north of the Polar Front near Drake Passage at a depth of 59–98 m. The green and red dashed lines are predictions of the QNSE theory for $\theta ={58}^{\circ }\mathrm{S}$ and ${\mathrm{\Pi }}_{\varepsilon }={10}^{-6}{\mathrm{m}}^2{\mathrm{s}}^{-3}$. For comparison, the microscale dissipation in the vicinity of Drake Passage is much lower, in the range of ${10}^{-8}\text{–}{10}^{-10}{\mathrm{m}}^2{\mathrm{s}}^{-3}$ [140]. (b) Results of llc4320 simulations with mitgcm with a resolution of $1/{48}^{\circ }$. The longitudinal and transverse spectra are the blue and red solid lines, respectively. The data and simulation results are from Rocha et al. [84].

Figure 10

Longitudinal (red) and transverse (blue) KE spectra for the subtropical North Pacific region in the mixed layer (50-m depth) and the thermocline (200-m depth) from ADCP observations [80] (thick solid lines). The dashed red and blue lines are Eqs. (17) and (18), respectively, with $\theta ={32}^{\circ }$ and ${\mathrm{\Pi }}_{\varepsilon }=6.0\times {10}^{-7}{\mathrm{m}}^2{\mathrm{s}}^{-3}$ for both depths.

Figure 11

Total KE spectrum at a 20-m depth for ADCP transects averaged over six lines of the CalCOFI data. The transects are season averaged for winter (December to February) and summer (June to August). Here $\theta ={30}^{\circ }\mathrm{N}$ and ${\mathrm{\Pi }}_{\varepsilon }=5\times {10}^{-6}{\mathrm{m}}^2{\mathrm{s}}^{-3}$. The data are from Chereskin et al. [145].

Figure 12

Longitudinal (red) and transverse (blue) velocity spectra for a partial segment south of $36.{64}^{\circ }\mathrm{N}$, outside of the GS meander, at a depth of 30 m from the Oleander project. The spectra are computed using the data of Wang et al. [125]. The thin dashed lines are Eqs. (17) and (18) with $\theta ={36}^{\circ }$ and ${\mathrm{\Pi }}_{\varepsilon }=1.5\times {10}^{-6}{\mathrm{m}}^2{\mathrm{s}}^{-3}$.

Figure 13

Longitudinal (red) and transverse (blue) velocity spectra at a depth of 150 m from the Oleander project [148]. The thin dashed lines are Eqs. (17) and (18) with $\theta ={37}^{\circ }$ and ${\mathrm{\Pi }}_{\varepsilon }=1.2\times {10}^{-5}{\mathrm{m}}^2{\mathrm{s}}^{-3}$.

Figure 14

Transverse velocity spectrum for the Greenland-Portugal transect. The ship-mounted ADCP spectrum (black solid line) and the 95% confidence interval estimated by the multitaper method (gray lines) are from [149]. The red line is the QNSE equation (18) with $\theta ={50}^{\circ }$ and ${\mathrm{\Pi }}_{\varepsilon }=2\times {10}^{-7}{\mathrm{m}}^2{\mathrm{s}}^{-3}$.