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Characteristics of oceanic mesoscale variabilities associated with the inverse kinetic energy cascade
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Mengmeng Li1, 2, Zhiliang Liu3, 4, *, Jianing Li5, 6, Chongguang Pang1, 6
Acta Oceanologica Sinica | 2021, 40(7) : 42 - 57
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Acta Oceanologica Sinica | 2021, 40(7): 42-57
• Physical Oceanography, Marine Meteorology and Marine Physics •
Characteristics of oceanic mesoscale variabilities associated with the inverse kinetic energy cascade
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Mengmeng Li1, 2, Zhiliang Liu3, 4, *, Jianing Li5, 6, Chongguang Pang1, 6
Affiliations
  • 1 Key Laboratory of Ocean Circulation and Waves, Institute of Oceanology, Chinese Academy of Sciences, Qingdao 266071, China
  • 2 University of Chinese Academy of Sciences, Beijing 100049, China
  • 3 Research Center for Marine Science, Hebei Normal University of Science and Technology, Qinhuangdao 066004, China
  • 4 Hebei Key Laboratory of Ocean Dynamics, Resources and Environments, Hebei Normal University of Science and Technology, Qinhuangdao 066004, China
  • 5 Physical Oceanography Laboratory, Ocean University of China, Qingdao 266100, China
  • 6 Pilot National Laboratory for Marine Science and Technology (Qingdao), Qingdao 266237, China
Published: 2021-07-25 doi: 10.1007/s13131-021-1814-2
Outline
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Oceanic geostrophic turbulence theory predicts significant inverse kinetic energy (KE) cascades at scales larger than the energy injection wavelength. However, the characteristics of the mesoscale variabilities associated with the inverse KE cascade in the real oceans have not been clear enough up to now. To further examine this problem, we analyzed the spectral characteristics of the oceanic mesoscale motions over the scales of inverse KE cascades based on high-resolution gridded altimeter data. The applicability of the quasigeostrophic (QG) turbulence theory and the surface quasigeostrophic (SQG) turbulence theory in real oceans is further explored. The results show that the sea surface height (SSH) spectral slope is linearly related to the eddy-kinetic-energy (EKE) level with a high correlation coefficient value of 0.67. The findings also suggest that the QG turbulence theory is an appropriate dynamic framework at the edge of high-EKE regions and that the SQG theory is more suitable in tropical regions and low-EKE regions at mid-high latitudes. New anisotropic characteristics of the inverse KE cascade are also provided. These results indicate that the along-track spectrum used by previous studies cannot reveal the dynamics of the mesoscale variabilities well.

mesoscale eddy  /  inverse cascade  /  scalar wavenumber spectrum  /  spectral slope  /  anisotropy
Mengmeng Li, Zhiliang Liu, Jianing Li, Chongguang Pang. Characteristics of oceanic mesoscale variabilities associated with the inverse kinetic energy cascade[J]. Acta Oceanologica Sinica, 2021 , 40 (7) : 42 -57 . DOI: 10.1007/s13131-021-1814-2
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1 Introduction
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Energy balance in ocean dynamics, in which ocean eddies play a key role, is crucial to predicting the state of ocean circulation and its evolution. To date, the generation mechanism and the balance process of eddies are still unclear. As predicted by the linear baroclinic instability theory, mesoscale eddies generate and obtain energy mainly from time-mean flows at the deformation scale, Lr (Ferrari and Wunsch, 2009; Kobashi and Kawamura, 2002; Wang et al., 2015; Salmon, 1998). However, the observed energy-containing scales inferred from diagnoses of spectral energy fluxes are approximately 100 km larger than that predicted by the linear theory (Smith, 2007). The inverse kinetic energy (KE) cascade process may be a bridge connecting this gap (Smith, 2007; Wang et al., 2015).
As is well known, two-dimensional (2-D) turbulence predicts two inertial ranges of energy spectra: an energy inertial range carrying KE from Lr to larger scales (inverse cascade) and an enstrophy inertial range carrying enstrophy to smaller scales (forward cascade) (Kraichnan, 1967; Vallis, 2007). After generating at Lr, most mesoscale eddies will grow through the inverse KE cascade process due to nonlinear triad interactions (Stewart et al., 1996; Scott and Wang, 2005; Kobashi and Kawamura, 2002) and equilibrate at some halting scale, Lequ (Vallis, 2007). The features observed by satellites should be around the scales of eddy equilibration near Lequ. Thus, inverse KE cascade within the scales of Lr < L < Lequ is important for understanding how mesoscale eddies interact among themselves and to what extent this interaction contributes to the eddy equilibration process (Qiu et al., 2008). Our fragmentary comprehension of the characteristics of the inverse KE cascade motivates this study.
Typical theories give different interpretations of the generation and evolution of eddies. The QG turbulence theory suggests that baroclinic instability is driven by mean quasigeostrophic potential vorticity (QGPV) gradient sign reversal and occurs in the deep ocean (~1 km) (Charney, 1971; Wang et al., 2019). It predicts a k–5 (k–11/3) power law for sea surface height (SSH) wavenumber spectra in the inverse KE cascade (forward enstrophy cascade) regime (Kraichnan, 1967; Leith, 1968; Batchelor, 1969). Nevertheless, the surface quasigeostrophic (SQG) theory demonstrates that baroclinic instability is driven by an interaction of the mean surface gradient with a constant interior potential vorticity (PV) and dominated by surface intensified modes (about 100 m to 200 m) (Blumen, 1978). This theory correspondingly predicts SSH spectra following a k–11/3 (k–3) power law (Held et al., 1995). Hence, calculating the SSH spectral slope and comparing the results with theoretical predictions could clarify the applicability of the QG and SQG theories in real oceans, leading to further understanding of the characteristics and dynamics of the inverse KE cascade.
Related studies have intensively emerged with the development of satellite observations (Fu, 1983; Scott and Wang, 2005; Wortham IV, 2013; Le Traon et al., 2008; Vergara et al., 2019; Wang et al., 2019). The results suggest that the mesoscale wavenumber spectra of SSH are somewhat diverse, with spectral slopes ranging from approximately –11/3 to –5, depending on the local energy level (Fu, 1983; Le Traon et al., 2008; Xu and Fu, 2011, 2012; Khatri et al., 2018). However, no uniform conclusions regarding the efficacies of the QG and SQG theories have been presented until now. Thus, the characteristics of the KE inverse cascade are still unclear. Furthermore, most efforts have used a fixed wavelength band (i.e., 100–250 km for Fu (1983); 100–300 km for Le Traon et al. (2008); or 70–250 km for Xu and Fu (2011, 2012)). Actually, the length bands corresponding to the two inertial ranges vary globally (Eden, 2007; Wang et al., 2015), and the SSHs observed by altimeters are in an enstrophy cascade regime from approximately 200 km to 100 km and in an inverse KE transfer regime for scales greater than approximately 200 km in the mid-latitude oceans (Khatri et al., 2018). Thus, most of the related studies using altimeter data have not considered the difference between the two inertial ranges; consequently, the interpretations of the SSH spectral slopes may not be correct.
To further understand the dynamics of the mesoscale motions associated with the inverse KE cascade in the real oceans, we use high-resolution globally gridded sea level anomaly (SLA) data to revisit the wavenumber spectral characteristics over the scales of the inverse KE cascade. To identify the scale band associated with inverse KE cascade more accurately, an improved geographically dependent wavenumber band is defined to compute the global pattern of the SSH wavenumber spectral slope for the inverse KE cascade. The applicability of the QG and SQG theories in real oceans and the correlation between the slope value and eddy-kinetic-energy (EKE) are further discussed. Some new anisotropy characteristics of the mesoscale variabilities are also provided by comparing the statistical characteristics between the zonal and meridional wavenumber spectra.
This paper is organized as follows. The data and methods are described in Section 2. The geographic variability in the SSH wavenumber spectral slope and the relationship between the slope and EKE are shown in Section 3. In Section 4, we compare the zonal and meridional SSH wavenumber spectral slopes to analyze the anisotropic characteristics of the inverse cascade. In Section 5, comparisons are made with previous studies, followed by a discussion. The conclusions are given in the last section.
2 Data and methods
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2.1 Description of the dataset
We used 20 years of the delayed gridded SLA product operated by the Data Unification and Altimeter Combination System (DUACS) and distributed by the Copernicus Marine Environment Monitoring Service (CMEMS). This dataset merges SSH measurements from 10 altimeter missions and deduces the long-term mean to yield the SLA data used in this research. Two versions of the dataset were used in this study. The version with a lower spatial ((1/3)° Mercator grid) and temporal resolution (7 d) but a lower error level was used to calculate the spectral KE flux, and the version with a higher spatial ((1/4)° Cartesian grid) and temporal resolution (1 d) but a higher error level (DT 2014) was used to calculate the SSH wavenumber spectra. The period of the two datasets ranges from 1993 to 2012.
The two datasets enable much better detection of mesoscale singles compared to using a single satellite (Pascual et al., 2006). One caveat is that the smoothing and interpolation techniques implemented during the gridding process cause energy attenuation at small scales, thereby limiting the capture capability of the gridded product (Le Traon et al., 1998). Thus, the spectra estimated using gridded data are steeper than those estimated using raw along-track data (Wortham IV, 2013; Wang et al., 2019). Fortunately, compared with the previous versions, the improvements in DT2014 reprocessing have led to a better effective resolution, reaching approximately 100 km at middle latitudes (Pujol et al., 2016). Furthermore, the noise in the measurements will shallow the spectra calculated using the raw along-track data (Xu and Fu, 2012). Indeed, the gridded data lead to SSH spectra with a similar shape to that obtained from along-track altimetry data, removing high-frequency noise (Zhou et al., 2015).
2.2 Preprocessing the SLA data
The data processing procedures are described as follows. First, 100-d high-pass filtering was used for the raw SLA to eliminate the wave signals (the reason for this operation is referenced in Appendix A). For the filtered SLA, a 10-d average was calculated first to remove high-frequency information that was not of interest. To avoid the influence of ice on the SSH measurements, the areas close to the polar sea (poleward of 60°S and 60°N) were excluded (Xu and Fu, 2011). Then, for each coordinate, a 10° × 10° box centered at the object point was selected for subsequent calculations (Qiu et al., 2008; Tulloch et al., 2011). The SLA field was detrended by fitting a linear plane by least squares and then subtracting the plane from the corresponding snapshot. To avoid slid-lobe power leakage, a Hanning window function was applied to all the detrended data in the box, such that each box tapered to zero at each edge. Points with more than 10% missing values in the corresponding box were omitted. Then, each box was cut into smaller boxes (9° × 9°), and each missing value was replaced by the average of the values of the 9 × 9 grid points centered on its position.
2.3 SSH wavenumber spectrum and EKE
2.3.1 Scalar SSH wavenumber spectrum and EKE
The 2-D SSH wavenumber spectra, ${E_2}({k_x},\,{k_y})$, at each grid point were computed by fast Fourier transformation (FFT) for SLA, where ${k_x}$ and ${k_y}$ are the two orthogonal wavenumber components. The scalar SSH wavenumber spectra, ${E_0}(K)$, were calculated using the following function:
$\int_0^\infty {{E_0}(K){\rm{d}}K = \int_{ - \infty }^{ + \infty } {\int_{ - \infty }^{ + \infty } {{E_2}({k_x},{k_y}){\rm{d}}{k_x}{\rm{d}}{k_y}} } }.$
The discrete format is described as follows:
${E_0}(K) = \sum\limits_{K = \sqrt {{k_x}^2 + {k_y}^2} } { {E_2}({k_x},\,{k_y})},$
where K is the scalar wavenumber, defined as $K = {({k_x}^2 + {k_y}^2)^{1/2}}$. Then, the time-averaged spectra, $E(K)$, at each grid point were computed.
Under the geostrophic balance, it was easy to calculate the geostrophic KE spectrum, $F(K)$, using the following relation (Fu, 1983):
$F(K) = \frac{{{{{g}}^2}}}{{2{f^2}}}{(2{\rm{\pi }}K)^2}E(K),$
where ${{g}}$ is the gravity constant, $\,f = 2\Omega \sin \theta $ is the Coriolis parameter, $\Omega $ is the angular velocity of Earth’s rotation, and $\theta $ is the geographical latitude. Integrating $F(K)$ over $K$ yields the total EKE in the box:
${\rm{EKE}} = \int {F(K){\rm{d}}K} = \int {\frac{{{{{g}}^2}}}{{2{f^2}}}{{(2{\rm{\pi }}K)}^2}E(K)} \;{\rm{d}}K.$
2.3.2 One-dimensional SSH wavenumber spectrum and EKE
The meridional (zonal) SSH wavenumber spectrum for each grid was obtained by summing all the 2-D spectra in the box with wavenumber ${k_y} < {k_x}$ (${k_x} < {k_y}$):
${E_{\rm{m}}}(K) = \sum\limits_{{k_y} < {k_x}} {{E_2}({k_x},{k_y}){\rm{d}}{k_x}{\rm{d}}{k_y}},$
${E_{\rm{z}}}(K) = \sum\limits_{{k_x} < {k_y}} {{E_2}({k_x},{k_y}){\rm{d}}{k_x}{\rm{d}}{k_y}}.$
The meridional and zonal KE wavenumber spectra and EKE (EKEm and EKEz) were computed using Eq. (4):
${\rm{EK}}{{\rm{E}}_{\rm{m}}} = \int {\frac{{{{{g}}^2}}}{{2{f^2}}}{{(2{\rm{\pi }}K)}^2}{E_{\rm{m}}}(K)} \;{\rm{d}}K,$
${\rm{EK}}{{\rm{E}}_{\rm{z}}} = \int {\frac{{{{{g}}^2}}}{{2{f^2}}}{{(2{\rm{\pi }}K)}^2}{E_{\rm{z}}}(K)} \;{\rm{d}}K.$
2.4 Scalar spectral KE flux
The geostrophic velocity anomalies can be estimated from the SLA data by assuming geostrophy:
$u' = - \frac{{{g}}}{f}\frac{{\partial {\rm{SLA}}}}{{\partial y}},$
$v' = \frac{{{g}}}{f}\frac{{\partial {\rm{SLA}}}}{{\partial x}}.$
Following Wang et al. (2015), the spectral energy transfer can be calculated (detailed derivation is referred to Wang et al. (2015)):
$\begin{split} T({k_x},{k_y},t) =& \Re \bigg[{\rm{fft}} {(u')^*}{\rm{fft}} \left(u'\frac{{\partial u'}}{{\partial x}} + v'\frac{{\partial u'}}{{\partial y}}\right) + \\& {\rm{fft}} {(v')^*}{\rm{fft}} \left(u'\frac{{\partial u'}}{{\partial x}} + v'\frac{{\partial u'}}{{\partial y}}\right)\bigg]\bigg/\Delta {k^2},\end{split}$
where $T({k_x},{k_y},t)$ represents the transmission of EKE between the different spatial modes. The star indicates a complex conjugate. $\Delta k = 1/N\Delta x$ cpkm (cycles per kilometer), $\Delta x$ is the grid spacing, and N is the number of grid points in each direction, as derived from the FFT. The KE flux at wavenumber $K'$ can be computed by summing all the spectral energy transfer at modes (${k_x}\,,\;{k_y}$) with $k_x^2 + k_y^2 > K{'^2}$:
${\pi}(K',t) = \sum\limits_{K'^2 < k_x^2 + k_y^2} {T({k_x},{k_y},t)} \Delta {k^2}.$
The ratio of the magnitude of the inverse cascade KE to that of the total cascade KE (forward plus inverse cascade KE) can be further calculated using the time-averaged ${{\pi }}(K',t) $ (denoted as $\prod (K)$):
$R = {\rm{ratio}}\;{\rm{of}}\;{\rm{inverse}}\;{\rm{KE}}\;{\rm{cascade}} = \frac{{\displaystyle\sum\limits_{\prod (K) < 0} {\left| {\prod (K)} \right|} \Delta K}}{{\displaystyle\sum {\left| {\prod (K)} \right|\Delta } K}}.$
Figure 1a shows $\prod (K)$ calculated over the Kuroshio, the Gulf Stream, and the Agulhas regions. Consistent with the results of previous research, all $\prod (K)$ results have a prominent negative lobe that indicate an inverse KE cascade to smaller wavenumbers (Scott and Wang, 2005; Tulloch et al., 2011; Wang et al., 2015). Moreover, the wavelength at which the spectral flux crosses zero with a positive slope can be thought of as the energy injection scale Linj, namely, the start of the inverse KE cascade. We also define the zero point at a large wavelength with a negative slope as the equilibration scale, Lequ, where the inverse KE cascade ends and eddies equilibrate. The percentage of the magnitude of the inverse cascade KE relative to that of the total cascade KE, R, is shown in Fig. 1b. Obviously, the inverse KE cascade is a robust feature of most oceans. It dominates the interscale energy transfer at high latitudes and the western boundaries, while it presents complex variation at low latitudes. It should be mentioned that limited by the resolution of the data, the domination of zero points in the spectral KE flux would produce large calculation errors. Thus, the spectral KE flux can only be used for qualitative analysis, and the Linj and Lequ obtained from it is not accurate enough for the SSH spectral slope calculation.
2.5 Variable wavelength range technique
The inverse cascade band should be rationally determined before calculating the SSH wavenumber spectral slopes. The length scale of mesoscale eddies varies geographically, especially in the meridional direction (Jacobs et al., 2001). Thus, using a variable wavelength range to calculate the spectral slope is more reliable.
2.5.1 Lower limit of the inverse cascade scales
Generally, the deformation scale Lr is considered to be the energy injection wavelength, namely, the start of the inverse cascade (Scott and Wang, 2005). Unfortunately, it is smaller than the effective resolution of DT2014 SLA at mid-high latitudes; hence, we can only use the real Lr value in equatorial areas. Using SSH fields observed by altimeters, Chelton et al. (2011) computed a speed-based eddy scale Ls. This observed scale is larger than Lr in the extratropics, nearly achieving an effective resolution in most oceans (Fig. 12 in Chelton et al., 2011). Moreover, Chelton et al. (2011) did not show precise Ls values in equatorial regions (5°S−5°N). In this context, Lr is replaced by Ls in the extratropics to ensure that the calculation band of the spectrum is not distorted. The shortening of the calculation band does not affect the results significantly because the slope of the inverse cascade is almost constant.
In short, the lower limit of the inverse cascade used in this paper, ${L_{\rm{low}}}$(the corresponding wavenumber is recorded as ${K_{\rm{low}}}$), is a combination of Ls calculated by Chelton et al. (2011) and Lr calculated by Chelton et al. (1998):
${L_{{\rm{low}}}}\; = \left\{ {\begin{aligned}&{{L_{\rm{r}}}},\qquad\;\;{{\rm{for}}\;{5^ \circ }{\rm{S}} - {5^ \circ }{\rm{N}}};\\&{{L_{\rm{s}}}},\qquad\;\;{{\rm{for}}\;{\rm{the}}\;{\rm{other}}\;{\rm{regions}}}.\end{aligned}} \right.$
2.5.2 Upper limit of the inverse cascade scales
At scales larger than the halting scale L0, the energy dissipated by the frictional effect is balanced by the energy transferred through the inverse cascade, such that the KE spectrum and the corresponding SSH spectrum peak at L0 (Vallis, 2007). In this paper, the wavelength corresponding to the spectral peak is used as the upper limit of the inverse cascade.
The global distribution of the magnitude of EKE (m2/s2), ${\rm{lg(EKE)}} $, is shown in Fig. 2. Notably, Eq. (3) is invalid in the equatorial area (5°S–5°N) because the motions there are ageostrophic (Tchilibou et al., 2018). Thus, the EKE value is not shown, but the SSH spectrum is credible here. Furthermore, baroclinic instability is inhibited in equatorial areas, while barotropic instability is more important (Qiu and Chen, 2004), and the motion is dominated by a wavelike regime (Fu, 2004; Theiss, 2004; Chelton et al., 2007).
As revealed in Fig. 2, the EKE around predominantly eastward flowing currents, such as the Kuroshio Extension, the Gulf Stream, the Subtropical Countercurrent (STCC), and the Antarctic Circumpolar Current (ACC), is generally 1 to 2 orders of magnitude higher than that in the eastern basin. The Northeast Pacific and the Southeast Pacific are two distinct low-value areas. The eastern subtropical Pacific, especially in the Northern Hemisphere, has a higher EKE level compared to the surrounding area. Based on this trait, several typical positions were selected for the analysis of the shape of the wavenumber spectrum: the local maximum points of the Kuroshio (KS), the Gulf Stream (GS) and the Agulhas Current (AS), which represent high-EKE regions; three equatorial points in the Indian Ocean (EI), the Pacific Ocean (EP) and the Atlantic Ocean (EA), which represent regions in which it is difficult to distinguish the signals of waves and eddies; and three common low-EKE points in the South Indian Ocean (SIL), South Pacific Ocean (SPL) and South Atlantic Ocean (SAL).
Figure 3 shows the SSH wavenumber spectra in the typical positions selected above. In the high (Fig. 3a)- and low (Fig. 3c)-EKE level regions, the spectra are characterized by a broad peak at wavelengths from 250 km to 350 km and a maximum at approximately 300 km, showing a “red” spectrum at shorter wavelengths and a slight “blue” spectrum at longer wavelengths1. In equatorial regions (Fig. 3b), the spectrum is “red” at all wavelengths without a peak. It is worth mentioning that positions with higher EKE levels show larger amplitudes in their SSH spectra, which can be understood based on Eq. (4). This positive relationship is obvious in the comparison between Figs 3a, b and c.
In conclusion, the maximum spectrum occurs at approximately 300 km, except in the equatorial region, which has a “red” spectrum at all wavelengths. In this case, we used the wavelength corresponding to the highest power density in the SSH wavenumber spectrum as the upper limit, Lup (the corresponding wavenumber is recorded as Kup), in equatorial areas (10°S–10°N) and a constant, 300 km, in other areas.
Based on the discussions above, the definition of a variable wavelength range is shown in Fig. 4. The observed and theoretical inverse KE cascade bands are also shown to confirm that they cover the band used in the following slope calculations. The observed start scale of the inverse KE cascade (Linj) is slightly larger than its theoretical value (the Rossby deformation scale Lr). The lower limit of the wavelength band used to calculate the spectral slope in this study is sandwiched between the observed and theoretical values and is closer to Linj, while the upper limit is lower than the end scale of the inverse KE cascade. Thus, it could be considered that the wavelength band used to calculate the spectral slope belongs to the inverse KE cascade band.
3 Characteristics of the SSH wavenumber spectral slope for the inverse KE cascade
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3.1 Global variability for the SSH wavenumber spectral slope
The slope of the SSH wavenumber spectrum calculated in Section 2 is an intuitive manifestation of the generation of mesoscale eddies and the strength of the inverse KE cascade. A higher absolute value of the slope (referred to as a “steeper slope” in the following text), namely, a steeper spectrum, implies that eddies will transfer more KE upscale per unit wavelength band, which is described as a faster (or stronger) inverse KE cascade in the following discussion. Figure 5a demonstrates the global distribution of the SSH wavenumber spectral slope for the inverse KE cascade. For the vast mid-latitude oceans, which have been suggested to be geostrophically turbulent oceans (Stammer, 1997), the slopes have a wide range of –1.00 to –5.50. The predominantly eastward flowing currents (such as the Kuroshio Extension and the GS, STCC and ACC), the AS Current, the South Indian Ocean and the Brazilian Current, in which the dynamics are highly nonlinear (Chelton et al., 2011), may exhibit a much stronger inverse KE cascade than other areas. Specifically, the spectral slopes are steeper than –5 in the core of the high-EKE regions and flatter than –3.50 in the low-EKE regions of the tropics, the eastern basin of the Pacific Ocean and areas of the Atlantic Ocean away from the mean currents.
Notably, unlike in previous works, we shifted our focus to those characteristics associated with the inverse KE cascade, for which the slope value predicted by the QG and SQG theory are –11/3 and –3, respectively. Figure 5b shows the areas where the spectral slopes follow the QG k–11/3 (red) and the SQG k–3 (blue) power law within the 95% confidence level. Thus, based on our research, the results for the inverse cascade process could be obtained. At mid-high latitudes with high-EKE, the isopycnals slope upward to the pole, and the QGPV gradient (Qy) changes its sign in the deep ocean (Tulloch et al., 2011). The QG turbulence theory is a better dynamic framework. Thus, the inverse KE cascade tends to present a k–11/3 power law in the SSH spectrum at the edges of these regions. In tropical regions, the isopycnals slope upward to the equator, and the β-effect is significant, baroclinic instability is driven by the interaction of the mean surface temperature gradient with β (Tulloch et al., 2011). Thus the SQG theory is a suitable dynamic, and the inverse KE cascade tends to present a k–3 power law in the SSH spectrum in these areas. However, the QG and SQG theories are highly simplified frames, which do not consider the complex processes in the real oceans such as stratification, the formation of coherent structures, the bottom topography, etc. (Charney, 1971; Blumen, 1978; Scott and Wang, 2005). Moreover, typical oceanic velocity profiles contain a mix of both surface and nonconstant interior gradients. Thus, the QG and SQG theory do not always apply.
Overall, the SSH wavenumber spectral slope for the inverse KE cascade band is positively correlated with the local EKE level. However, there are two particular regions in the Northeast and Southeast Pacific, respectively, which have low EKE levels but steep slopes (framed with blue boxes in Fig. 5a). The two regions fit well with the “eddy desert” reported by Chelton et al. (2007), whose central position lies at (50°N, 160°W) and (50°S, 95°W), respectively. They are regions lacking mesoscale eddies and dominated by a wave regime (Stammer, 1997; Liu and Pang, 2017). The inverse KE cascade in these regions is consistent with the QG theory at the edge (Fig. 5b).
3.2 Relationship between SSH spectral slope and EKE
Figures 2 and 5a show an obvious relationship between the value of the SSH spectral slope and the local EKE. To further explore their relationship, the SSH wavenumber spectra at high (solid)- and low (dotted)-EKE level positions were compared in Fig. 6a. The spectra in “eddy desert” regions are also shown to explain the particular phenomenon that occurs (Fig. 6b). It should be mentioned that the values of the EKE and slope are shown in the figures for clear analysis. In most oceans, the power densities at the lower limit scales ($E({K_{\rm{low}}})$) are similar in different EKE-level regions, and the power density at the upper limit scales ($E({K_{\rm{up}}})$) is higher in high-EKE regions. In “eddy desert” regions, the energy injection scales contain lower power densities compared to the adjacent regions with higher EKE levels, and the equilibration scales contain similar power densities in different EKE-level regions.
The SSH spectral slope is recorded as $ - S(K)$, and
$S(K) = - \frac{{{\rm{d}}(\lg E(K))}}{{{\rm{d}}(\lg K)}}.$
For the inverse KE cascade band (${K_{\rm{up}}} < K < {K_{\rm{low}}}$),
$ - S(K) = - {S_0} = \frac{{\lg E({K_{\rm{low}}}) - \lg E({K_{\rm{up}}})}}{{\lg {K_{\rm{low}}} - \lg {K_{\rm{up}}}}} < 0,$
where ${S_0}$ is the slope of the inverse KE cascade that we calculated in Section 3, which is a constant for different wavenumbers at a specified location.
3.2.1 For most of the oceans
As mentioned above, $E({K_{\rm{low}}})$ values are very similar in different EKE-level regions. Thus, for $K \in [{K_{\rm{up}}},{K_{\rm{low}}}\;]$:
$\left\{ {\begin{aligned} & \lg E(K) = \lg E({K_{\rm{low}}}) + {S_0}(\lg {K_{\rm{low}}} - \lg K),\\ & E(K) = E({K_{\rm{low}}}){\left(\frac{{{K_{\rm{low}}}}}{K}\right)^{{S_0}}} .\end{aligned}}\right.$
Then,
$E({K_{\rm{up}}}) = E({K_{\rm{low}}}){\left(\frac{{{K_{\rm{low}}}}}{{{K_{\rm{up}}}}}\right)^{{S_0}}}.$
By substituting Eqs (17) and (18) into Eq. (4), and taking the integral from the minimum wavenumber observed in the box, ${K_{\min }}$, to the maximum one, ${K_{\max }}$, the following is obtained:
$\begin{split} {\rm{EKE}} =& \int_{{K_{\min }}}^{{K_{\max }}} {\frac{{{{\rm{g}}^2}}}{{2{f^2}}}{{(2{\rm{\pi }}K)}^2}E(K){\rm{d}}K} \; \\ = & 2{\left(\frac{{{\rm{\pi }} {\rm{g}}}}{f}\right)^2}\bigg\{ \int_{{K_{\min }}}^{{K_{\rm{up}}}} {{K^2}E(K){\rm{d}}K} + \int_{{K_{\rm{up}}}}^{{K_{\rm{low}}}} {K^2}E({K_{\rm{low}}}) \times \\& {{\left(\frac{{{K_{\rm{low}}}}}{K}\right)}^{{S_0}}}{\rm{d}}K+\int_{{K_{\rm{low}}}}^{{K_{{\rm{max}}}}} {{K^2}E(K){\rm{d}}K} \bigg\} \\ =& 2{\left(\frac{{{\rm{\pi }} {\rm{g}}}}{f}\right)^2}\bigg\{ \int_{{K_{\min }}}^{{K_{\rm{up}}}} {{K^2}E(K){\rm{d}}K} + {K_{\rm{low}}}^{ - 2}E({K_{\rm{low}}}) \times \\&\int_{{K_{\rm{up}}}}^{{K_{\rm{low}}}} {{{\left(\frac{{{K_{\rm{low}}}}}{K}\right)}^{{S_0} + 2}}{\rm{d}}K} + \int_{{K_{\rm{low}}}}^{{K_{{\rm{max}}}}} {{K^2}E(K){\rm{d}}K} \bigg\} . \end{split}$
From the SSH spectrum (Fig. 3, Fig. 6), it can be regarded that $E(K) \approx E({K_{\rm{up}}})$ for $K < \;{K_{\rm{up}}}$, and $\int_{{K_{\rm{low}}}}^{{K_{{\rm{max}}}}} {{K^2}E(K){\rm{d}}K} $ is a small term compared with other two and is ignored here. Therefore,
$\begin{split} \lg {\rm{EKE}} \approx& \lg \bigg\{ 2{\left(\frac{{{\rm{\pi }} {\rm{g}}}}{f}\right)^2}\bigg[\int_{{K_{\min }}}^{{K_{\rm{up}}}} {{K^2}E({K_{\rm{up}}}){\rm{d}}K} + \\ & {K_{\rm{low}}}^{ - 2}E({K_{\rm{low}}})\int_{{K_{\rm{up}}}}^{{K_{\rm{low}}}} {{{\left(\frac{{{K_{\rm{low}}}}}{K}\right)}^{{S_0+2}}}{\rm{d}}K\bigg]} \bigg\} \\ =& \lg \bigg\{ 2{\left(\frac{{{\rm{\pi g}}}}{f}\right)^2}\bigg[\frac{1}{3}({K_{\rm{up}}}^3 - {K_{{\rm{min}}}}^3)E({K_{\rm{low}}}){\left(\frac{{{K_{\rm{low}}}}}{{{K_{\rm{up}}}}}\right)^{{S_0}}} +\\ & {K_{\rm{low}}}^{ - 2}E({K_{\rm{low}}})\int_{{K_{\rm{up}}}}^{{K_{\rm{low}}}} {{\left(\frac{{{K_{\rm{low}}}}}{K}\right)}^{{S_0} + 2}}{\rm{d}}K\bigg]\bigg\} . \end{split}$
Let
$A = \frac{2}{3}{\left(\frac{{{\rm{\pi }} {\rm{g}}}}{f}\right)^2}({K_{\rm{up}}}^3 - {K_{{\rm{min}}}}^3)E({K_{\rm{low}}}) > 0,$
$B = 2{\left(\frac{{{\rm{\pi }} {\rm{g}}}}{f}\right)^2}{K_{\rm{low}}}^{ - 2}E({K_{\rm{low}}}) > 0,$
$\alpha = A{\left(\frac{{{K_{\rm{low}}}}}{{{K_{\rm{up}}}}}\right)^{{S_0}}},$
$\beta = B \int_{{K_{\rm{up}}}}^{{K_{\rm{low}}}} {{{\left(\frac{{{K_{\rm{low}}}}}{K}\right)}^{{S_0} + 2}}{\rm{d}}K},$
thus,
$\begin{split} \lg {\rm{EKE}} \approx & \lg \bigg\{ 2{\left(\frac{{{\rm{\pi }} {\rm{g}}}}{f}\right)^2} \left[ \int_{{K_{\min }}}^{{K_{\rm{up}}}} {{K^2}E({K_{\rm{up}}}){\rm{d}}K} +\right.\\ & {K_{\rm{low}}}^{ - 2}E({K_{\rm{low}}})\int_{{K_{\rm{up}}}}^{{K_{\rm{low}}}} {\left.{{\left(\frac{{{K_{\rm{low}}}}}{K}\right)}^{{S_0}}}{\rm{d}}K \right]} \bigg\} \\ =& \lg \left[A{\left(\frac{{{K_{\rm{low}}}}}{{{K_{\rm{up}}}}}\right)^{{S_0}}} + B\int_{{K_{\rm{up}}}}^{{K_{\rm{low}}}} {{\left(\frac{{{K_{\rm{low}}}}}{K}\right)}^{{S_0} + 2}}{\rm{d}}K\right] \\ =& \lg (\alpha + \beta) .\end{split}$
We have $\left(\dfrac{{{K_{\rm{low}}}}}{{{K_{\rm{up}}}}}\right) > 1$ and A > 0; thus, $\alpha \propto {S_0}$. For $K \in [{K_{\rm{up}}},{K_{\rm{low}}}]$, $\dfrac{{{K_{\rm{low}}}}}{K} > 1$; thus, $\,\beta \propto {S_0}$.
In summary,
$\lg {\rm{EKE}} \propto {S_0},$
namely, lg(EKE) is positively correlated with ${S_0}$.
3.2.2 For the “eddy desert” regions
In the “eddy desert” regions, $E({K_{\rm{low}}})$ is different, while $E({K_{\rm{up}}})$ shows similar values. Thus, for $K \in [{K_{\rm{up}}},{K_{\rm{low}}}\;]$,
$\begin{cases} & \lg E(K) = \lg E({K_{\rm{up}}}) - {S_0}(\lg K - \lg {K_{{\rm{up}}}}),\\& E(K) = E({K_{\rm{up}}}){\left(\dfrac{{{K_{{\rm{up}}}}}}{K}\right)^{{S_0}}}.\end{cases}$
By substituting Eqs (27) and (18) into Eq. (4) and taking integral of the wavenumber, the following is obtained:
$\begin{split} \lg {\rm{EKE}} \approx & \lg \bigg\{ 2{\left(\frac{{{\rm{\pi }} {\rm{g}}}}{f}\right)^2}\bigg[E({K_{\rm{up}}})\int_{{K_{\min }}}^{{K_{\rm{up}}}} {{K^2}{\rm{d}}K} + \\ & \int_{{K_{\rm{up}}}}^{{K_{\rm{low}}}} {{K^2}E({K_{\rm{up}}}){{\left(\frac{{{K_{\rm{up}}}}}{K}\right)}^{{S_0}}}{\rm{d}}K} \bigg]\bigg\} \\ =& \lg \bigg\{ 2{\left(\frac{{{\rm{\pi }} {\rm{g}}}}{f}\right)^2}\bigg[\frac{1}{3}({K_{\rm{up}}}^3 - {K_{\min}}^3)E({K_{\rm{up}}}) +\\ & {K_{\rm{up}}}^{ - 2}E({K_{\rm{up}}})\int_{{K_{\rm{up}}}}^{{K_{\rm{low}}}} {{{\left(\frac{{{K_{\rm{up}}}}}{K}\right)}^{{S_0} + 2}} {\rm{d}}K} \bigg]\bigg\} , \end{split}$
$\lg {\rm{EKE}} \propto \lg \left(\frac{{{K_{\rm{up}}}}}{K}\right) ({S_0} + 2).$
For $K \in [{K_{\rm{up}}},{K_{\rm{low}}}\;]$, $\dfrac{{{K_{\rm{up}}}}}{K} < 1$; thus, $\lg \dfrac{{{K_{\rm{up}}}}}{K} < 0$, and ${\rm{lg(EKE)}} $ is negatively correlated with ${S_0}$.
Dynamically, nearly all of the world ocean is baroclinically unstable (Spall, 2000; Arbic and Flierl, 2004; Smith, 2007), through which large-scale flows feed energy into mesoscale eddies. The eddies at the injection scales contain similar EKE levels, while steeper spectral slopes imply more intense inverse KE cascades, through which the nonlinear triad interaction causes greater KE transfer within the phase space. Thus, larger eddies obtain more KE from the inverse KE cascade, and the integral EKE in the regions with a steeper slope is larger. In contrast, linear dynamics (wave motions) are dominant in the “eddy desert” regions, in which the eddies are prevented from growing larger (Chelton et al., 2007, 2011; Liu and Pang, 2017). Thus, weaker inverse KE cascades and flatter slopes are expected in these areas, which is inconsistent with Fig. 6b. A possible explanation for this phenomenon comes from Fig. 6b and Eq. (29). The “eddy desert” regions are dominated by the wave regime. The relatively larger-scale motions there contain similar energy levels. Thus, the lower is the smaller-scale energy, the steeper is the spectral slope.
The scatter diagram between ${\rm{lg(EKE)}} $ and ${S_0}$ shows the linear correlation clearly (Fig. 7). Notably, the values in the equatorial area (10°S to 10°N), where the quasigeostrophic relationship is invalid, and the polar regions (with latitudes higher than 60°), where the ice influence on observations could not be ignored, were not considered. Figure 7 shows that the SSH spectral slope is strongly related to the lg(EKE), as Eqs (26) and (29) show. The correlation coefficient at the 95% confidence level (represented by “r” in the following discussion) is approximately 0.67, which indicates a significant correlation. The first fitting was performed for the scatters, with a root mean square error (RMSE) of 0.60.
The differently colored scatter points in Fig. 7a are obviously inconsistent with the fitted curve. The scatter diagrams of the “eddy desert” in the Northeast and Southeast Pacific are shown in Figs 7c and d, respectively, which suggest that the scatter points for those regions are contrary to the global trend. The scatter diagram in other low-EKE (<103.2 m2/s2) areas (Fig. 7e) indicates that the spectral slope shows no obvious dependence on the $ {\rm{lg(EKE)}}$. Figure 7b shows a scatter diagram in the global ocean excluding the scatters shown in Figs 7c, d, and e, and r improves to 0.71, while the RMSE of the first fitting is reduced to 0.57.
In conclusion, the absolute magnitude of the slope and lg(EKE) show an obvious linear positive correlation in most of the oceans, except in the “eddy desert” regions, where they are negatively correlated, and in the areas with very low EKE levels (<103.2 m2/s2), where they have no significant correlation.
3.3 Spectral slope in the eastern subtropical Pacific
The eastern subtropical Pacific deserves a special discussion because it has the locally steepest slope (framed with red boxes in Fig. 5a), which was significant in our result but not in previous works (Xu and Fu, 2011, 2012; Vergara et al., 2019). As presented in Fig. 2, the eastern subtropical Pacific, especially the portion in the Northern Hemisphere, has a higher EKE level compared to the surroundings. This area has been suggested to have strong baroclinic instability and energetic nonlinear motions (Chelton et al., 2011; Wang et al., 2015). Hence, the inverse cascade of eddies is energetic in the eastern subtropical Pacific, which implies steep slopes that are consistent with our results. For further exploration, the SSH wavenumber spectra in the eastern subtropical Pacific are compared with the SSH wavenumber spectra in the adjacent regions with common EKE levels in Fig. 8.
According to Eq. (16),
${S_0} = \frac{{\lg E({K_{\rm{up}}}) - \lg E({K_{\rm{low}}})}}{{\lg {K_{\rm{low}}} - \lg {K_{\rm{up}}}}}.$
The eastern subtropical Pacific has the same value of $\lg {K_{\rm{low}}} - \lg {K_{\rm{up}}}$(Fig. 8). The lower power densities at injection scales ($E({K_{\rm{low}}})$) and the higher power densities at equilibration scales ($E({K_{\rm{up}}})$) work together to cause a high value of $\lg E({K_{\rm{up}}}) - \lg E({K_{\rm{low}}})$ and, thus, a steep slope. Based on Eq. (4), the higher $E({K_{\rm{up}}})$ might be the main factor affecting the high EKE level here.
4 Anisotropy of the wavenumber spectrum
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The inverse KE cascade becomes anisotropic when the change in the Coriolis parameter with latitude, namely, the β-effect, is introduced (Rhines, 1975, 1979; Theiss, 2004; Galperin et al., 2010; Wang et al., 2015). Hence, the along-track spectrum used by previous studies cannot fully reveal the dynamics of mesoscale variabilities, especially for scales large enough to feel the β-effect. This section will discuss the anisotropic characteristics of the entire SSH wavenumber spectrum and only the inverse cascade band. The meridional and zonal SSH wavenumber spectra were calculated as the method of Section 2.3.2. Figures 9a and b depict a comparison of the zonal and meridional spectral slopes (recorded as Sz and Sm, respectively). Note that these one-dimensional spectral slopes were computed at the same variable wavelength band used for the scalar spectrum. Here, the relative differences, Sr_diff and EKEr_diff, are shown simultaneously (Figs 9c and d), and they were computed as follows:
${S_{\rm{r\_diff}}} = ({S_{\rm{z}}} - {S_{\rm{m}}})/{S_{\rm{z}}},$
${\rm{EK}}{{\rm{E}}_{\rm{r\_diff}}} = ({\rm{EK}}{{\rm{E}}_{\rm{z}}} - {\rm{EK}}{{\rm{E}}_{\rm{m}}})/{\rm{EK}}{{\rm{E}}_{\rm{z}}}.$
Clearly, anisotropy of the inverse cascade exists in the global ocean and is significant in the tropics, consistent with Stewart et al. (2015) (Fig. 9c). Specifically, Sz is slightly steeper than Sm at latitudes higher than 30° and significantly smoother at low latitudes, especially near 10° (Fig. 9c), implying that the zonal (meridional) inverse cascade of the baroclinic KE is stronger at high (low) latitudes. Furthermore, the geographical distributions of the zonal and meridional spectral slopes share a similar spatial pattern with the gridded spectral slopes, except in subtropical areas, where the meridional slope is significantly steeper. This effect might be due to the seasonal meridional oscillation of the equatorial current system (Peterson and Stramma, 1991; Bourles et al., 1999), which increases the large-scale meridional energy and leads to the overestimation of the meridional slope. Focusing on the entire spectra, EKEz > EKEm at low latitudes (5° to 15°) and a few spots around the ACC and western boundary currents, whereas EKEz < EKEm at mid-high latitudes away from zonal currents. Interestingly, the entire spectra tend to be isotropic in high-EKE regions.
To explore the reasons for the anisotropic characteristics, a comparison between zonal and meridional SSH spectra in typical regions is shown in Fig. 10. In the tropics, the meridional energy is lower than the zonal energy at eddy-dominant scales. This finding implies that eddies are stretched zonally and contribute to flattening the zonal spectrum (Figs 10d, e and f). Indeed, observations have demonstrated that mesoscale signals tend to become zonally elongated due to the β-effect (Wang et al., 2015). Around the strong zonal currents in the tropics and western boundaries, the meridional energy is higher than the zonal energy at large scales (Figs 10a–f). This phenomenon might be caused by the meridional oscillation of predominant zonal currents, which could also be observed as large-scale meridional energy in altimeter observations. These meridionally elongated large-scale signals also contribute to steepening of the meridional spectrum in the tropics. However, in the high-EKE areas around the western boundary currents, the meridional slope is not affected because the meridionally elongated scales exceed the slope calculation band. In terms of EKE, in the tropics, the higher zonal energy at eddy-dominant scales not only offsets the higher meridional energy at large scales but also ultimately results in a higher zonal integral EKE. Meanwhile, in the high-EKE areas around the western boundary currents, the zonal elongation of eddies is not so significant that the integral EKE is higher in the meridional direction. Determining the dynamics of the anisotropic characteristics is an interesting task and requires further research.
Moreover, the anisotropy of the inverse cascade might cause a loss of information from the one-dimensional spectra. Hence, the along-track spectrum used by previous studies cannot reveal the dynamics of the mesoscale variabilities well, and the scalar spectrum is more suitable for answering this question.
5 Comparison and discussion
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5.1 Wavelength band selection for slope calculation
In recent years, the importance of globally variable mesoscale wavelength ranges for mesoscale eddy research has been gaining recognition. Vergara et al. (2019) used the minimum value between the local Rhines scale (LR) and Lr as the lower wavelength limit in a related study. However, LR is regarded to be the equilibrated scale between the nonlinear turbulence and linear Rossby waves, at which the inverse KE cascade is halted (Liu and Pang, 2017; Vallis and Maltrud, 1993). Considering LR as the lower limit of the slope calculation band at which the inverse KE cascade initially starts might be physically unreasonable. Wang et al. (2019) used an artificially lower limit defined as a function of the upper limit, which is somewhat subjective. More realistically and objectively, the slope calculation wavelength band used in this paper has an upper limit that is similar to those in previous studies and a lower limit that is replaced by the eddy scale observed by Chelton et al. (2011).
5.2 Analysis of the applications of the QG and SQG theories
Before comparison, it should be reiterated that previous spectral studies focused on the forward cascade of enstrophy, whereas this study focused on the inverse cascade of baroclinic KE. We could not compare the slope value directly because typical theories provide different slope predictions in the two inertial ranges.
Using 24 days of Seasat data, Fu (1983) suggested that the QG theory is an appropriate dynamics framework in high-EKE areas. Le Traon et al. (1990) used a much longer altimetric record (two years of Geosat data) to investigate the mesoscale variability in the North Atlantic and obtained similar results. Wang et al. (2010) calculated the KE wavenumber spectra in the GS region from ADCP measurements, which again supported the QG theory. On the contrary, recent studies based on more accurate altimeter observations concluded that the SQG theory, with a flatter SSH spectrum, is more suitable in high-EKE areas (Le Traon et al., 2008; Xu and Fu, 2011). However, Xu and Fu (2012) suggested that altimeter instrument noise correction would flatten the SSH wavenumber spectrum, such that the real slope value might be steeper than the SQG prediction.
Based on our results (Fig. 5b), however, even when compared with the QG theory predictions, the SSH spectrum is significantly steeper in high-EKE regions. This discrepancy can be attributed to the quality difference between the gridded and along-track data. The smoothing and filtering in the gridding process might reduce small-scale signals and cause a steeper spectrum, while the measurement noise of the along-track data might flatten the spectrum (Xu and Fu, 2012; Wang et al., 2019). Both of these factors could cause the gridded spectra to be steeper than the corresponding along-track results.
5.3 Map of the SSH wavenumber spectral slopes
The geographical distribution of the gridded SSH wavenumber spectral slopes in the inverse cascade inertial range (Fig. 5a) shares a similar spatial pattern to that of the along-track spectral slopes in the forward cascade range (Xu and Fu, 2012; Vergara et al., 2019; Wang et al., 2019). Furthermore, our map depicts clearer structures of the major currents and a more obvious relationship between the spectral slope and EKE. Moreover, the detailed differences between our results and those of previous studies (in the eastern subtropical Pacific and “eddy deserts”) are shown in Sections 3.2 and 3.3. Vergara et al. (2019) obtained results from SARAL/AltiKa with a high-value center in the eastern subtropical Pacific, which is consistent with our results. In terms of the “eddy deserts”, Wang et al. (2019) showed high-value centers in the scalar spectral slope map but low-value centers in the along-track spectral slope map. Thus, the discrepancy in “eddy deserts” might result from the information mission of the along-track spectrum compared with the scalar spectrum.
5.4 Anisotropy
Wang et al. (2019) obtained zonal and meridional SSH wavenumber spectra using globally gridded SLA data. Compared with the meridional spectra, their zonal spectra are steeper (flatter) at low (high) latitudes, which seems to be perpendicular to but essentially consistent with our results. This apparent discrepancy is attributed to the different methods used to calculated the wavenumber spectrum. They computed zonal (meridional) SSH wavenumber spectra by averaging all the SSH spectra from the zonal (meridional) SLA as follows (Wang et al., 2019):
${E_{\rm{zonal}}}(kx) = {\rm{fft}} ({\rm{SSH}}(x,\,{y_0})),$
${E_{\rm{meridional}}}(kx) = {\rm{fft}} ({\rm{SSH}}({x_0},\,y)).$
Meanwhile, from the geostrophic equation, the following equations can be obtained:
$v = \frac{{g}}{f}\frac{{\partial \eta }}{{\partial x}},$
$u = - \frac{g}{f}\frac{{\partial \eta }}{{\partial y}}.$
Hence, the one-dimensional SSH wavenumber spectra that they computed are perpendicular to the KE wavenumber spectra. This paper computes the zonal (meridional) SSH spectra by summing all the 2-D spectra in a box with wavenumbers ${k_x} < {k_y}$ (${k_y} < {k_x}$), such that they are parallel to the KE wavenumber spectra. To verify this finding, we calculated the one-dimensional spectrum using Eqs (33) and (34) and obtained similar results to those of Wang et al. (2019) (not shown here).
6 Conclusions
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This study turns its attention from the forward enstrophy cascade to the inverse KE cascade. Through the spectral KE flux spectrum, the inverse KE cascade is verified to be a robust feature of most oceans, which is consistent with previous studies (Scott and Wang, 2005; Tulloch et al., 2011; Wang et al., 2015). Using 20-year gridded SLA data observed by multisatellite altimeters, the spectral characteristics of the inverse KE cascade are explored by discussing the geographic patterns of the SSH wavenumber spectral slope. The slope is calculated in a geographically variable wavelength band, which was defined following the speed-based eddy scale Ls calculated by Chelton et al. (2011), the first baroclinic Rossby radius of deformation Lr calculated by Chelton et al. (1998), and the shape of the SSH spectrum itself.
Based on our research, the results of the inverse KE cascade can be obtained. At mid-high latitude regions with high EKE, baroclinic instability occurs in the deep ocean (Tulloch et al., 2011). The QG turbulence theory is a better dynamic framework at the edges of these regions, and the inverse KE cascade in these regions tends to present a k–11/3 power law in the SSH spectrum. In tropical regions, the β-effect is significant, resulting in baroclinic instability surface intensification (Tulloch et al., 2011). Thus, the SQG theory is a suitable dynamic, and the inverse KE cascade tends to present a k–3 power law in the SSH spectrum in these areas. Moreover, the QG and SQG theories do not always apply because the complex processes in the real ocean are not considered and typical oceanic velocity profile contain a mix of both surface and nonconstant interior gradients.
The correlation analysis indicates that the spectral slope and $ {\rm{lg(EKE)}}$ are first-order linearly related with a high correlation coefficient of 0.67 at mid-latitudes. In particular, “eddy deserts” show an exactly opposite correlation, and regions with very low EKE levels (<103.2 m2/s2) show no significant correlation between the spectral slope and ${\rm{lg(EKE)}} $.
The anisotropic characteristics of the inverse cascade were also explored. The anisotropy is obvious in the tropics. There, mesoscale eddies are zonally stretched due to the β-effect, whereas large-scale signals are meridionally elongated due to the meridional oscillation of the predominant zonal currents. The combination of these two factors produces a steeper spectral slope at meridional scales and a higher EKE at zonal scales. In the high-EKE regions around the western boundary currents and the ACC, meridionally elongated large-scale signals do not affect the slope calculation because they exceed the inverse cascade band. Meanwhile, the zonal elongation of mesoscale eddies is not so significant that the integral EKE is higher in the meridional direction. Moreover, because of the widespread existence of anisotropic characteristics in mesoscale eddies, the along-track spectrum used by previous studies cannot reveal the dynamics of the mesoscale variabilities well, and the scalar spectrum is more suitable for this question. The dynamics of the anisotropic characteristics are beyond the scope of this study, constituting an interesting question that requires further research.
Acknowledgements
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The altimeter products are created by the Data Unification and Altimeter Combination System (DUACS) and distributed by the Copernicus Marine Environment Monitoring Service (CMEMS), which can be downloaded from http://marine.copernicus.eu/. The surface current velocity data are referenced in http://apdrc.soest.hawaii.edu/. The global first baroclinic Rossby radius, calculated by Chelton et al. (1998), can be downloaded from http://www-po.coas.oregonstate.edu/. The speed-based eddy scale was obtained and downloaded from http://wombat.coas.oregonstate.edu/eddies/nc_data.html.
Funding
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  • The National Key R&D Program of China under contract Nos 2016YFC0301203 and 2019YFC1407903; the Natural Science Foundation of Hebei Province under contract No. D2019407046; the Hebei Science and Technology Project under contract No. 19273301D; the NSFC-Shangdong Province Joint Fund under contract No. U1406401.
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Article Info
doi: 10.1007/s13131-021-1814-2
  • Receive Date：2020-10-06
  • Online Date：2026-03-03
  • Published：2021-07-25
Article Data
Affiliations
History
  • Received：2020-10-06
  • Accepted：2021-01-09
Funding
The National Key R&D Program of China under contract Nos 2016YFC0301203 and 2019YFC1407903; the Natural Science Foundation of Hebei Province under contract No. D2019407046; the Hebei Science and Technology Project under contract No. 19273301D; the NSFC-Shangdong Province Joint Fund under contract No. U1406401.
Affiliations
    1 Key Laboratory of Ocean Circulation and Waves, Institute of Oceanology, Chinese Academy of Sciences, Qingdao 266071, China
    2 University of Chinese Academy of Sciences, Beijing 100049, China
    3 Research Center for Marine Science, Hebei Normal University of Science and Technology, Qinhuangdao 066004, China
    4 Hebei Key Laboratory of Ocean Dynamics, Resources and Environments, Hebei Normal University of Science and Technology, Qinhuangdao 066004, China
    5 Physical Oceanography Laboratory, Ocean University of China, Qingdao 266100, China
    6 Pilot National Laboratory for Marine Science and Technology (Qingdao), Qingdao 266237, China

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表12种不同金属材料的力学参数
科
Family		 属数
Number of
genus		 种数
Number of
species		 占总种数比例
Percentage of
total species (%)		 属
Genus		 种数
Number of
species		 占总种数比例
Percentage of total
species (%)
鹅膏菌科Amanitaceae 2 11 5.26 鹅膏菌属 Amanita 10 4.78
小菇科 Mycenaceae 2 12 5.74 丝盖伞属 Inocybe 5 2.39
多孔菌科 Polyporaceae 8 14 6.70 蜡蘑属 Laccaria 5 2.39
红菇科 Russulaceae 3 23 11.00 小皮伞属 Marasmius 6 2.87
小菇属 Mycena 11 5.26
光柄菇属 Pluteus 5 2.39
红菇属 Russula 17 8.13
栓菌属 Trametes 5 2.39
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