1. Introduction
Submesoscale processes (SMPs) with typical horizontal scales of Ο(10) km and temporal scales of Ο(1) day (Thomas et al. 2008; Capet et al. 2008b; McWilliams 2016) are ubiquitous in the upper ocean. The Rossby number (Ro = ζ/f, here ζ is the vertical component of relative vorticity, and f is the Coriolis parameter) is ∼Ο(1) for SMPs, indicating the possibility of ageostrophic motions that potentially account for a downscale energy cascade (Boccaletti et al. 2007; Capet et al. 2008d; Molemaker et al. 2010). The strong vertical velocity caused by SMPs has important influences on the heat budget and biogeochemical characteristic of the upper layer (Lévy et al. 2001; Klein and Lapeyre 2009; McWilliams 2016; Mahadevan 2016; Su et al. 2018). Furthermore, SMPs can suppress the deepening of the mixed layer by restratification (Hosegood et al. 2008; Mahadevan et al. 2010; Parekh et al. 2015). SMPs have been extensively studied over the past two decades, especially regarding their generation mechanisms (McWilliams 2016), instabilities (Thomas et al. 2013; Dewar et al. 2015), and spatiotemporal characteristics (Mensa et al. 2013; Qiu et al. 2014; Callies et al. 2015; Buckingham et al. 2016; Sasaki et al. 2017; Wang et al. 2018; Zhang et al. 2020). In upper open oceans, mixed layer instability (MLI) and frontogenesis are two important mechanisms converting available potential energy (APE) into kinetic energy (KE), favoring generation of SMPs (Boccaletti et al. 2007; Capet et al. 2008b; Thomas et al. 2008; Callies et al. 2016; McWilliams 2016; Cao et al. 2021). Symmetric instability (SI) is another frontal instability that transfers KE from geostrophic flow to submesoscale (Thomas et al. 2008, 2013).
Seasonality of submesoscale motions has been extensively studied in regional and global oceans (e.g., Mensa et al. 2013; Qiu et al. 2014; Callies et al. 2016; Rocha et al. 2016; Thompson et al. 2016; Wang et al. 2018; Luo et al. 2016; Yu et al. 2019; Dong et al. 2020a,b, 2021). Due to complex dynamical background and hydrological characteristics in the Bay of Bengal (BoB), SMPs exhibit unique regional characteristics, which are quite different from other regions. Exploring SMPs in varied geographical and dynamical settings can incrementally extend the frontiers of knowledge in this research field.
Located in the northeast Indian Ocean, the BoB is subject to semiannually reversing monsoonal wind forcing and is characterized by abundant multiscale processes, such as seasonal basin circulation, mesoscale eddies, and tides (Yu et al. 1991; Durand et al. 2009; Chen et al. 2012; Mukherjee et al. 2014; Cheng et al. 2013, 2017, 2018; Mohanty et al. 2018; Jithin et al. 2020). The BoB receives large freshwater influxes from precipitation and runoff (Sengupta et al. 2016; Sree Lekha et al. 2018), which lead to a shallow salinity-controlled mixed layer that influences the upper-ocean heat content. The southern BoB is the key area that links the BoB to adjacent seas. During June–October, the Southwest Monsoon Current (SMC) flows northeastward around Sri Lanka and bring saltier Arabian Sea Water northward into the BoB. The Northeast Monsoon Current (NMC) flows westward across the basin during December–April, carrying fresher water from the BoB to the Arabian Sea (Jensen 2001; Wijesekera et al. 2015; Jensen et al. 2016). The SMC is accompanied by a cyclonic eddy in the northwest, known as the Sri Lanka Dome (SLD), and an anticyclonic eddy (AE) in the southeast (Wijesekera et al. 2015; Pirro et al. 2020). The AE is larger and stronger and dominates the open ocean east of Sri Lanka. The southern bay is also affected by equatorial remote forcing. Coastal Kelvin waves propagate northward and subsequently radiate as westward Rossby waves that influence the interior bay (Vialard et al. 2009; Cheng et al. 2013; Suresh et al. 2013). In addition, the southern BoB is replete with mesoscale eddies generated from nonlinear unstable Rossby waves reflected from the eastern boundary (Cheng et al. 2017).
Observations and model results have revealed that submesoscale fronts are active in the BoB, with sharper gradients and smaller widths than those in midlatitudes (Sarkar et al. 2016; Ramachandran et al. 2018; Pham and Sarkar 2019; Li et al. 2022). Strong submesoscale fronts lead to pronounced frontal instabilities capable of driving strong ageostrophic motions. Based on mooring observations, Sengupta et al. (2016) reported frequent sharp salinity jumps when submesoscale fronts passed by the mooring. They suggested that submesoscale fronts are likely playing an important role in sustaining near-surface stratification in the northern BoB. Jensen et al. (2018) found that atmospheric convection can drive submesoscale flows in the mixed layer and that the injection of freshwater into the northern BoB influences the variability of SMPs.
A large number of studies examined the BoB in terms of large-scale circulation, mesoscale processes, tides, freshwater plumes, wave dynamics, air–sea interactions, etc., but few explored the SMPs. The existing works, albeit surely expanded our understanding, are mostly confined to a short period and/or a small area. The spatiotemporal characteristics and related dynamical mechanisms still remain unclear.
High-resolution images from the Medium Resolution Imaging Spectrometer (MERIS) reveal fine filaments (10–50 km) of high chlorophyll-a concentrations in summer (Figs. 1a,b) and winter (Fig. 1c). The sea surface height anomaly (SSHA) contours in the figures show the location of mesoscale eddies. The filaments are located between the AE and the SLD during the summer of 2009. In December 2010, a large anticyclonic eddy occupied the region, while weaker filaments appeared along the rim of the eddy. Figure 1 presents rich submesoscale features in the southern BoB during summer and winter.
The present work aims to analyze the spatiotemporal features and possible mechanisms of SMPs in the southern BoB. Potential generation mechanisms like frontogenesis, MLI, and SI will be discussed specifically. The characteristics of SMPs in the southern BoB are supposed to be quite different from other regions due to significant seasonal variations of the monsoonal currents in this region. The paper is organized as follows: section 2 describes the details of the model configuration and diagnostic methods. Section 3 presents the spatiotemporal features of SMPs. Section 4 analyzes the underlying mechanisms. A brief conclusion and a discussion are provided in section 5.
2. Data and methods
a. Data
1) MITgcm LLC4320 outputs
The output of the Massachusetts Institute of Technology general circulation model (MITgcm) on a latitude–longitude–polar cap (LLC) grid (LLC4320) is used to investigate the features of SMPs in the southern BoB. LLC4320 has 13 faces, and each face has 4320 × 4320 grids, which is from an ocean-only numerical model. The initial conditions of LLC4320 are from the Estimating the Circulation and Climate of the Ocean, Phase II (ECCO2), project (Menemenlis et al. 2008). It is forced by 6-hourly ERA-Interim atmospheric fields and by hourly tides containing the 16 most significant tidal components. Detailed descriptions of the spinup of this simulation can be found in Menemenlis et al. (2008). This model has a horizontal resolution of 1/48°, i.e., nearly 2 km in the BoB. In the vertical, LLC4320 includes 90 levels with ∼1-m vertical resolution at the surface. The model was run for approximately 14 months, from September 2011 to November 2012. This model run has been previously used to study the seasonality of SMPs in the Kuroshio Extension, Gulf of Mexico, and global oceans (Rocha et al. 2016; Su et al. 2018; Dong et al. 2020a; Yang et al. 2021). The daily mean outputs including temperature, salinity, and horizontal and vertical velocity are analyzed in this study.
2) Observation data
The observed sea surface salinity (SSS), SSHA, and geostrophic current anomaly are used to assess the simulation skills of the model. Detailed information for these observation datasets is listed in Table 1.
Observation data used to assess the model simulation skills.
3) Model validation
It is necessary to examine whether the LLC4320 can simulate the basic dynamic features in the BoB before performing a detailed analysis. Because SMPs in the mixed layer in this region are closely related to salinity fronts, the seasonal mean SSS from LLC4320 is compared with Aquarius SSS during 2011/12 (Fig. 2). In this study, spring is from April to June, summer is from July to September, fall is from October to December, and winter is from January to March. The pattern of SSS of LLC4320 is in agreement with that of Aquarius, with fresher water in the northeastern side and salter water around Sri Lanka. Their consistency is much pronounced in winter and summer, although the LLC4320 result is somewhat fresher than that from Aquarius.
The seasonal SSHA and current anomalies from the model are compared with the altimetry data from the AVISO project (Fig. 3). The modeling results during 2011/12 largely resemble the observations (left and center columns in Fig. 3), well representing the SMC in summer and NMC in winter. Overall, the model is capable of reproducing the basic circulations in the southern BoB quite well. The simulated and observed patterns of SSHA and current anomalies during 2011/12 are similar to the climatological mean, except that the dipole pattern in the SSHA during winter of 2011/12 is absent in AVISO winter climatology. This discrepancy is reasonable for that the dipole pattern is generally driven by local wind, which varies from year to year. The overall consistency between LLC4320 and AVISO climatology indicates that the seasonal features in the LLC4320 is universal. Thus, it can be inferred that the seasonal features of SMPs that will be presented in the following sections are unlikely occasional.
b. Methods
1) Decomposition of spatial scales
Previous studies have introduced several methods to decompose variables into different spatial scale components. In this study, the decomposition scale is derived from the theoretical spatial scale of disturbances developed from baroclinic instabilities, i.e., the local deformation radius L = NH/f, where N is the buoyancy frequency averaged within the mixed layer, H is the mixed layer depth (MLD), and f is the Coriolis parameter. In our study domain, the estimated values of the parameters are N ≈ 0.013 s−1, H ≈ 30 m, f ≈ 1.77 × 10−5 s−1, and the resultant decomposition scale L ≈ 30 km (each parameter was averaged over the entire simulation period and the region within 78°–94°E, 4°–13°N). We also estimate L following the spectral method proposed by Cao and Jing (2022). In this method, L is defined as the scale at which the ratio of divergence (δ = ux + υy) and rotation (ζ = υy − ux) contribution to kinetic energy is close to 0.1. In this way, the decomposition scale is estimated at about 33 km, which is very close to the former method. Besides, the spectral slope of surface kinetic energy versus radial wavenumber approaches k−2 at scales less than 30 km (Fig. 4), also confirms the chosen scale (30 km) is reasonable.
The length scale of SMPs in the BoB varies with latitude, ranging from ∼30 km at 4°N to ∼10 km at 15°N (figure not shown). Wang et al. (2018) suggested that the horizontal scale of submesoscale eddies in the equatorial region is more than an order of magnitude greater than that at midlatitudes. In the southern BoB, the typical spatial length of SMPs is larger than that at midlatitudes and smaller than that in the eastern tropical Pacific (ETP). The typical spatial scale of submesoscale motions in the ETP is about 80–500 km (Wang et al. 2018). The LLC4320 output with horizontal resolution of about 2 km can better capture SMPs in the southern BoB than at higher latitudes.
Using 30 km as the decomposition scale and applying a high-pass filter in the spatial domain on the original fields, the physical variable V (e.g., velocity, density, buoyancy) are decomposing into the large-scale/mesoscale fraction
2) Diagnostic analysis of SMPs
SMPs are highly ageostrophic, characterized by a high Ro of order O(1) (Thomas et al. 2008). Therefore, the characteristic of high Ro is used as an indicator for SMPs in this study. Here,
3) Multiscale window transform
To further investigate the characteristics of SMPs and their generation mechanisms, scale decomposition is also performed using the multiscale window transform (MWT) method (Liang and Anderson 2007; Liang 2016). The MWT strictly ensures the orthogonal decomposition of an original flow field into multiscale windows, and the resulting energetics retain both spatial and temporal dependence. Based on two predefined critical scales, the original u = u(t) can be decomposed into three parts: the large-scale window, the mesoscale window, and the submesoscale window, which are denoted by superscripts 0, 1, 2, respectively: u(t) = u∼0(t) + u∼1(t) + u∼2(t) (appendix A). In this study, the critical scales are selected as 128 days and 8 days, thus attributing intraseasonal signals to the mesoscale and synoptic signals to the submesoscale. In the following, superscript 0 represents the large-scale window (>128 days), superscript 1 represents the mesoscale window (8–128 days), superscript 2 represents the submesoscale window (<8 days).
Mathematical form and meaning of the energy terms in Eqs. (8) and (9). For details, see Liang (2016) and Yang et al. (2021).
3. Spatiotemporal characteristics
a. Spatial distribution of SMPs
To explore the geographic characteristics of SMPs, the annual means of |Ro| at the depth of 5 m for LLC4320 is displayed in Fig. 5. In the model, SMPs are relatively weak in the central bay, but much more active in certain regions, such as along the western boundary, near the islands, and particularly in the southern BoB. Strong SMPs around the islands of the Andaman Sea are likely caused by the interaction between currents and topography, as suggested by Gula et al. (2016). Lin et al. (2020) showed that strong internal tide side in the Luzon Strait is accompanied by the increased vertical velocity. We speculated that the SMPs around the Andaman–Nicobar (AN) Ridge may also be associated with the internal tides. Semidiurnal solitary waves propagating from the Andaman Sea into the southern BoB can also induce submesoscale variability (Jackson et al. 2012; Jensen et al. 2018).
b. Seasonal features of SMPs in the southern BoB
Previous studies demonstrated that local mesoscale dynamic processes significantly impact variations of SMPs (Capet et al. 2008b; Yang et al. 2017; Zhang et al. 2020). The presence of local complex large-scale and mesoscale processes in the southern BoB provides the generation conditions for the vigorous SMPs. Figure 6 shows snapshots of Ro in the southern BoB on a typical winter monsoon day (15 February, Fig. 6a) and summer monsoon day (15 August, Fig. 6c), and two monsoon transition days (15 May and 15 November, Figs. 6b,d) from the LLC4320 simulation. Large Ro values, in the form of submesoscale eddies and filaments, prevail at the depth of 5 m in all seasons. It is generally weak in May and November, the monsoon transition period, with only a few submesoscale eddies and filaments appearing around topography. High Ro values are ubiquitous in February and August.
The seasonal mean |Ro| patterns of the model are shown in Fig. 7 to further illustrate the seasonal features of SMPs. For comparison, we also show the depth-integrated submesoscale KE (SMKE) based on the MWT method (Fig. 8). Like in the snapshots in Fig. 6, SMPs in the LLC4320 are strong around the monsoonal currents during the monsoon periods, especially the summer monsoon, characterized by larger |Ro| and SMKE amplitudes (Figs. 7c and 8c).
To visualize and quantify SMPs, diagnostic variables are calculated in the SMC region east of Sri Lanka (R1, black box in Fig. 7), where SMPs are evident. The vertical distribution of |Ro| averaged over R1 is shown in Fig. 9a. Generally, high values are concentrated in the upper mixed layer (MLD indicated by the black line) and gradually decreased with depth. In this study, MLD is defined as the depth at which the density differs from that at the surface value by 0.2 kg m−3 (Narvekar and Kumar 2014). The mean |Ro| in R1 shows a clear seasonal cycle, being the highest during summer and early fall and weakest in spring and late fall, with a secondary peak signal in late winter. The seasonal cycle of SMPs in the southern BoB is quite different from that in northern BoB. The latter is much stronger in winter than in summer (Li et al. 2022). In the ETP, SMPs are the strongest and weakest in the autumn (September–October) and spring (April–May), respectively (Wang et al. 2018). At midlatitudes such as around the Kuroshio extension, the strength of SMPs is large during February–May (Qiu et al. 2014; Rocha et al. 2016). In the southern BoB, the SMPs are relatively strong during August–October and weak during April–May, which is similar to the seasonal cycle of SMPs in the ETP, except that SMPs in the southern BoB has a secondary peak during February–March.
4. Potential generation mechanisms
a. Frontogenesis
The time series of the regionally averaged F,
It is noticeable that
MLD is another factor that modulates the development of frontogenesis. McWilliams et al. (2009) found that frontogenesis tends to be strengthened in correspondence to the MLD deepening in both idealized and realistic models dominated by mesoscale eddies and fronts. Our results also show that the strong frontogenesis accompanies the deepening of MLD during the southwest monsoon period (Figs. 9a,b). During the northwest monsoon, the NMC is weaker than the SMC (Fig. 3; Wijesekera et al. 2015; Jensen et al. 2016). The shallower MLD and weaker fronts associated with the NMC are responsible for the weaker SMPs. During the monsoon transition period (spring and later fall), the moderate current field and the shallowest MLD both acts to suppress the generation of SMPs.
b. Mixed layer instabilities
Mixed layer instability can generate SMPs by the release of APE to KE, which can be represented by VBF. As expected, high VBF is mainly concentrated in the mixed layer and becomes small underneath (Fig. 9d), as MLI are expected to occur mainly in the mixed layer (Fox-Kemper and Ferrari 2008). As shown in Fig. 9f, mixed layer averaged VBF (denoted by PK) is always positive, indicating the release of APE to EKE. PK is enhanced in summer and winter monsoons but weakened in spring. As mentioned above, MLD and
Destabilizing atmospheric forcing can lead to generation of submesoscale motions by means of reducing the upper-ocean stratification, Richardson number and the PV, and thus favors the occurrence of frontogenesis, MLI and SI (Thomas et al. 2008). The EBF along the northern periphery of the anticyclonic eddy east of Sri Lanka is mainly negative during the southwest monsoon (Fig. 11). The Ekman flow advects lighter water over denser, effectively stratifying the water column and preventing convective mixing. In the southern side of the eddy, the EBF is positive. The Ekman flow is conducive to the deepening of mixed layer and APE generation. During winter, the EBF is positive along the NMC, which tends to increase the buoyancy loss and generate negative PV. During spring, the EBF east of Sri Lanka is negative, which is not conducive to the generation of SMPs. The EBF during fall is somewhat weaker and has no significant effect on the SMPs.
c. Symmetric instability in the southern BoB
The negative Ertel PV is a necessary condition for the occurrence of the SI. Figures 12a and 12b display geographic distributions of surface Ertel PV during winter and summer, respectively. Pronounced negative Ertel PV appears around monsoon currents where large |Ro| and SMKE exist (Figs. 7 and 8). Dong et al. (2021) estimated the spatial scales of SI using LLC4320 and showed that the SI scale generally decreases with latitudes and is uniform zonally. Following Dong et al. (2021), we calculated the SI scale in the southern BoB. As shown in Fig. 13, the SI scale ranges from ∼3 to ∼10 km in the study area, which indicates that the LLC4320 can resolve part of SI in the southern BoB. By employing the balanced RiB criterion introduced in Thomas et al. (2013), the types of instability that play leading roles in southern BoB can be identified. The proportion during study period when the necessary conditions for SI are satisfied (
The above analyses suggest that the ocean conditions and atmospheric forcing along the rim of the anticyclonic eddy during summer favor the occurrence of frontogenesis, MLI, and SI. We further provide more comprehensive and detailed evidence of the generation mechanisms of SMPs by showing snapshots of relevant variables in a front segment on 15 August 2012 (black rectangle in Fig. 6c). Strong velocity lateral shear is found in the periphery of the anticyclonic eddy, which can induce strong stretching and deformation, and therefore strong strain (Fig. 14c). The strain rapidly intensifies frontal density gradients via positive frontogenesis tendency (Figs. 14a,b). Meanwhile, the enhanced lateral buoyancy gradients reinforce the frontal baroclinicity (Fig. 14d), accounting for large negative horizontal component of frontal PV (Fig. 14e). As a result, these dynamical processes provide favorable conditions for SI. A cross-frontal ageostrophic secondary circulation (ASC) is excited due to these frontal instabilities (Fig. 14f), which makes a large contribution to the vertical materials exchange in the upper ocean. The ASC can usually slump isopycnals to restratify the surface mixed layer and restore geostrophic balance by SI.
d. Source of kinetic and available potential energy for the SMPs
The above analyses indicate that SMPs generated in the southern BoB are closely related to larger-scale processes. To better understand the source and sink of energy for the SMPs and the relationship between submesoscale and larger-scale processes, the MWT method is used here to diagnose the energy budget. Figure 15 shows the spatial distributions of the energy terms b∼2,
The large-to-submesoscale KE transfer
The APE transfer terms
The APE and KE budgets based on Eqs. (8) and (9) over the R1 region are shown in Fig. 16. Each energy term is vertically averaged over the upper 50 m and temporally averaged over spring, summer, fall and winter, respectively. For the KE budgets,
Previous studies suggested that the barotropic (baroclinic) energy conversion in the ETP (at midlatitudes) plays dominant roles in the annual cycle of SMKE (Wang et al. 2018; Qiu et al. 2014; Rocha et al. 2016). In the southern BoB, the barotropic and baroclinic energy conversions are comparable to SMKE during summer (
5. Conclusions and discussion
In this article, we investigated the spatiotemporal characteristics and potential mechanisms of SMPs in the southern BoB based on the results of LLC4320. It is found that the southern BoB is rich of SMPs, which mainly occur in the mixed layer. Temporally, SMPs exhibit a significant seasonal cycle during 2011/12, being strong during monsoon periods and weak during monsoon transition periods. Spatially, strong SMPs are primarily found along the SMC and the NMC. The generation of SMPs is associated with frontogenesis, MLI and SI, all closely related to the monsoon currents. During summer monsoon period, as the SMC develops, abundant sharp fronts and strong strains are generated, favoring the occurrence of frontogenesis. Besides, the effects of EBF and AE during summer monsoon period favor deepening MLD, which promotes excitement of MLI and further modulates the SMPs generation. During winter monsoon period, surface cooling can deepen MLD which is conducive to MLI. Surface cooling, together with the increasing buoyancy gradients and strain due to the NMC, enables the existence of abundant SMPs in wintertime. In other times of the year, background large-scale and mesoscale activities are suppressed, and SMPs are therefore also much weaker. The scale of SI derived from LLC4320 ranges from ∼3 to ∼10 km in the southern BoB. SI is more active during summer and winter, with a proportion of 40%–80% during study period when the necessary conditions for SI are satisfied. The negative Ertel PV and positive EBF favor the occurrence of SI.
Energetics diagnosis based on the MWT is employed to further investigate the submesoscale energy budgets. The primary energy source of SMPs is from the background and mesoscale flows. The SMKE are dominantly dissipated through the internal dissipations and inverse cascade toward the background flow, while submesoscale APE is mostly depleted by diffusions.
Based on numerical results, this work qualitatively analyzes the variations of SMPs in the southern BoB. The findings shed some lights to our understanding on SMPs. The seasonal variations and dynamics of SMPs in the southern BoB show some differences from other regions. Even in the BoB, the features of SMPs in the southern part is quite different from the northern part. Li et al. (2022) suggested MLI plays a crucial role in generating SMPs in the northern BoB, while frontogenesis is week. Unlike the southern BoB, SMPs in the northern BoB is much stronger in winter than in summer.
There are still some limitations in this study. First, the model simulations with ∼2-km resolution can only partially resolve the SMPs, missing out some high-wavenumber unbalanced motions like internal gravity waves, which may cause underestimation of the forward energy cascade. Second, although the major source and sink of the submesoscale have been preliminarily analyzed, how exactly are the energies transferred to SMPs still requires further process-based investigation. Higher spatiotemporal resolution and more comprehensive simulations are needed in future work.
Acknowledgments.
This study was supported by the National Natural Science Foundation of China (42276014, 41876002), and supported by High Performance Computing Platform, Hohai University.
Data availability statement.
The chlorophyll data were provided by the MERIS Level 2 FRS dataset (https://merisfrs-merci-ds.eo.esa.int/merci/queryProducts.do). The sea surface height data are distributed at Copernicus and can be downloaded from http://marine.copernicus.eu/, and Aquarius sea surface salinity data can be obtained from http://apdrc.soest.hawaii.edu. The LLC4320 output can be accessed at https://xmitgcm.readthedocs.io/en/latest/llcreader.html.
APPENDIX A
Multiscale Window Transform
The multiscale window transform (MWT) is a functional analysis tool that was developed by Liang and Anderson (2007) to satisfy the scale decomposition. The MWT splits a function space into a direct sum of several orthogonal scale windows while retaining its local properties (temporal or spatial locality). Originally, the MWT was developed to faithfully represent multiscale energies on the corresponding scale windows (Liang and Anderson 2007; Liang 2016). Liang and Anderson (2007) found that there is a transfer–reconstruction pair for certain type of specially designed orthogonal filters, that is, MWT and its associated multiscale window reconstruction (MWR). Generally speaking, MWR functions just like a traditional filter for generating decomposed fields in physical space, while MWT gives rise to transform coefficients in phase space to represent multiscale energy on the resulting scale windows.
APPENDIX B
Canonical Transfer
The multiscale interaction is characterized by the energy transfer across different scale windows, which is closely related to barotropic and baroclinic instability processes. However, a detailed description for the energy transfer process is essential. Fortunately, it has faithfully been revealed in a canonical transfer theory which was first proposed by Liang and Robinson (2005). The form of canonical transfer was later strictly established by Liang (2016). Next, we only give a brief introduction for the theory of canonical transfer. For more details, we refer readers to Liang (2016).
APPENDIX C
Derivation of Multiscale Energy Equations
For a detailed derivation process for multiscale KE and APE budget equations, we refer the readers to Liang (2016).
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