Seasonal Features and Potential Mechanisms of Submesoscale Processes in the Southern Bay of Bengal during 2011/12

Xuhua Cheng aCollege of Oceanography, Hohai University, Nanjing, China
bSouthern Marine Science and Engineering Guangdong Laboratory (Zhuhai), Zhuhai, China

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Lanman Li aCollege of Oceanography, Hohai University, Nanjing, China

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Zhiyou Jing cState Key Laboratory of Tropical Oceanography, South China Sea Institute of Oceanology, Chinese Academy of Sciences, Guangzhou, China

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Haijin Cao aCollege of Oceanography, Hohai University, Nanjing, China

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Guidi Zhou aCollege of Oceanography, Hohai University, Nanjing, China

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Wei Duan aCollege of Oceanography, Hohai University, Nanjing, China

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Yifei Zhou aCollege of Oceanography, Hohai University, Nanjing, China

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Abstract

This study investigates the seasonal features and generation mechanisms of submesoscale processes (SMPs) in the southern Bay of Bengal (BoB) during 2011/12, based on the output of a high-resolution model, LLC4320 (latitude–longitude–polar cap). The results show that the southern BoB exhibits the most energetic SMPs, with significant seasonal variations. The SMPs are more active during the summer and winter monsoon periods. During the monsoon periods, the sharpening horizontal buoyancy gradients associated with strong straining effects favor the frontogenesis and mixed layer instability (MLI), which are responsible for the SMPs generation. The symmetric instability (SI) scale is about 3–10 km in the southern BoB, which can be partially resolved by LLC4320. The SI is more active during summer and winter, with a proportion of 40%–80% during the study period when the necessary conditions for SI are satisfied. Energetics analysis suggests that the energy source of SMPs is mainly from the local large-scale and mesoscale processes. Baroclinic instability at submesoscales plays a significant role, further confirming the importance of frontogenesis and MLI. Barotropic instability also has considerable contribution to the submesoscale kinetic energy, especially during summer.

Significance Statement

Submesoscale processes (SMPs) are ubiquitous in the Bay of Bengal (BoB). Affected by the seasonally reversing monsoon, abundant rainfall and runoff, and equatorial remote forcing, the upper circulation in the BoB is complex, featuring active mesoscale eddies and rich submesoscale phenomena, making the BoB a “natural test ground” for submesoscale studies. It is found in this work that characteristics of SMPs in the BoB are quite different from other regions. In the southern bay, SMPs are most active during the summer and winter monsoons due to the frontogenesis, enhanced mixed layer instability (MLI), and symmetric instability. These findings could deepen our understanding on multiscale dynamic processes and energy cascade in the BoB and have implications for the study of marine ecology and biogeochemical processes.

© 2023 American Meteorological Society. For information regarding reuse of this content and general copyright information, consult the AMS Copyright Policy (www.ametsoc.org/PUBSReuseLicenses).

Corresponding author: Xuhua Cheng, xuhuacheng@hhu.edu.cn

Abstract

This study investigates the seasonal features and generation mechanisms of submesoscale processes (SMPs) in the southern Bay of Bengal (BoB) during 2011/12, based on the output of a high-resolution model, LLC4320 (latitude–longitude–polar cap). The results show that the southern BoB exhibits the most energetic SMPs, with significant seasonal variations. The SMPs are more active during the summer and winter monsoon periods. During the monsoon periods, the sharpening horizontal buoyancy gradients associated with strong straining effects favor the frontogenesis and mixed layer instability (MLI), which are responsible for the SMPs generation. The symmetric instability (SI) scale is about 3–10 km in the southern BoB, which can be partially resolved by LLC4320. The SI is more active during summer and winter, with a proportion of 40%–80% during the study period when the necessary conditions for SI are satisfied. Energetics analysis suggests that the energy source of SMPs is mainly from the local large-scale and mesoscale processes. Baroclinic instability at submesoscales plays a significant role, further confirming the importance of frontogenesis and MLI. Barotropic instability also has considerable contribution to the submesoscale kinetic energy, especially during summer.

Significance Statement

Submesoscale processes (SMPs) are ubiquitous in the Bay of Bengal (BoB). Affected by the seasonally reversing monsoon, abundant rainfall and runoff, and equatorial remote forcing, the upper circulation in the BoB is complex, featuring active mesoscale eddies and rich submesoscale phenomena, making the BoB a “natural test ground” for submesoscale studies. It is found in this work that characteristics of SMPs in the BoB are quite different from other regions. In the southern bay, SMPs are most active during the summer and winter monsoons due to the frontogenesis, enhanced mixed layer instability (MLI), and symmetric instability. These findings could deepen our understanding on multiscale dynamic processes and energy cascade in the BoB and have implications for the study of marine ecology and biogeochemical processes.

© 2023 American Meteorological Society. For information regarding reuse of this content and general copyright information, consult the AMS Copyright Policy (www.ametsoc.org/PUBSReuseLicenses).

Corresponding author: Xuhua Cheng, xuhuacheng@hhu.edu.cn

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.

Fig. 1.
Fig. 1.

Daily 300-m resolution MERIS chlorophyll (color; mg m−3) in the southern BoB on (a) 15 Jun 2009, (b) 27 Aug 2009, and (c) 12 Dec 2010. The black contours indicate the sea surface height anomaly (SSHA; m). The dashed contours represent negative values, and the solid contours represent positive or zero values.

Citation: Journal of Physical Oceanography 53, 4; 10.1175/JPO-D-22-0078.1

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.

Table 1

Observation data used to assess the model simulation skills.

Table 1

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.

Fig. 2.
Fig. 2.

Seasonal mean sea surface salinity (color shading) and surface current (vectors; m s−1) for (left) LLC4320 and (right) Aquarius sea surface salinity and AVISO geostrophic current during 2011/12.

Citation: Journal of Physical Oceanography 53, 4; 10.1175/JPO-D-22-0078.1

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.

Fig. 3.
Fig. 3.

Seasonal mean sea surface height anomalies (color shading; m) and current anomalies (vectors; m s−1) for (left) LLC4320 during 2011/12, (center) AVISO during 2011/12, and (right) AVISO climatological mean.

Citation: Journal of Physical Oceanography 53, 4; 10.1175/JPO-D-22-0078.1

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 = υyux) 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.

Fig. 4.
Fig. 4.

Horizontal wavenumber spectra of surface kinetic energy in the southern BoB averaged over the whole LLC4320 simulation period. The lighter green shading denotes 95% confidence intervals for the spectral estimates, and the gray lines represent the spectral slopes of k−2 and k−3.

Citation: Journal of Physical Oceanography 53, 4; 10.1175/JPO-D-22-0078.1

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 V¯, and the submesoscale fraction V′, i.e., V=V¯+V. Here the spatial filtering procedure is on the radial direction of each variable. The submesoscale component V′ is obtained by the high-pass filter, and the deviation of the unfiltered field from the submesoscale is the large-scale/mesoscale component V¯.

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, Ro=ζ/f=[(υ/y)(u/x)]/f, where u and υ are the unfiltered velocity.

We analyze three main generation mechanisms for SMPs, i.e., frontogenesis, MLI, and SI. Frontogenesis can usually be expressed in terms of the frontal tendency function F,
F=D|hρ¯|2Dt=Qhρ¯,
where ρ is the density and Q is the Q vector, Q=(Q1,Q2)=[(u¯/x)hρ¯,(u¯/y)hρ¯] (Hoskins and Bretherton 1972; Hoskins 1982; Capet et al. 2008c). The term hρ¯ is the horizontal density gradient, and u¯ represents the low-pass spatial-filtered velocity field. Positive values of F indicate enhancement of horizontal density gradients, which means frontogenesis. Negative values indicate frontolysis.
Frontogenesis can be intensified due to the mesoscale strain (Hoskins 1982), which is measured by the mesoscale strain rate (MSR),
MSR=(u¯/xυ¯/y)2+(u¯/x+υ¯/y)2.
Here we use low-pass spatial filtered velocity field as an approximation of “mesoscale” velocity. From the above equations, large MSR and horizontal density gradient ( |hρ¯|) are preconditions for the generation of SMPs (Gula et al. 2014; McWilliams 2016; Dong and Zhong 2018). Once the fronts are intensified and lateral density gradients are enhanced, submesoscale baroclinic disturbances develop and restratify the mixed layer by slumping the gradients from the horizontal to vertical, which are termed MLI (Boccaletti et al. 2007). Here, we diagnose MLI in terms of their net effect, i.e., the rate of conversion of APE into KE,
PK=1MLDMLD0wbxydz,
where w′ is the submesoscale vertical velocity, b′ is the submesoscale buoyancy, b is defined as b = −/ρ0, ρ0 being a reference density and g the gravity acceleration. The wb′ indicates the vertical buoyancy flux (VBF), and 〈〉xy signifies horizontal averaging over the area (Boccaletti et al. 2007; Fox-Kemper and Ferrari 2008; Mensa et al. 2013). The scaling of VBF for MLI in the boundary layer follows the relationship (Capet et al. 2008a; Luo et al. 2016),
PK=wbxyz|b¯|xyz2MLDxy21f,
where 〈〉xyz indicates the 3D volume average over the domain and within the mixed layer. In this paper, the domain for both 2D and 3D averaging operators is 82°–88°E, 6°–11°N (black box in Fig. 7a). This formula shows that the APE-to-KE conversion by MLI is proportional to the buoyancy gradient, as well as to the MLD. Therefore, the presence of strong horizontal buoyancy gradients and a deep MLD are conducive to the formation of SMPs.
The SI is also a key mechanism for the development of SMPs (Thomas et al. 2013), which extracts geostrophic kinetic energy from fronts to feed small-scale processes (Peng et al. 2020; Jing et al. 2021). One criterion for the development of SI is associated with the Ertel potential vorticity (PV), which is defined as q=(fk^+×u)b. The SI occurs when the sign of q is opposite to the local Coriolis parameter f, i.e., fq < 0 (Hoskins 1974). Here, k^ is a vertical unit vector, and u is the velocity vector. Therefore, in the Northern Hemisphere, the SI can grow in fronts when q < 0. Under thermal wind balance, q can be simplified as
q=(f+ζ)N2M4/f,
where ζ = ∂υ/∂x − ∂u/∂y is the vertical component of relative vorticity, N2 = ∂b/∂z is the buoyant frequency, and M2 = |∇hb| is the amplitude of lateral buoyancy gradient. Strong fronts combined with weak stratification facilitate negative q, providing favorable conditions for SI. Negative Ertel PV, however, can develop a variety of instabilities besides SI. Under the geostrophic assumption, SI develops only when the value of balanced Richardson angle ( ϕRiB), and the critical angle (ϕc) meets the following terms (Thomas et al. 2013; Bachman and Taylor 2014; Bachman et al. 2017; Thompson et al. 2016):
90°<ϕRiB<ϕc,
ϕRiB=tan1(RiB1)=tan1(|hb|2f2N2),
ϕc=tan1(ζgf).
Here, ζg is the vertical component of absolute vorticity of the geostrophic flow. Additionally, both gravitational instability and SI codevelop when the following term is satisfied,
135°<ϕRiB<90°.
Following Dong et al. (2021), the SI scale in the surface mixed layer is estimated as
L=0.45f2H0|hb|dz,
where H is determined as the depth where the bulk PV changing sign traversing downward in the water column from the surface, indicating the deepest penetration depth of the unstable SI modes.
Downfront winds act to reduce the stratification and RiB in the surface layer, and thus favor the development of SMPs (Thomas et al. 2008). The wind effect can be represented by the Ekman buoyancy flux (EBF), which can drive frontogenesis and SI. The EBF is the dot product of the Ekman transport and the surface buoyancy gradient and is defined as
EBF=(τ×k^)hb/(ρ0f),
where τ is wind stress. Positive EBF indicates a destabilizing state in the boundary layer.

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).

By using MWT, the KE and APE budgets for the submesoscale window can be derived from the primitive equations as (see appendix C for a derivation)
K2t=ΓK2+b2+ΔQK2+ΔQP2+FK2,
A2t=ΓA2b2+ΔQA2+SA2+FA2.
The expressions and meanings of the above energy terms are listed in Table 2. For rigorous derivation and detailed description of these equations, we refer the reader to Liang and Anderson (2007) and Liang (2016). It should be noted that the term ΓK2 not only contains energy transfer from the larger and mesoscale, but also from submesoscale itself ( ΓK22) that do not involve instability and are thus has no contribution to the K2/t. Interaction analysis gives that ΓK02 ( ΓA02) represents the KE (APE) transfer between large-scale and submesoscale. Positive values of ΓK02 ( ΓA02) indicate the energy transfer from large-scale to submesoscale. It is also called the canonical KE (APE) transfer (appendix B). Similarly, ΓK12 ( ΓA12) indicates the energy transfer between mesoscale and submesoscale. The b∼2 represents the conversion between APE and KE on submesoscale. Positive b∼2 means the submesoscale APE converting to KE, and vice versa.
Table 2

Mathematical form and meaning of the energy terms in Eqs. (8) and (9). For details, see Liang (2016) and Yang et al. (2021).

Table 2

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).

Fig. 5.
Fig. 5.

The annual mean |Ro| at 5 m for LLC4320.

Citation: Journal of Physical Oceanography 53, 4; 10.1175/JPO-D-22-0078.1

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.

Fig. 6.
Fig. 6.

Snapshots of Ro at the depth of 5 m on (a) 15 Feb, (b) 15 May, (c) 15 Aug, and (d) 15 Nov for the LLC4320. The black rectangle shown in (c) is chosen for specific investigation in section 4.

Citation: Journal of Physical Oceanography 53, 4; 10.1175/JPO-D-22-0078.1

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).

Fig. 7.
Fig. 7.

Distribution of the seasonal mean |Ro| over (a) winter, (b) spring, (c) summer, and (d) fall at the depth of 5 m simulated by LLC4320. The black box represents the R1 region.

Citation: Journal of Physical Oceanography 53, 4; 10.1175/JPO-D-22-0078.1

Fig. 8.
Fig. 8.

Horizontal maps of submesoscale kinetic energy (K∼2; m3 s−2) integrated over the upper 50 m and averaged over (a) winter, (b) spring, and (c) summer for LLC4320 based on MWT method.

Citation: Journal of Physical Oceanography 53, 4; 10.1175/JPO-D-22-0078.1

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.

Fig. 9.
Fig. 9.

Time series of the (a) |Ro|, (b) frontogenesis F (s−5), (c) horizontal buoyancy gradients |hb¯| (s−2), and (d) vertical buoyancy flux (wb′; m2 s−3) as a function of depth, (e) |hb¯| (black; s−2) and mesoscale strain rate (MSR; blue; s−1) averaged over the MLD, (f) PK (black; m2 s−3) and |b¯|2xyzMLD2xy (blue; m2 s−4) averaged over the MLD. All of these values are horizontally averaged over R1 defined in Fig. 7. The black curve in (a) and (b) denotes the regionally averaged MLD.

Citation: Journal of Physical Oceanography 53, 4; 10.1175/JPO-D-22-0078.1

4. Potential generation mechanisms

a. Frontogenesis

The time series of the regionally averaged F, |hb¯|, and MSR derived from LLC4320 are shown in Figs. 9b, 9c, and 9e. Positive F appears in the upper layer, indicating the presence of frontogenesis there. Strong frontogenesis is present during the southwest monsoon and midnortheast monsoon (Fig. 9b), in agreement with the seasonal variations of |Ro| (Fig. 9a), implying that rich SMPs are partly due to the active frontogenesis.

It is noticeable that |hb¯| and MSR (Figs. 9c,e) are also the strongest during summer monsoon, associated with the SMC. It is confirmed that the SMC passing south of Sri Lanka transports saltier water from the Arabian Sea into southwest of the BoB, and the salty water meets freshwater there, forming sharp salinity fronts (Fig. 2c). The large strain can be explained by the horizontal velocity shear and the deformation of the SMC, especially along the rim of the mesoscale eddy. Strong strain rate and sharp fronts associated with the SMC are favorable for frontogenesis.

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 |hb¯| affect the variations of PK, both having a good correspondence with PK (Figs. 9c,e). During the summer monsoon, the deepening mixed layer is driven by the strongest wind forcing, the weaker stratification due to the intrusion of the salty Arabian Sea Water, as well as the westward propagating Rossby waves (Thadathil et al. 2007; Narvekar and Kumar 2014). In addition, the enhancement of |hb¯| at this time (Figs. 9c,e) is partially related to the active anticyclonic circulation with high velocity shear and abundant fronts around the SMC. The |b¯|xyz2MLDxy2 term closely follows the PK all year-round, indicating the combined effects of strong lateral buoyancy gradient and deep mixed layer on APE-to-KE conversion during summer and winter. The deeper MLD contributes to greater APE stored in the upper layer, while the sharp fronts are more unstable, therefore resulting in a stronger PK and additional SMPs during this period.

Both MLI and strain-induced frontogenesis can create strong VBF (McWilliams 2016; Barkan et al. 2017), which can be measured by
wb¯MLIhb2|hb|2f,
wb¯STRAINShb2|hb|2f2,
where hb is the surface boundary layer depth and S represents large-scale geostrophic strain rate. Subscripts MLI and STRAIN indicate VBF induced by MLI and strain-induced frontogenesis, respectively. The patterns of VBF are consistent with those of |Ro| (Figs. 7 and 10), suggesting that MLI and frontogenesis play an important role in the generation of submesoscale motions, and that both of them can cause a positive restratifying eddy buoyancy flux. The VBFMLI and VBFSTRAIN have similar patterns, except that the former is stronger than the latter (Fig. 10). In the northern BoB, MLI also plays a more important role than frontogenesis (Li et al. 2022).
Fig. 10.
Fig. 10.

Distribution of the seasonally mean vertical buoyancy flux (VBF; m2 s−3) integrated within the MLD associated mixed layer instability in (a) winter and (b) summer. (c),(d) As in (a) and (b), but for VBF associated with frontogenesis.

Citation: Journal of Physical Oceanography 53, 4; 10.1175/JPO-D-22-0078.1

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.

Fig. 11.
Fig. 11.

The time-mean Ekman buoyancy flux (EBF; m2 s−3) in (a) winter, (b) spring, (c) summer, and (d) fall for LLC4320. The gray contours and black vectors show the SSH anomalies (m) and surface wind stress (N m−2), respectively.

Citation: Journal of Physical Oceanography 53, 4; 10.1175/JPO-D-22-0078.1

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 ( 90°<ϕRiB<ϕc) in the southern BoB can reach 40%–80% (Figs. 12e,f), which corresponds roughly with the positive EBF (Figs. 11a,c). The positive EBF tends to deepen the mixed layer and increase APE, which favors the occurrence of SI. By contrast, the proportion for mixed gravitational/symmetric instability ( 135°<ϕRiB<90°) is relatively small, with maximum proportion of about 20%–40% (Figs. 12c,d). Other types of instability, such as centrifugal/inertial instability, pure gravitational instability, and inertial/symmetric instability, rarely occur in the southern BoB (figure not shown).

Fig. 12.
Fig. 12.

Ertel PV (color shading; s−3) in (a) winter and (b) summer. The proportion (%) when mixed gravitational/symmetric instability is satisfied during (c) winter and (d) summer. (e), (f) As in (c) and (d), but for the proportion (%) when symmetric instability is satisfied. Note that the proportion is defined as the number of days when mixed gravitational/symmetric instability or SI is satisfied among days in winter or summer.

Citation: Journal of Physical Oceanography 53, 4; 10.1175/JPO-D-22-0078.1

Fig. 13.
Fig. 13.

Spatial distribution of symmetric instability scale (km) in the southern BoB in (a) winter and (b) summer.

Citation: Journal of Physical Oceanography 53, 4; 10.1175/JPO-D-22-0078.1

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.

Fig. 14.
Fig. 14.

The snapshots of front structure in edge of the mesoscale eddy (as shown in the black rectangle of Fig. 6). (a) Surface density (colors shading; kg m−3) and currents (vectors; m s−1) on 15 Aug 2012. The gray contours denote the sea surface height anomalies (m). (b) Frontogenesis tendency (color shading; s−5) at the surface; (c) strain rate (color shading; s−1) at the surface; (d) frontal sharpness M4 (color shading; s−4) at the surface; (e) Ertel PV (color shading; s−3) at the surface; (f) depth–distance distribution of density (colors shading and gray contours; kg m−3) and cross-front ageostrophic secondary circulation (vectors) along across-front transect [black dashed line in (a)]. The black solid line refers to mixed layer depth.

Citation: Journal of Physical Oceanography 53, 4; 10.1175/JPO-D-22-0078.1

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, ΓK02, ΓK12, ΓA02, and ΓA12 that are vertically integrated over the upper 50 m during each season. Energy transfer primarily occurs along the SMC (NMC) during summer (winter) monsoon, suggesting that the development of SMPs is closely related to the background currents (Figs. 15a–e,k–o). Along the central SMC, the b∼2 exhibits mostly positive values, indicating the conversion of submesoscale APE to KE and the generation of SMPs (Fig. 15a). During winter, positive values also appear along the NMC (Fig. 15k). Similar patterns of K∼2 and b∼2 further confirm the role of MLI and frontogenesis in the SMPs generation (Figs. 8 and 15).

Fig. 15.
Fig. 15.

Spatial distribution of (a),(f),(k),(p) energy conversion from submesoscale APE to KE (b∼2); (b),(g),(l),(q) kinetic energy transfer from large-scale to submesoscale (ΓK02); (c),(h),(m),(r) kinetic energy transfer from mesoscale to submesoscale (ΓK12); (d),(i),(n),(s) available potential energy transfer from large-scale to submesoscale (ΓA02); and (e),(j),(o),(t) available potential energy transfer from mesoscale to submesoscale (ΓA12) based on the LLC4320 output. The values are vertically integrated over the upper 50 m in (first row) summer, (second row) fall, (third row) winter, and (fourth row) spring (m3 s−3).

Citation: Journal of Physical Oceanography 53, 4; 10.1175/JPO-D-22-0078.1

The large-to-submesoscale KE transfer ΓK02 is characterized by interlacing negative and positive values along the SMC in summer, which is indicative of bidirectional KE transfer between large-scale KE and SMKE (Fig. 15b). Overall, positive energy transfer prevails (Figs. 15b and 16a), hence SMPs obtain KE from background currents in total. Between mesoscale and submesoscale processes, energy mainly transfers from the former to the later during summer, indicated by the dominate positive values of ΓK12(Fig. 15c). During winter, on the other hand, ΓK02 and ΓK12 are relatively weak (Figs. 15l,m), which indicates that negligible KE is transferred from the larger scale to submesoscale.

Fig. 16.
Fig. 16.

Energy budgets of (a) KE (m3 s−2) and (b) APE (m3 s−2) vertically integrated over the upper 50-m depth based on the LLC4320 output. The values are horizontally averaged over R1 defined in Fig. 7, and temporally averaged over spring (green), summer (blue), fall (yellow), and winter (red). The meaning of each term can be found in Table 2 and appendix C.

Citation: Journal of Physical Oceanography 53, 4; 10.1175/JPO-D-22-0078.1

The APE transfer terms ΓA02 and ΓA12 in summer (Figs. 15d,e) and winter (Figs. 15n,o) exhibit large and positive values, indicating that submesoscale APE is mainly from large-scale and mesoscale flows, especially around the SMC during summer and around the NMC during winter. The magnitude of the energy transfer terms is noticeably smaller in the monsoon transition period, in agreement with the seasonal variations of SMPs.

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, ΓK02,ΓK12 and b∼2 are the main source terms of SMKE in summer. Most of the KE is depleted by strong internal dissipation indicated by the negative ΔQK2, ΔQP2, and FK2, the three sink terms. Energy terms of the APE budget are generally of a larger magnitude than the KE budget (Fig. 16b). Submesoscale APE is mainly originated from the forward cascade from large-scale background flows and mesoscale eddies ( ΓA02 and ΓA12). Both KE and APE budgets exhibit obvious seasonal cycles that are stronger in summer and winter monsoon periods and weaker in nonmonsoon periods, in agreement with the seasonal cycle of K∼2.

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 ( ΓK02 and ΓK12 versus b∼2), while baroclinic instability (b∼2) dominates during winter. The origin of SMKE in the southern BoB is different from other regions, including the ETP at the same latitudes.

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.

In this study, we will perform a three-scale-window decomposition: the large-scale window, the mesoscale window, and the submesoscale window, simply which are denoted by ω = 0, 1, 2, respectively. Here, the MWT is used in the frequency domain. These windows can be divided on the wavelet spectrum by the wavelet scale levels: j0, j1, and j2. Namely, given a time series T(t) whose duration is τ, then the three wavelet scale levels correspond to periods 2j0τ,2j1τ, and 2j2τ, respectively. Given a field U = U(t) which is a square integrable function defined on [0, 1] and suppose that {φnj(t)}n is an orthonormal translation invariant scale sequence (Liang and Anderson 2007; Yang et al. 2020). where n is the discrete time step. Then, using {φnj(t)}n as a basis, we apply a scaling transform
U^nj=01U(t)φnj(t)dt.
Then U can be reconstructed via the three windows ω = 0, 1, 2:
U0(t)=n=02j01U^nj0φnj0(t),
U1(t)=n=02j11U^nj1φnj1(t)U0(t),
U2(t)=U(t)U0(t)U1(t).
The three-scale-window reconstructions correspond to low-pass, bandpass, and high-pass filtering fields, that is, MWR. Through these reconstructions, the MWT of U is defined as
U^nω=01Uω(t)φnj2(t)dt,
Here, n = 0, 1, …,  2j2 − 1. Equations (A2)(A4) can be uniformly written as
U(t)=n=02j21U^nωφnj2(t),ω=0,1,2.
Therefore, the energy on windows (ω = 0, 1, 2) is (U^nω)2 in the MWT framework. For the detailed derivation process, we refer the reader to Liang and Anderson (2007).

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).

Given a scalar field K in an incompressible flow u, the governing equation of K is
Kt+(uK)=0.
For simplicity, we have neglected the diffusion term. The interactions between different scale windows are expected to happen due to the nonlinear term ∇ ⋅ (uK). Using MWT on both sides of Eq. (B1) and multiply transform coefficient K^nω, we can get the energy equation on window ω:
[(K^nω)2/2]t=K^nω(uK)^nω,
where (K^nω)2/2 is the energy on window ω at time step n. The term K^nω(uK)^nω represents the nonlinear processes, containing a transport process term Qnω that indicates the redistribution of energy in physical space and a transfer process term Γnω that denotes energy exchanges across different scale windows. Based MWT method, Liang (2016) proved and given interscale transfer expression, namely, canonical transfer:
Γnω=12[(uK)^nωK^nωK^nω(uK)^nω].
Notably, the transfer expression meets a property that does not hold in other traditional energetics formalisms, that is,
ωnΓnω=0.
Physically, Eq. (B4) indicates that canonical transfers are interscale processes without energy generation and destruction.

APPENDIX C

Derivation of Multiscale Energy Equations

For a Boussinesq and hydrostatic flow, the primitive equations are
uht+uhhuh+wuhz+fk×uh=1ρ0hP+Fm,
Pz=ρg,
huh+wz=0,and
ρt+uhhρ+wρz=ρ0N2gw+Fρ,
where uh is horizontal components of the three-dimensional velocity vector u, w is the vertical velocity, P is the dynamic pressure, and Fm and Fρ are terms for forcing and dissipation.
The kinetic energy (KE) on scale window ω, based on the MWT, is
Kω=12(u^hω)2,
where the operator ()^ω represents MWT on window ω. Using MWT on both sides of Eq. (C1), and multiplying by transform coefficient u^ω, then separating the nonlinear advection term into a canonical transfer term and a transport term, the multiscale KE budget equation can be obtained:
Kωt=12[(uuh)^ω:u^hω(uuh)^ωu^h2]12[(uuh)^ωu^hω]1ρ0(u^ωP^ω)gρ0ρ^ωw^ω+FKω=ΓKωQKωQPω+bω+FKω,
where QKω is the convergence of the Kω flux on window ω and QPω is the pressure working rate on window ω. Note that QKω and QPω will be written as ΔQKω and ΔQPω in this paper, respectively. The term ΓKω is the transfer of KE, from the other windows to window ω, and bω is the buoyancy conversion rate on window ω.
Similarly, the multiscale available potential energy (APE) budget equation can be written as
Aωt=c2[(uρ)^ωρ^ωρ^ω(uρ)^ω]12[cρ^ω(uρ)^ω]+12ρ^ω(uρ)^ωcz+gρ0ρ^ωw^ω+FAω=ΓAωQAω+SAωbω+FAω,
where Aω is the available potential energy on scale window ω, defined to be
Aω=12c(ρ^ω)2,c=g2ρ02N2.
Also, QAω is the convergence of the Aω flux on window ω and will be denoted as ΔQAω. The term ΓAω is the transfer of APE, from the other windows to window ω, and SAω is the apparent source/sink of Aω due to the vertical stratification.

For a detailed derivation process for multiscale KE and APE budget equations, we refer the readers to Liang (2016).

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