The Seasonality of Submesoscale Energy Production, Content, and Cascade

Submesoscale processes in the upper ocean vary seasonally, in tight correspondence with mixed layer thickness variability. Based on a global high‐resolution MITgcm simulation, seasonal evaluation of strong vorticity and spectral analysis of the kinetic energy in the Kuroshio Extension System show the strongest submesoscales occur in March, implying a lag of about a month behind mixed layer thickness maximum in February. An analysis of spectral energy sources and transfers indicates that the seasonality of the submesoscale energy content is a result of the competition between the conversion of available potential energy into submesoscale kinetic energy via a buoyancy production/vertical buoyancy flux associated with mixed layer instability and nonlinear energy transfers to other scales associated with an energy cascade. The buoyancy production is seasonally in phase with the mixed layer depth, but the transfers of energy across scales makes energizing the reservoir of submesoscale kinetic energy lag behind by a month.

Three-dimensional turbulence transfers KE to smaller and smaller scales where it is ultimately dissipated (Kolmogorov, 1991); Two-dimensional and quasi-geostrophic theory predicts an inverse cascade in the ©2020. The Authors. This is an open access article under the terms of the Creative Commons Attribution License, which permits use, distribution and reproduction in any medium, provided the original work is properly cited. inertial range above the scales at which baroclinic instability converts available PE to KE and a downscale enstrophy cascade below (Charney, 1971;Kraichnan, 1967). However, submesoscale processes are not isotropically three-dimensional, nor purely two-dimensional or quasi-geostrophic due to the moderate symmetry-breaking effects of rotation and stratification implied by order one Rossby and Richardson numbers. Many studies have demonstrated a significant role of submesoscales in ocean energy cascade after Capet et al. (2008b). The contribution to inverse and forward energy cascades from submesoscales is studied successively in different regions based on high-resolution observations and simulations, such as the subtropical North Pacific (Qiu et al., 2014), Drake Passage , Kuroshio extension (Sasaki et al., 2017), South Indian Ocean (Schubert et al., 2019) and South China Sea (Zhang et al., 2016;Zhang et al., 2020). All of these works emphasize the importance of submesoscales in the ocean energy cascade, and some use the same model simulations analyzed in this study.
In this work, the seasonality of submesoscale processes in the Kuroshio Extension System (KES) region is investigated based on a high-resolution global model simulation (MITgcm LLC4320). The seasonality of KE cannot be explained simply by the variations in the rate of PE released associated with the mixed layer depth variations. Instead, the contribution of the energy cascade to the submesoscale seasonality is also required and emphasized in this study. The paper is organized as follows: Section 2 introduces the model output and methods that used here; Section 3 describes the submesoscale seasonality in the KES, and discusses the mechanism that modulates the seasonality; the last section summarizes the work.

Model Description
The model simulation analyzed is the Massachussetts Institute of Technology general circulation model (MITgcm) on a Latitude-Longitude polar Cap (LLC) grid (Forget et al., 2015;Menemenlis et al., 2008;. The model simulation is named as LLC4320, which has a horizontal resolution of 1/48 o with 90 vertical layers over the globe. The model is forced by six-hourly atmospheric forcings from the 0.14°European Centre for Medium-Range Weather Forecasting and 16 most significant tidal constituents. The vertical mixing is accomplished by using the K-profile Parameterization (KPP). The initial model state was derived from a 1/6°global ocean state estimate generated by the Estimating the Circulation and Climate of the Ocean (ECCO), Phase II project, and then a set of simulations were successively developed with increasing resolution of 1/12°, 1/24°and finally 1/48°. The LLC4320 is simulated for 14 months, from Sep 2011 to November 2012. The essential model states are stored at hourly snapshot. This model's output has been widely applied, especially for studying submesoscale processes (e.g., Su et al., 2018;Viglione et al., 2018). The work by  is of special note as inspiration for this study as it focuses on the submesoscale seasonality in the KES region, and demonstrates that the upper ocean density structure and eddy KE are well captured by this model simulation.

Temporal and Spatial Filtering
The barotropic tides in the model can generate strong internal waves under the interaction with topography. And the high-frequency wind forcing can also generate inertial internal waves. These internal waves have a spatial scale similar to the submesoscale currents, which potentially impacts the analysis result. Hence, the velocity and tracer (temperature, salinity) fields in KES region are firstly filtered by using a temporal lowpass filter with a cutoff period of 30 hours. The period is little larger than that of the local near-inertial internal waves in the KES region, ensuring that most internal waves are excluded. Then submesoscales are extracted by using a spatial highpass filter. Here, the cutoff scale for the filtering is chosen to be 80 km based on a correlation analysis of the unfiltered variability identifying submesoscales, which will be explained in the next section.

The Spectral Kinetic Energy Balance
The application of a horizontal spatial Fourier transform to the primitive kinematic equation yields spectral KE balance equation (Capet, McWilliams, et al., 2008b), Here, the kinetic energy KE k Þu·k, and the production from PE or vertical buoyancy flux PK k ð Þ ¼ b w * b b·k are the critical terms to analyze to understand the seasonality of KE. The 3D velocity is u, w is the vertical component of velocity, b is the buoyancy defined as − g ρ ρ , k is the wavenumber interval corresponding to the horizontal resolution. The caret denotes a horizontal spatial Fourier transform after removing the areal mean and asterisk denotes complex conjugate. The residual Res(k) includes pressure work, horizontal and vertical dissipation, and is small and so will be not discussed here. The wind work in Res(k) is an important energy source for ocean KE, and demonstrated to be able to interact with submesoscales (McWilliams, 2016;Renault et al., 2019;Thomas & Lee, 2005). A rough evaluation using daily snapshots of this contribution is made in this model and is found to be two orders of magnitude smaller than other terms in the energy budget. Thus, a more accurate evaluation is unlikely to show this effect changes the storyline here, and in any case we are unable to do the evaluation of wind-work at high-frequency based on what fields are saved of the LLC4320 results. Actually, Res(k) is beyond the scope of this paper, as the submesoscale seasonality is explained fully by the other two terms.
Only the real part is considered in the expression, so all phases in space (and time) are co-located (and simultaneous). Equation (1) provides the rate of change of KE density at a given wavenumber, which is mostly determined by the combined effects of the advective spectral flux and production. The advective spectral flux underlies the energy cascade and transfers energy conservatively between different scales, while the production or vertical buoyancy flux transforms available PE to KE.

Seasonal Variability
Submesoscale processes tend to have a significant ageostrophic component with Ro~1, so following  the normalized vertical vorticity (ζ/f, ζ = v x − u y , f is the local planetary vorticity) is a good indicator for the submesoscale activity, and can be regarded as the Rossby number. A magnitude in this ratio near unity implies submesoscale processes. The Rossby number at the sea surface calculated based on the original velocity field at different times shows the spatial distribution of submesoscale processes in the northwest Pacific Ocean (Figure 1a, b). The Rossby number is generally relative weak in September in this simulation, and only few submesoscale filaments and eddies can be observed in the KES region. In contrast, the Rossby number is enhanced remarkably with magnitude up to 1.5 including ubiquitous submesoscale filaments and eddies, especially in the KES and subtropical regions. Zooming into rectangular regions shows more detailed vortical structures (Figure 1c, d). Stronger submesoscale filaments and eddies are found in March than September. Positive vorticity is stronger than the negative, as large anticyclonic vorticities (i.e. ζ/f < −1) are quickly destroyed by inertial instability (Capet et al., 2008c;Shcherbina et al., 2013).
The vorticity derived from the original unfiltered velocity field also shows stripes and patches, in addition to the submesoscale filaments and eddies (such as close to the Japanese archipelago and subtropical region in Figure 1c). These features are more obvious in September compared with March and are attributed to the inertial-gravity waves . The enhanced stratification in summer favors the generation of internal waves. The application of a 30-hr temporal lowpass filter (Figure 1e, f) greatly diminishes these features. Previous works have applied a daily-averaging method to remove internal waves Su et al., 2018). However, the lowpass filtering method is chosen here to retain submesoscale energy to a larger extent than with daily averaging.
Based on the lowpass filtered velocity field, the probability of the Rossby number larger than 0.8 (|ζ/f| > 0.8, an indicator for the submesoscale activity) in different months from September 2011 to August 2012 is calculated (blue line in Figure 2a). The Rossby number probability has a strong seasonal variability with a peak in March. The probability variation indicates the submesoscales are more active in winter, which is consistent with the root mean square variation of vertical vorticity from the works of  and Sasaki et al. (2017). This seasonality is usually attributed to more PE storage due to a deeper mixed layer depth in winter. The mixed layer depth derived from the model simulation is thick in winter up to 140 m, 10.1029/2020GL087388

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but shallow in summer-often less than 10 m (blue line in Figure 2b). The Rossby number probability and mixed layer depth show a good consistency in seasonality.
However, the Rossby number probability and submesoscale KE occur somewhat after the mixed layer depth thickens and the production term energizes. The mixed layer depth peaks in February, a month earlier than the probability. The lag of the vorticity peak can be noted as well in  and Sasaki et al. (2017). Corresponding with the probability, the submesoscale energy with scales below 80 km is isolated based on spatial highpass filtering to make the correlation maximal between the submesoscale KE (KE ¼ ρ u′ 2 þv′ 2 2 , u′ and v′ are the submesoscale zonal and meridional velocities after spatial highpass filtering) and the Rossby number probability (red line in Figure 2a). After this filter-based definition (spatial highpass at 80 km, temporal lowpass at 30 hr) is applied, the maximum submesoscale KE in winter is 5 times larger than in summer. The PE production or vertical buoyancy flux is calculated as the average of the product of the analogous submesoscale vertical velocity w′ and buoyancy perturbation b′ using the same spatial filter scale of 80 km and temporal filter scale of 30 hr. This covariance is the physical space equivalent to PK in spectral space in Equation 1. The monthly-averaged vertical buoyancy flux at 25 m depth is shown (red line in Figure 2b). It is almost zero in summer and largest in winter. The buoyancy flux corresponds in phase with the mixed layer depth, peaking in February, one month ahead of the Rossby number probability and submesoscale KE. The mean buoyancy flux is always positive, so it tends to make the submesoscale KE increase at all times. As a result, the peak of Rossby number probability and submesoscale KE cannot be explained solely by the buoyancy flux.

Mixed Layer Instability Vs. Energy Cascade
As shown by Equation 1, another term that affects the submesoscale KE is the energy transfer between scales, namely, advective spectral flux. Both terms, PK and ADV, are calculated along horizontal wavenumber in Figure 3. ADV and PK are strong in the small wavenumber range, 1x10 −6~1 x10 −5 cycles m −1  (corresponding to large horizontal scale of 100~1000 km) and weak in large wavenumber (below 100 km; subpanels in Figure 3). ADV is generally larger than PK, since PK is amplified by 10 times in the subpanels for comparison. However, ADV and PK become comparable in wavenumber larger than 1x10 −5 cycles m −1 (below 100 km).
PK is weaker in September compared with ADV ( Figure 3a). The generally positive PK still implies the positive contribution of PK to the KE change. The weak PK in summer has been revealed in the preceding section ( Figure 2b). In contrast, ADV is much stronger and large negative value can be observed especially around the scale of 100 km, which indicates that the advective spectral flux transfers KE to larger scales, i.e., inverse cascade. But as the wavenumber increases to about 1.7x10 −5 cycles m −1 (about 60 km), weakly positive ADV increases the KE at that scale. Overall, the summation of these two terms is dominated by ADV.
The circumstance becomes quite different in winter when submesoscales are active (Figure 3b). On one hand, PK is enhanced remarkably in the submesoscale range indicating conversion of PE into KE, as is ADV indicating transfer of KE across scales. The strong positive PK provides the energy source for the generation of submesoscales. On the other hand, ADV becomes more complicated due to the variety of submesoscale processes combining with mesoscale processes. The strong negative ADV on scales with wavenumber larger than 0.9x10 −5 cycles m −1 (>110 km) and between 1.5x10 −5~2 .3x10 −5 cycles m −1 (45~65 km) moves energy toward smaller wavenumbers (larger scales), which is more complicated compared with that in summer. The positive ADV is enhanced (note different vertical axis scales) and strong positive ADV is scattered across the wavenumber range of 1x10 −5~1 .5x10 −5 cycles m −1 (65~100 km). The summation of ADV and PK leads to a generally positive spectral flux in the submesoscale range 8x10 −64 x10 −5 cycles m −1 (25~125 km), indicating an increasing tendency of KE across these scales. The positive ADV in winter associated with active submesoscales implies a forward energy cascade in submesoscale range, especially at smaller scale, potentially contributing to local mixing (Capet, McWilliams, et al., 2008b;Yang et al., 2017).
The energy fluxes integrated from the spectral fluxes of PK and ADV on wavenumbers larger than 1.25x10 −5 cycles m −1 (80 km length scale) are investigated, since the submesoscale KE below 80 km was identified to correlate with the Rossby number probability (Figure 4). The integrated energy fluxes indicate the contributions from these two terms to the KE in the submesoscale range. The energy conversion from potential to KE integrated from PK over the submesoscale has an obvious seasonal variability and peaks in February, which is consistent with that calculated directly from the physical space estimate (green line in Figure 4) and this timing coincides with the deepest seasonal mixed layers. The spectrally integrated value is smaller than the physical estimate, since it is an average over a wider depth range. The integration of ADV (blue lines in Figure 4) is divided into two parts, the integral over wavenumbers with positive ADV (scales to which energy is delivered) and those with negative ADV (scales from which energy is extracted). The positive contribution is enhanced in winter, indicating that submesoscale processes receive KE via nonlinear interactions. The negative contribution of ADV peaks in April, lagging behind the peak of the positive contribution slightly. Overall, the negative ADV dominates the positive contribution, so the submesoscale 10.1029/2020GL087388

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acts to energize other length scales (e.g., Callies et al., 2016), supporting an inverse energy cascade transferring energy from submesoscales to larger scales. As a result, the submesoscales lose energy due to the nonlinear advective spectral flux, but this withdrawal occurs after a few months' delay behind the production of submesoscale KE from PE.
The summation of the integrated ADV and PK shows a transition between positive submesoscale energy tendency and negative submesoscale energy decline due to the positive PK and negative ADV (red line in Figure 4). The total source is positive from September to March, steadily increasing over the seasons of mixed layer deepening from September to February and then decreasing as the production drops off during springtime re-stratification. From April to August, the already energized submesoscale begins to lose energy as the total becomes a negative sink that peaks in May, and then decreases to null in August. The seasonality of the total flux indicates that submesoscale processes obtain energy from September to March, but lose energy after April. Hence, the seasonality and peak of submesoscale activity in March is a result of the competition between sources and sinks of submesoscale energy, and because the sinks lag behind production, the maximum submesoscale energy content occurs after the maximum of production which coincides with the deepest mixed layers (Figure 2), but before the season of maximum energy transfer to larger scales ( Figure 4).

Conclusions and Discussion
The seasonal variability is an important dynamic feature of submesoscale processes, which implies potential generation mechanisms. The seasonal variability has been attributed to the vertical buoyancy flux and mixed layer thickness variability (Callies et al., 2015;Mensa et al., 2013;Sasaki et al., 2017). However, buoyancy flux variability fails to explain the submesoscale energy seasonality completely, since the buoyancy flux is always positive (i.e., re-stratifying and extracting PE) in the submesoscale range. The sinks of submesoscale energy are an important consideration, and this work shows that they may be dominated by advective transfer to larger scales which may lag a few months behind the production of energy. In the KES study region, the submesoscale activity (high Rossby number probability and strong submesoscale energy) peaks in March, a month behind the vertical buoyancy flux and mixed layer thickness which peak in February.
Recently, a variety of parameterizations of eddies based on energy content have been proposed (e.g., Bachman et al., 2017;Jansen et al., 2019). Here it is important to note that the re-stratification effect, which is parameterized in Fox-Kemper et al. (2008, 2011 is not temporally coincident with the submesoscale energy content, indicating that a deeper analysis of these effects is needed. It may be that distinguishing lateral transport from vertical re-stratification as a function of time is needed to explain both phenomena consistently. In this work, a 1-month lag is found between the submesoscale activity and mixed layer depth. However, the correlation between the submesoscale activity and mixed layer depth varies regionally. Examining a broader swath of the North Pacific Ocean, submesoscale energy and vorticity tends to lag production in most regions, but an in-phase correlation occurs in the Subarctic Western Pacific region ( Figure 6 of Sasaki et al., 2017). Mensa et al. (2013) in the Gulf Stream also find a lagged correlation (Figure 18 of Mensa et al., 2013). The submesoscale activity peak is determined by the time of sign reversal of the net energy tendency, which in this analysis mainly depends on the seasonality of the ADV. One possible explanation for the lag of the negative ADV peak behind the PK peak is the seasonal strength difference between submesoscales and larger scales (above 80 km). The larger scale KE is stronger in summer while submesoscale energy is accumulated in winter ( Figure S1 in the supporting information; e.g., Sasaki et al., 2017). The remarkable increase of the relative energy content of larger scales in spring affects the nonlinear interactions between submesoscales and larger scales, manifested in the inverse cascade and weakening submesoscales. So, the negative ADV is inferred to be strong in spring, but relatively weak at other times of year. Other sources and sinks of energy, such as riverine inputs of PE (Barkan et al., 2017), bottom drag or topographic interactions in shelf seas (Gula et al., 2016), and possibly forward energy cascades under other forcing conditions (Capet, McWilliams, et al., 2008b) will lead to other distinct regimes of energetic seasonality. In any case, sinks of submesoscale energy should be considered alongside sources since the conversion of PE by vertical buoyancy flux and re-stratification does not reduce the submesoscale KE.