Energetics of Eddy-Mean Flow Interactions in the Amery Ice Shelf Cavity

Previous studies demonstrated that eddy processes play an important role in ice shelf basal melting and the water mass properties of ice shelf cavities. However, the eddy energy generation and dissipation mechanisms in ice shelf cavities have not been studied systematically. The dynamic processes of the ocean circulation in the Amery Ice Shelf cavity are studied quantitatively through a Lorenz energy cycle approach for the first time by using the outputs of a high-resolution coupled regional ocean-sea ice-ice shelf model. Over the entire sub-ice-shelf cavity, mean available potential energy (MAPE) is the largest energy reservoir (112 TJ), followed by the mean kinetic energy (MKE, 70 TJ) and eddy available potential energy (EAPE, 10 TJ). The eddy kinetic energy (EKE) is the smallest pool (5.5 TJ), which is roughly 8% of the MKE, indicating significantly suppressed eddy activities by the drag stresses at ice shelf base and bottom topography. The total generation rate of available potential energy is about 1.0 GW, almost all of which is generated by basal melting and seawater refreezing, i.e., the so-called “ice pump.” The energy generated by ice pump is mainly dissipated by the ocean-ice shelf and ocean-bottom drag stresses, amounting to 0.3 GW and 0.2 GW, respectively. The EKE is generated through two pathways: the barotropic pathway MAPE→MKE→EKE (0.03 GW) and the baroclinic pathway MAPE→EAPE→EKE (0.2 GW). In addition to directly supplying the EAPE through baroclinic pathway (0.2 GW), MAPE also provides 0.5 GW of power to MKE to facilitate the barotropic pathway.


INTRODUCTION
The Amery Ice Shelf (AIS) is the third largest Antarctic ice shelf located in East Antarctica, with an area of about 62,000 km 2 (Herraiz-Borreguero et al., 2016a). The AIS is mainly fed by the Lambert-Mellor-Fisher tributary glacial systems (Wen et al., 2007;Yu et al., 2010), and it contains some of the deepest Antarctic floating ice (more than 2,500 m below sea level; Fricker et al., 2000). Antarctic ice shelves are key for global sea level rise and Southern Ocean circulation, by exchanging mass, heat, and salt with the Southern Ocean (Heil et al., 1996;Williams et al., 2002;Shepherd et al., 2010;Joughin and Alley, 2011;Pritchard et al., 2012). In addition, the ocean circulation under the AIS also has great impacts on marine bio-geochemical and ecological processes (Williams et al., 2010;Post et al., 2014;Herraiz-Borreguero et al., 2016b).
To better understand the important impacts of ocean-ice shelf interaction on the Antarctic climate system and ecosystem, a deeper understanding of ocean dynamics in the sub-ice-shelf cavity is essential. For this purpose, we conducted a thorough examination of the eddy-mean flow interaction by analyzing the Lorenz energy cycle (LEC). LEC is a useful method to analyze the oceanic and atmospheric energy cycle and corresponding dynamic processes (Lorenz, 1955;Oort and Peixoto, 1983;Oort et al., 1994;Wunsch and Ferrari, 2004). For example, using a number of ocean observations, Oort et al. (1994) analyzed the energetics of the global ocean including energy generation, conversion and dissipation terms, and Wunsch and Ferrari (2004) depicted major energy reservoirs and energy dissipation mechanisms. Based on fully compressible Navier-Stokes equations, Tailleux (2009) developed the framework of available potential energy (APE), despite the challenging nature of the accurate definition of APE (Huang, 2005); and the importance of buoyancy power input has been highlighted (Tailleux, 2010). In the sub-ice-shelf cavity, for example, the buoyancy power input supplies almost all the energy to drive the circulation as wind stress cannot be exerted on sea water (Foldvik and Kvinge, 1974;Lewis, 1985;Schodlok et al., 2016).
Benefitting from high-resolution numerical models, detailed investigations of the global ocean energy cycle have been advanced (Cox, 1985;Robinson, 2005, 2007;Maltrud et al., 2010;Lucarini and Ragone, 2011;Olbers et al., 2012;von Storch et al., 2012;Chen et al., 2014;Kang and Curchitser, 2015;Zemskova et al., 2015;Yang and Liang, 2016;Wu et al., 2017aWu et al., , 2021. For example, Wu et al. (2017a) analyzed the response of the LEC for the Southern Ocean to intensified westerlies. They found that all energy conversions are enhanced under stronger wind forcing, and the conversion rates in the Southern Ocean are strongly influenced by large topography where energy is converted from eddy kinetic energy (EKE) to mean kinetic energy (MKE). In addition, a number of studies (Eden and Boning, 2002;Xie et al., 2007;Shore et al., 2008;Zhai et al., 2010;Yang et al., 2013;Curchitser, 2015, 2016;Zhan et al., 2016) have also investigated the energetics of a regional ocean using a similar method as that used in von Storch et al. (2012) and Wu et al. (2017a). These investigations found that the eddies are generated by two pathways: the barotropic pathway (where MKE transfers energy directly to EKE) and the baroclinic pathway (where eddies gain energy from available potential energy), with the baroclinic pathway being dominant (Kang and Curchitser, 2015;Wu et al., 2017aWu et al., , 2021. Recently, model results and observations have demonstrated that eddy processes play an important role in transporting warm Circumpolar Deep Water (CDW) onto the continental shelf where it can enter the ice shelf cavity (Moffat et al., 2009;Martinson and McKee, 2012;Stewart and Thompson, 2015;Couto et al., 2017). St-Laurent et al. (2013) also found that eddies are necessary to represent the interaction between a Rossby wave along the shelf break and a bathymetric trough, leading to trough-induced intrusions of CDW. The importance of eddy-mediated heat transfer onto the continental shelf has also been found in the East Antarctic, Amundsen Sea, western Antarctic Peninsula, and Prydz Bay by realistic ocean models (Hattermann et al., 2014;Nakayama et al., 2014;St-Laurent et al., 2015;Graham et al., 2016;Liu et al., 2017). In addition, a comprehensive modeling study of on-shelf heat transport around Antarctica found that eddies drive the net shoreward heat transport onto the continental shelves (Stewart et al., 2018). However, a detailed investigation on the eddy-mean flow interaction in an Antarctic sub-iceshelf cavity has not been conducted so far. One principal difficulty arises from the limited spatial and temporal coverage of oceanographic observational data underneath Antarctic ice shelves. Another difficulty is that the typical value of Rossby deformation radius is about 5 km on the Antarctic continental shelves (Hallberg, 2013;Nurser and Bacon, 2014;Mack et al., 2019) and hence most models with coarse spatial resolutions cannot fully resolve eddies on the Antarctic continental shelves and in sub-ice-shelf cavities (Wu et al., 2017a;Mack et al., 2019). Closer to the ice shelf front, the brine rejection induced by sea ice formation, especially in polynya regions, is key to the water mass transformation by increasing salinity and mediating surface freezing point (Naughten et al., 2019). Thus, the detailed investigation of eddy-mean flow interaction in a sub-ice-shelf cavity requires a high-resolution coupled ocean-sea ice-ice shelf model.
There are also a number of differences between the energy cycles in a sub-ice-shelf cavity and the open ocean. In the open ocean, the ocean general circulation is driven by external forcing such as the winds, tides, the exchanges of heat and freshwater with the atmosphere; particularly, the wind supplies most energy to the ocean circulation (Wunsch, 1998;Huang et al., 2006;von Storch et al., 2007;Hughes and Wilson, 2008;Scott and Xu, 2009;Wu et al., 2016Wu et al., , 2017b, and most of that is deposited into the Southern Ocean (Roquet et al., 2011;Zhai et al., 2012;Wu et al., 2017aWu et al., , 2020. However, as mentioned above, wind power input cannot be exerted on sea water beneath an ice shelf. Instead, the ice pump mechanism mainly drives the circulation in the vast majority of big sub-ice-shelf cavities (Foldvik and Kvinge, 1974;Lewis, 1985;Schodlok et al., 2016). The ice pump mechanism involves freshwater generation by ice shelf melting resulting in an increase of buoyancy and then generating kinetic energy. The increase in the freezing with the ascending water can lead to direct marine ice accretion with subsequent brine rejection which might in turn modify the in situ ocean condition. Hence, a quantitative analysis of the ice pump mechanism through investigating the eddy-mean flow interaction is useful for understanding the unique dynamics in a sub-ice-shelf cavity.
In Prydz Bay, the main ocean circulation is characterized by a large cyclonic gyre, centered in the Amery Depression (Smith et al., 1984;Nunes Vaz and Lennon, 1996). Previous numerical studies have simulated the ocean circulation in Prydz Bay and beneath the AIS (Williams et al., 2001;Galton-Fenzi et al., 2012;Liu et al., 2017). The cyclonic Prydz Bay gyre is reproduced in the previous numerical simulations; these simulations further suggest the existence of cyclonic circulation beneath the ice shelf, i.e., inflow of dense shelf waters in the eastern flank of the cavity and outflow of cold and low salinity Ice Shelf Water (ISW) in the western flank (Figure 1). The inflow of modified CDW along the eastern flank of the AIS can induce a basal melting rate of up to 2 m yr −1 at the northeastern corner of the AIS (Herraiz-Borreguero et al., 2015).
In this paper, we used the outputs of a recently developed eddy-resolving ocean-sea ice-ice shelf model (Liu et al., 2017) to conduct a quantitative study on the LEC in the sub-ice-shelf cavity of the AIS. The remainder of this paper is structured as follows. Sections "Materials and Methods" and "Results" provide brief descriptions of the theoretical framework and the numerical model used in this study. The detailed analysis of the eddy-mean flow energetics under the AIS is presented in section "Results". This paper is summarized with a conclusion and discussion in section "Concluding Remarks and Discussion".

Diagnostic Framework
Following von Storch et al. (2012) and Wu et al. (2017a), the four energy reservoirs, namely, MKE ( k m ), EKE ( k e ), mean available potential energy (MAPE, P m ), eddy available potential energy (EAPE, P e ), are defined as follows: where ρ 0 is a constant reference density (1,038 kg m −3 ), u is the zonal velocity, and v is the meridional velocity, with the over bar denoting monthly means over the 5-year period and the prime denoting the deviation from these time means, as the general life period of the eddy activities in the AIS cavity is less than 1 week. It will be shown later that the annual cycle makes large contributions to the eddy reservoirs and the associated energy conversion terms in the AIS cavity especially near the ice front region , if the annual cycle is included in the eddy terms. n 0 is the vertical gradient of the time (monthly) and area mean (over the whole AIS cavity) of local potential density ρ, which can be split into two parts, as followingρ(x, y, z, t)=ρ * (x, y, z, t)+ < ρ(x, y, z, t) >; and, < · > represents the total area mean at a given depth. < ρ(x, y, z, t) > is the reference density defined by the whole area average of the monthly density over the AIS cavity. The value of n 0 is different for each month. It is noted that this definition of APE is just an approximation, particularly in the regions with weak stratification and big topography (Stewart et al., 2014;Zemskova et al., 2015). The MAPE field is found to be sensitive to the choice of density reference stratification (ρ r ). Both its spatial distribution and integrated value change largely when evaluated with alternative ρ r profiles, especially for the one over the region near the ice shelf front (not shown). However, the density reference stratification has a negligible influence on EAPE and the associated conversion rates ( MAPE→EAPE and MAPE→MKE ; not shown). Briefly, the dissipation terms of KE induced by the drag stress at ice shelf base and bottom topography, and the generation terms of APE are given by and where τ IS and τ B denote the horizontal ice shelf-ocean drag stress and bottom drag stress, u IS and u B are the ice shelfocean interface horizontal velocity and the bottom horizontal velocity, B 0 = g ρ 0 ( αQ * C w + ρ 0 βSE * ), α = ( ∂ρ ∂θ ) S,z , β = ( ∂ρ ∂S ) θ,z , θ and S denote the potential temperature and salinity, C w is the heat capacity of seawater, Q * and E * are the net ice shelf-ocean heat and freshwater fluxes. Note that wind stress does not supply wind power to the sea water beneath the AIS; conversely, the ice shelf-ocean and the bottom drag stresses (Eqs. 5, 6), which are estimated using the quadratic bulk formula (Hibler and Bryan, 1987;McPhee, 2008), extract KE from the ocean. And, the quadratic drag coefficients at the ice shelf and sea bed are set as 2 × 10 −3 . Note that the dissipation terms of KE are just approximations, as it can also be dissipated through interior vertical viscosity and biharmonic viscosity in this model (Wu et al., 2017a).
In addition, the conversion terms are the following: and C(P e , k e ) = −gρ w , where w is the vertical velocity, u and u h denote the threedimensional and horizontal velocity vectors. The positive value of C (A, B) means the energy is converted from A to B. Conversely, a negative value of C (A, B) means the energy is converted from B to A. The lateral transport terms of MKE ( B (k m ) ), EKE ( B (k e ) ), MAPE ( B (P m ) ), and EAPE ( B (P e ) ) are given by Frontiers in Marine Science | www.frontiersin.org and where p is the pressure, N denotes the northern boundary at the AIS calving front. We refer readers to Wu et al. (2017a) for the detailed derivation of the energy budget equations used here.

Model
A high-resolution coupled regional ocean-sea ice-ice shelf model (Liu et al., 2017) is used to investigate the energetics under the AIS based on the equations given in section "Materials and Methods." The simulation is performed using the Massachusetts Institute of Technology General Circulation Model (MITgcm; Marshall et al., 1997a,b;Losch, 2008), which solves the primitive equations with the Boussinesq approximation and hydrostatic assumption. The model domain extends from 74 to 65 • S and from 60 to 90 • E with a spherical grid projection, including Prydz Bay and the AIS cavity (Figure 1). The mean zonal and meridional grid spacings of the model are 0.045 • and 0.014 • , respectively. Hence, the zonal grid spacing is between 1.4 and 2.2 km at the southern and northern boundaries, respectively, and the meridional grid spacing is 1.5 km. In addition, there are 70 unevenly spaced vertical levels whose thickness increases from 10 m near the surface to 250 m near the ocean bottom. This high-resolution horizontal grid spacing is necessary for resolving meso-scale eddies, as the Rossby radius of deformation here is only a few kilometers (Mack et al., 2019). Note that it can become less than 1 km, especially in destratified water column as commonly observed under cold ice shelves in polar winter condition ( The horizontal sub-grid-scale viscosity and diffusion is set as strain rate-dependent diffusivity (Smagorinsky, 1963). The background mixing of potential temperature and salinity is realized by a vertical diffusivity of 5 × 10 −7 m 2 s −1 , supplemented by the K-Profile Parameterization (Large et al., 1994). Bottom stress is parameterized using the quadratic drag law; and the quadratic drag coefficients at the ice shelf and sea bed are set as 2 × 10 −3 (Griffies and Hallberg, 2000). The dissipation rate of kinetic energy will change with different quadratic drag coefficients. The oceanic equation of state is the same as that used in Jackett and McDougall (1995). The parameterization used to model the ice-ocean thermodynamic exchanges are obtained from solving a system of three equations that is derived from the heat and freshwater balance at the ice ocean interface (Hellmer and Olbers, 1989;Holland and Jenkins, 1999;Jenkins et al., 2001;Losch, 2008;Naughten et al., 2019). The transfer coefficients for temperature and salinity at the ice shelf-ocean interface are functions of the friction velocity (Holland and Jenkins, 1999;Dansereau et al., 2014). This model does not explicitly include tides. This flaw may lead to a weaker eddy activity and lower freezing/melting rates (Liu et al., 2017;Mueller et al., 2018). The vertical diffusivity and viscosity coefficients of 10 −5 m 2 s −1 and 10 −4 m 2 s −1 are applied, respectively. It can partially mimic the tide mixing, as used in Naughten et al. (2019). Details of the model configuration and the model assessments can be found in Liu et al. (2017).

Experiments
The model was spun up for 20 years using the monthly climatology (averaged over 1979-2012) of the atmospheric data from the Japanese Reanalysis (JRA-55) dataset at a spatial resolution of 1.125 • × 1.125 • (Kobayashi et al., 2015). After the 20-yr spin-up phase, the kinetic energy in the sub-ice-shelf cavity reaches a steady state (Liu et al., 2017). Afterward, the simulation (CONTROL) is integrated for another 5 years (1979)(1980)(1981)(1982)(1983) driven by the 6-hourly atmospheric forcing and the 1-day mean outputs for these 5 years were analyzed in this study. The JRA-55 used here includes net long wave radiation, net shortwave radiation, humidity, 2 m air temperature, precipitation, and 10 m winds. The monthly eastern, western and northern boundary conditions are specified by the outputs from the Estimating the Circulation and Climate of the Ocean phase II: high resolution global-ocean and sea-ice data synthesis (ECCO2) (Menemenlis et al., 2008). Benefiting from hydrographic and satellite observations, the model performance has been assessed in Liu et al. (2017).
To further diagnose the influence of the ice pump mechanism on the LEC beneath the AIS, another experiment (Expshutdown) has been conducted. The only difference from the CONTROL experiment is that the thermodynamic influence of the AIS on the ocean is excluded in Exp-shutdown, i.e., the heat and freshwater fluxes at the ice shelf-ocean interface are set to zero. Unless otherwise stated, model outputs from CONTROL over the last 5 years are used for this study. And, the annual means of energy reservoirs and conversion rates are analyzed.  Figure 2D). Two subdomains representing the inner (AIS1) and outer (AIS2) parts of the sub-ice-shelf cavity are outlined and labeled in Figure 2B. All four energy components exhibit distinctive spatial patterns. The distribution of MKE is dominated by two narrow strips of large MKE along the eastern and western flanks of the AIS cavity, with the MKE on the eastern side being weaker than that on the western side in the AIS2 region (Figure 2A). As the kinetic energy mainly concentrates in the ocean circulation, the distribution of MKE reflects the main cyclonic circulations beneath the ice shelf, i.e., inflow of shelf water in the east and outflow of ISW in the west (Williams et al., 2001(Williams et al., , 2002Hemer et al., 2006;Galton-Fenzi et al., 2012). The relatively large MKE in the eastern flank in the outer part (AIS2) reflects the simulated intrusion of the outside water until 70 • S, while the re-emerging of large MKE in the inner part (AIS1) is apparently driven by the ice pump mechanism (Figure 2A).

Energy Reservoirs
The EKE shows a different pattern from MKE, characterized by large and well spread EKE in the interior areas, especially in regions AIS1 ( Figure 2B). Large EKE on the eastern side of AIS2 is induced by the flow of EKE from the outside. Large EKE also exists in region AIS1, especially near the grounding line where the maximal melt rate is about 30 m yr −1 (Wen et al., 2007;Galton-Fenzi et al., 2012). The remarkable release of freshwater near the grounding line makes the seawater buoyant; hence, APE is available to generate vigorous eddy activity there. When the seawater ascends along the upward-sloping base of the ice shelf, refreezing of this water diminishes buoyancy through brine rejection. Notably, the large EKE is also concentrated in the depression region near 72 • S (Figure 1) where MKE (Figure 2A), MAPE (Figure 2C), and EAPE ( Figure 2D) are also large, implying strong barotropic and baroclinic instability there. In addition, the relative vorticity shows that abundant eddies exist in the AIS cavity, especially in the regions near the boundaries and the ice front (not shown). There are substantial high-frequency variabilities in time series of temperature, salinity and velocities at the top two model levels in the AIS cavity (not shown). Power spectra analysis of these high-frequency fluctuations show that the most visible period is from 2 to 6 days (not shown).
Figures 2C,D show the distributions of vertically integrated MAPE and EAPE, respectively. Broadly, the pattern of EAPE is similar to that of MAPE. The MAPE and EAPE distributions feature larger values in region AIS2 and the region around 72 • S than in region AIS1. The ocean gains large MAPE in the eastern AIS calving front caused by basal melt of up to 2 m yr −1 (melting causes an increase in MAPE in the form of buoyancy release; Herraiz-Borreguero et al., 2015); Another region of high MAPE is near the grounding line which is induced by basal melting as large as 30 m yr −1 , consistent with previous modeling studies ( Galton-Fenzi et al., 2012). Also, the pattern of EAPE is basically similar to that of EKE, especially in region AIS1. The resemblance of the horizontal structure between EAPE and EKE was also found in the global ocean and the Southern Ocean (Roullet et al., 2014;Wu et al., 2017a). It will be shown later that the similar spatial distributions of these three reservoirs (MAPE, EAPE, and EKE) imply that the energy under the ice shelf mainly transfers through the baroclinic pathway, especially near 72 • S.
The eddy term is redefined to examine the effect of annual cycle on the reservoirs and the associated energy conversion terms. The "mean" denotes time mean over the 5-yr model outputs and the "eddy" denotes the deviation from this annual mean. Variables included the annual cycle are defined as "variable _annual ." When the seasonal cycle is included, the patterns of eddy reservoirs change slightly except the AIS2 region (Figure 3). Also, the life period of the eddies in the AIS cavity is less than 1 week (not shown). Hence, the monthly mean is chosen to decompose the "mean" and the "eddy" in this study. In addition, the volume integrated values of EKE and EKE _annual are 5.5 TJ to 15.7 TJ, respectively; the volume integrated EAPE and EAPE _annual increase from 10.1 to 34.9 TJ when the annual cycle is included. Additionally, the new defined eddy term shows that the annual cycle accounts for roughly 60% of variability in the G(EAPE _annual ), D(EKE _annual ) and the associated conversion terms ( MAPE _annual →EAPE _annual →EKE _annual and MKE _annual →EKE _annual Frontiers in Marine Science | www.frontiersin.org ), especially in the AIS front region (not shown). These results further indicate suppressed eddy activities in a sub-ice-shelf cavity as both the stresses at the ice shelf base and bottom topography dissipate the kinetic energy.
The sections across the areas with large reservoirs are chosen to illustrate the vertical distributions of the four energy components; both the MKE and EKE extend throughout the whole water column (Figures 4A,B). Larger MKE concentrates in the inner part of the ocean cavity from the ice shelf draft to bedrock ( Figure 4A); large EKE can also be identified throughout the whole water column near the grounding line ( Figure 4B). In contrast, large MAPE and EAPE concentrate near the base of the AIS. It is interesting to note that the relatively large MAPE and EAPE in the region from 69 to 70 • S (Figures 4C,D) result from strong transports of MAPE and EAPE from near the grounding line where large buoyancy power input exists (see below). The corresponding EKE increase shifts to the eastern side of this selected cross section (Figure 2). The basal melting only occurs at the interface between the ice shelf base and ocean water. Hence, APE is confined mostly to the upper water column (Figures 4C,D). When the APE is transferred to the kinetic energy, the vertical structure of the mean circulation and eddy activity extends through the column (Figures 4A,B). The down sloping feature at 71.5 • E in the EAPE is most likely induced by large eddy activity present in this location ( Figure 4D).
The magnitudes of averaged MKE over the whole AIS domain generally decrease toward the deeper ocean ( Figure 5A), except in the upper 200 m. The profile of EKE is visibly different from that of MKE in such a way that the EKE decreases dramatically in the upper 600 m, and then changes slightly below 1,000 m. Another feature is that the area average MKE is larger than EKE in almost all the model layers except the upper layers. This is contrast to earlier findings that EKE is much more dominant than MKE in most areas of the Southern Ocean (Wu et al., 2017a) and the global ocean (von Storch et al., 2012), reflecting again that eddies are greatly suppressed by both the stresses at ice shelf base and bottom topography. The larger MKE reservoir than EKE reflects that the current under the AIS, driven by the ice pump mechanism, is steadier than that in the open ocean. It is noted that EKE is larger than MKE in the upper tens of meters which is presumably influenced by the combined effects of CDW, dense shelf water (DSW), ISW, the formation of marine ice, and the vertical shear. The area average MAPE and EAPE (Figure 5A) over the AIS cavity are different from each other: above 150 m, FIGURE 3 | The annual mean distributions of the vertically integrated EKE _annual (J m −2 ). Note that the time-mean used here is the 5-year mean which is different from monthly mean used in Figure 2B.
MAPE increases with depth and decreases dramatically below, whereas EAPE decreases with depth dramatically above 200 m and shows a moderate decrease below; in addition, MAPE is much larger than EAPE. The vertical profile of EAPE is roughly similar to that of EKE, implying that the energy exchange between these reservoirs is dominated by the baroclinic pathway (i.e.,

MAPE→EAPE→ EKE ).
Furthermore, the vertical profiles of the four reservoirs are separated into regions AIS1 and AIS2 (Figures 5B,C). In region AIS1, the MKE profile is different from that of MAPE in such a way that the values of MKE increase gradually from 500 to 1,000 m and decrease slightly from 1,000 m to the bottom (Figure 5B). The vertical profiles of MAPE and EAPE in region AIS1 have similar patterns featured by the larger values rapidly decreasing in the upper 800 m. The averaged value of EKE is much smaller than the other reservoirs in region AIS1 with larger values in the depth from 600 to 800 m ( Figure 5B). The profiles of energy components in region AIS2 are similar to those over the whole AIS cavity, featured by large values concentrating in the upper 200 m and a large decrease below ( Figure 5C).

Generation of APE
As revealed by Herraiz-Borreguero et al. (2015, 2016a, the modified CDW intrusions are found at the eastern flank of the AIS calving front in the early austral winter. The ice shelf base melting/freezing rate is determined by the temperature difference between the seawater contacting with the ice shelf and the in situ freezing point. Generally, freezing occurs along the west and  north of the AIS, and melting occupies the east and south of the AIS, consistent with previous studies (Galton-Fenzi et al., 2012;Depoorter et al., 2013;Rignot et al., 2013;Herraiz-Borreguero et al., 2015). The simulated net basal melt rate is relatively larger than previous modeling studies and observations (Hellmer and Jacobs, 1992;Galton-Fenzi et al., 2012;Depoorter et al., 2013;Rignot et al., 2013), and it might be due to the absence of a frazil ice parameterization in our model (Galton-Fenzi et al., 2012;Herraiz-Borreguero et al., 2013) and the relatively thick vertical layers at the base of the ice shelf in our simulation (Schodlok et al., 2016). The southernmost ISW is formed by the mixing of the inflow water with ice shelf melt water. The maximal melt rate amounts to 30 m yr −1 at the grounding line. The high melt rate at the grounding line agrees with estimations from previous studies (Wen et al., 2007;Galton-Fenzi et al., 2012). In the western flank of the ocean cavity, ISW ascends along the AIS base and gradually becomes super cooled (cooler than the in situ freezing point temperature), leading to areas of re-freezing (Figures 6A,B). Note that the generation of APE is the total effect including the ice shelf melting and the seawater re-freezing. Figures 6E,F show the distributions of the MAPE and EAPE generated by total buoyancy flux, respectively. The pattern of MAPE generation is similar to that of freshwater flux at the ocean-ice shelf interface. Similarly, the pattern of EAPE generation features larger values near the grounding line, and spreads much more evenly over region AIS1. As expected, the APE generation is dominated by the freshwater flux, as salinity is a dominant factor in determining the potential density of cold seawater with temperature close to the in situ freezing point (Figures 6C,D). These regional patterns of MAPE and EAPE generation are important for driving ocean circulation.

Dissipation of KE
The time-mean ice shelf-ocean drag stress and bottom drag stress are shown in Figures 7A,D, which are estimated using the quadratic bulk formula (Hibler and Bryan, 1987;McPhee, 2008). The spatial distributions of ice shelf-ocean drag stress and bottom drag stress are characterized by large values in the eastern and western edges (Figures 7A,D). These patterns reflect that the stresses are induced by the strong inflow and outflow in the eastern and western flanks. The distributions of energy extracted from the ocean by the time-mean ice shelf-ocean drag stress and the bottom drag stress are very similar to the distributions of time-mean drag stresses (Figures 7B,E), featured by notably negative values near the eastern and western flanks. In addition, the patterns of energy dissipation induced by time-varying drag stress (Figures 7C,F) are similar to the distribution of EKE ( Figure 2B). The EKE is significantly suppressed by the stresses both at ice shelf base and the bottom topography. Note that the KE in this model can also be dissipated through interior vertical viscosity and biharmonic viscosity, which varies at each time step. Thus, the dissipation terms of KE (D terms) presented here are just approximations, as the focus of this study is to investigate the important effect of the ice pump mechanism on driving the circulation beneath the AIS.

Energy Conversions
According to equation (10), MAPE is converted into EAPE when the eddy density flux ρ u h is directed down the mean density gradient. Figure 8A presents the vertically integrated distribution of the conversion from MAPE to EAPE. A positive value indicates energy being transferred from MAPE to EAPE, and a negative value indicates transfer from EAPE to MAPE. The distribution is featured by large positive values concentrated in the western flank of the ocean cavity and in the eastern flank from 69 • S to the calving front, while some negative values exist in the central region of AIS2. Similarly, the vertically integrated distribution of energy conversion between EAPE and EKE is characterized by the relatively small positive values across the majority of the AIS cavity and large positive values toward the grounding line ( Figure 8B). These results indicate that substantial eddy activities are generated in the AIS1 and the western boundary current in the AIS cavity. In region AIS1, the spatial pattern of energy conversion from EAPE to EKE is featured with mixed positive and negative values. The excess buoyancy obtained through the interactions between the ocean and ice shelf is released by the baroclinic instability, as the buoyant water ascends along the upward-sloping base of the ice shelf. The energy conversions between MAPE, EAPE and EKE indicate a baroclinic pathway in the AIS cavity, which is especially visible in AIS1 and western boundary current regions. Figure 8C gives the vertically integrated distribution of energy conversion between MAPE and MKE, featured by mixed large positive and negative values along the western flank of the AIS cavity in region AIS2 and large positive values in region AIS1, and similar to that between EAPE and EKE ( Figure 8B). The large MAPE generation rate in region AIS1 (Figure 6C) is generally balanced by the large conversion rate from MAPE to MKE (Figure 8C), leading to very small MAPE there ( Figure 2C). In contrast to the patterns of energy conversions between MAPE, EAPE, and EKE, the energy conversion between MKE and EKE is characterized by mixed positive and negative values over region AIS2 (Figure 8D). However, in region AIS1, the energy is mainly transferred from MKE to EKE especially in the west flank of region AIS1. The energy conversions from MAPE to MKE and then to EKE indicates a different energy conversion pathway rather than the baroclinic one, which is particularly strong over region AIS1. The large energy conversion rate from MAPE to MKE generates much stronger and wider mean current concentrated in the western and eastern boundary regions of AIS1 and western boundary region of AIS2 which is strikingly different from the open ocean. However, the spread and strength of mean current in the open ocean are much narrower and weaker than those of eddy activities (von Storch et al., 2012;Wu et al., 2017a). In the AIS cavity, the buoyancy forcing (i.e., ice pump) is major factor driving the circulation. Hence, the significant positive energy conversion rate from MAPE to MKE is confined in the western AIS cavity (Figur 1). As APE is the only generating source of eddy activity, the energy conversion rate from EAPE to EKE is positive under most of the shelf. The heterogenous pattern of energy conversion rate from MKE to EKE is induced by the interaction between the ocean circulation and the topography as shown in previous studies Curchitser, 2015, 2016;Wu et al., 2017a). Figure 9 presents the vertical distributions of energy conversion rates between the four energy reservoirs. The vertical pattern of the energy conversion between MAPE and EAPE ( Figure 9A) is similar to that of the energy conversion between EAPE and EKE (Figure 9B), featured by large values near the interface between the ice shelf and the ocean, especially in region AIS1 (Figures 9A,B). The vertical structure of the energy conversion from MAPE to MKE shows a pattern of large positive values in the upper water column and large negative values in the lower water column from 72 to 73 • S ( Figure 9C). These large negative values indicate that the MAPE extracts energy from the mean current in the lower water column, similar to the pathway of energy conversion in open ocean. The lower density water from ice shelf melting leads to a more tilted isopycnal surface. Consequently, the energy is transferred from MKE to MAPE in the lower water column. This suggests that stronger currents arise from here than in other areas of the AIS cavity. The vertical distribution of energy conversion rate between MKE and EKE ( Figure 9D) is much more homogeneous. In region AIS1, the conversion rate from MKE to EKE is largely positive in the entire water column, except for some negative values close to the ice shelf base (Figure 9D). The positive values mean that the eddy activity is generated by the shear instability there. In region AIS2, there are mixed large positive and negative values but vertically homogenous ( Figure 9D). The positive values indicate that the eddy activity is generated by the mean flow through the shear instability. Negative values mean that eddy activity can also supply energy to the mean flow through the energy inverse cascade, such as by the interaction between the mean flow and the topography Curchitser, 2015, 2016;Wu et al., 2017a). For all conversion terms it holds that they are dominated by smaller scale structures of positive and negative values. Consequently, only in an integrated sense can a sign or direction be associated with each of the conversions (von Storch et al., 2012;Wu et al., 2017a).

Energy Budget
In this section, we present the eddy-mean flow energy budgets integrated over the entire ocean cavity and over two subdomains indicated in Figure 2B. We first examine the energy reservoirs. Over the whole AIS domain, MAPE is the largest energy reservoir (111.6 TJ, 1 TJ = 10 12 J) which is almost two times as large as MKE (70.1 TJ) and more than eleven times larger than EAPE (10.1 TJ). The ratio of EKE to MKE is roughly 8% (Figure 10A), in contrast to that for the Southern Ocean and global ocean, where ratios of EKE to MKE are almost 200 and 280% times, respectively (von Storch et al., 2012;Wu et al., 2017a). However, MKE is the leading energy reservoir in region AIS1 which is 1.4, 8 and 12.2 times as large as MAPE, EAPE, and EKE, respectively. These results indicate a strong conversion of MAPE to MKE and significant suppression of EKE over the whole AIS cavity, especially in region AIS1 (Figures 10A,B). After the much larger MAPE reservoir (63.5 TJ), the MKE reservoir (26.6 TJ) contains much more energy than EAPE and EKE which is roughly 7.8 and 16.7 times as large as EAPE and EKE ( Figure 10C) in region AIS2.
Integrated over the AIS cavity, the total MAPE generation rate is 0.9 GW (1GW = 10 9 W), dominated by the freshwater forcing with a very small contribution from the heat flux forcing indicating that the thermal contribution to density modification is almost negligible at near freezing point temperature (Figures 6C,E). Similarly, the total EAPE generation rate of 26.9 MW (1MW = 10 6 W) is composed of a dominant contribution from time-varying freshwater flux ( Figure 6D). These results reveal the basal melting is the dominant forcing to drive the current under the AIS cavity. This ice pump generated energy is mainly dissipated by the ice shelf-ocean drag stress and bottom drag stress. The integrated values of D(MKE) and D(EKE) induced by ice shelf-ocean drag stress (bottom drag stress) are −0.25 and −0.07 GW (−0.0.15 and −0.03 GW), respectively ( Figure 11A).
Regarding the energy conversions among the reservoirs for the entire AIS cavity, the energy is transferred away from the mean flow to the eddy field with an APE conversion rate of 0.2 GW and a KE conversion rate of 0.03 GW. The EKE is generated through two pathways: the barotropic pathway MAPE→MKE→EKE (0.03 GW) and the baroclinic pathway MAPE → EAPE → EKE (0.18 GW), with the baroclinic pathway being dominant. The ratio of barotropic to baroclinic contribution to EKE production is 17%, similar to the ratio for the global ocean, but quite different from the ratio (about 7%) for the Southern Ocean (von Storch et al., 2012;Chen et al., 2014;Wu et al., 2017a). In addition, the MAPE reservoir also supplies 0.5 GW of energy to MKE and facilitates the barotropic pathway ( Figure 11A). Region AIS1 has the same energy conversion pathway as the whole AIS cavity, showing two major energy transferring pathways from MAPE: MAPE → EAPE → EKE and MAPE → MKE → EKE with conversion rates of 0.1 GW and 0.03 GW to the EKE, respectively ( Figure 11B). In region AIS2, despite the baroclinic pathway having a rate of 0.04 GW, no energy is transferred from MKE to EKE; instead, there is a weak energy transfer from EKE to MKE (Figure 11C). These results indicate that the eddies, on average, are mainly generated in region AIS1, and then act to accelerate the mean flow slightly in region AIS2. The energy flows at the AIS calving front are significantly weaker than the energy conversion rates in the AIS cavity. The circulation in the AIS cavity slightly loses APE and KE at the coastal-portion of the AIS (Figure 11A). The generation of EKE is dominated by the baroclinic pathway, which is driven by the buoyancy input of basal melting in the AIS cavity. This is evidence that general circulation models need to include ice/ocean interaction, with full descriptions of cavity dynamics, to fully measure the energy budgets of the ocean.

The Ice Pump Mechanism
Unlike the open-ocean circulation driven by the atmospheric forcing, the circulation beneath the AIS is mainly driven by the buoyancy power input supplied by the ice shelf basal melting i.e., the ice pump mechanism (Foldvik and Kvinge, 1974;Lewis, 1985;Schodlok et al., 2016). To quantify the impact of ice pump mechanism on the LEC for the circulation in the AIS cavity, we conducted a sensitivity experiment in which the heat and freshwater fluxes at the ice shelf-ocean interface are set to zero. When the buoyancy effects of basal melting and freezing are removed, all the energy reservoirs become very small (Figure 10). The inflow and outflow also become very weak, except for near the eastern calving front (Figure 10), stressing the dominant role of the ice pump mechanism in regulating the cyclonic circulations beneath the ice shelf. The integrated values of the four energy reservoirs, energy conversions and dissipations are also summarized in Figure 11. Again, these results present the dominate role of basal melting in driving the circulation in the AIS cavity. Note that the energy reservoirs in the eastern part of AIS2 are induced by the intrusion of the ocean circulation, and the values of MKE and EAPE in the deepest regions of the ice shelf cavity are probably from the model initial condition. If the model runs for an enough long time, these energy reservoirs will be dissipated.

CONCLUDING REMARKS AND DISCUSSION
In this study, we conducted a detailed energetic analysis for the circulation in the sub-ice-shelf cavity of the AIS, by using the daily outputs of a high-resolution coupled regional ocean-sea ice-ice shelf model. We examined the characteristics of the four energy components, their generations, dissipations and conversions in the AIS cavity. In particular, the critical role of the ice pump mechanism in driving the energy cycle for the ocean circulation beneath the AIS is investigated quantitatively. By analyzing the model outputs, we obtained the following results: • The MKE shows a distinct spatial pattern, with the most energetic motions being concentrated in the eastern and western flanks of the AIS cavity, while the EKE is more smoothly spread. The total EKE (5.5 TJ) is just 8% of the MKE (70.1 TJ), indicating a visible suppression of eddy by the stresses of ice shelf and bottom topography in the AIS cavity, which is different from the results for the Southern Ocean and global ocean where the EKE is much larger than MKE. The total EAPE (10.1 TJ) is about 2 times larger than the total EKE. • The total generation rate of APE amounts to 1.0 GW, almost all of which is generated by the ice pump mechanism. This ice pump generated energy is mainly dissipated by the ice shelf-ocean drag stress and bottom drag stress, amounting to 0.32 and 0.18 GW, respectively. • The energy is transferred away from MAPE by two pathways: the barotropic pathway MAPE → MKE → EKE and the baroclinic pathway MAPE → EAPE → EKE. The conversion rates from MAPE to MKE and EAPE are 0.5 and 0.2 GW, respectively. In the inner region near the grounding line, the energy pathway is the same as that in the entire AIS region, while in the outer region near the calving front, the eddies act to accelerate the mean flow slightly. • EKE is mainly generated by baroclinic instability, with 0.18 GW being converted into the EKE from the EAPE, while the barotropic pathway is rather weak, with only 0.03 GW being converted into the EKE from MKE.
These model results elucidated the dynamic processes of eddy generation in the AIS, which differs but complements the results from the recent observational and modeling studies on the AIS and other ice shelves (Galton-Fenzi et al., 2012;Depoorter et al., 2013;Herraiz-Borreguero et al., 2013;Rignot et al., 2013;Schodlok et al., 2016;Liu et al., 2017;Mack et al., 2019). Furthermore, we explored the underlying physical processes quantitatively and gave a description of dynamics in AIS cavity. The distinct characteristics of the energy cycle in the AIS cavity are revealed. The results obtained in this study offer useful benchmarks for analyzing energy cycles in other sub-ice-shelf cavities and provide a more complete understanding of the ocean dynamics in the sub-ice-shelf cavities. The potential energy is stored in the titled isopycnal surfaces and it is increased by adding buoyancy from lighter water masses or by extracting buoyancy from denser water masses. Likewise, potential energy is decreasing by extracting buoyancy from lighter water masses or by adding buoyancy to denser water masses. The ice shelf basal melting supplies most potential energy in the AIS cavity by influencing the sea water density. The density of sea water in the ice shelf cavity is strongly influenced by buoyancy flux through heat and freshwater exchanges between ice shelf and sea water. The relaxation of isopycnal surfaces is mainly through the processes associated with topography features (Wu et al., 2017a). The released energy is used to generate eddy activities. This energy pathway is usually named baroclinic pathway which is the dominated one to generate eddy activity. The other energy pathway of eddy activity generation is named barotropic pathway that MKE supplies energy to the eddy activity by the shear instability. The patterns of energy conversion terms are dominated by smaller scale structures of positive and negative values. Consequently, an integrated sense is needed to find the energy conversion pathway. The ice shelf base and bottom topography both dissipate the kinetic energy of ocean circulation in the ice shelf cavity. Hence, the eddy activities are suppressed significantly in the ice shelf cavity. This is different from the open ocean where the wind can supply energy to the eddy activity.
We acknowledge that this study has some limitations. Firstly, the definition of APE is an approximation, especially in the high-latitude deep convection regions (Zemskova et al., 2015). Secondly, there are several numerical model limitations, such as the uncertain geometry of the sub-ice-shelf cavity (Galton-Fenzi et al., 2012), the uncertain treatments of heat and freshwater exchanges at the interface of ocean-ice shelf, and the coarse model resolution which cannot fully capture the eddy activity in the cavity, especially in destratified water column as commonly observed in polar winter conditions and under cold ice shelves it can become very small < 1 km (Nurser and Bacon, 2014;Mack et al., 2019). For example, Mack et al. (2019) calculated a mean deformation radius for on-shelf winter conditions in the Ross Sea of 1.7 km. They noted a resolution of 350 m or less would be required to capture 95% of the eddy activity especially in the cavity. However, it is not computationally feasible at this time (Mack et al., 2019). Lastly, our model does not include a frazil ice parameterization.
It is also noted that the frazil-ice formation process was found important in determining the basal melting/freezing rate of the AIS (Galton-Fenzi et al., 2012;Herraiz-Borreguero et al., 2013). The frazil ice formation can lead to increased salinity, and hence moderate the circulation pattern and the maximum amount of super-cooling beneath the ice shelf. The frazil crystals can also input more buoyancy to the ocean to accelerate the current under the AIS (Jenkins and Bombosch, 1995). The lack of this process leads to a decrease of marine ice formation (Galton-Fenzi et al., 2012;Herraiz-Borreguero et al., 2013;Hughes et al., 2014), influences the ISW outflow properties (Jordan et al., 2015). Our model also does not include the tidal forcing which is important in modulating the ice shelf basal melting and altering the thermohaline circulation in the sub-ice-shelf cavity by pumping warm water into the sub-ice-shelf cavity and increasing the friction at the ice water interface (Holland and Jenkins, 1999;Padman et al., 2003;Robertson, 2013;Jendersie et al., 2018). Hence, the lack of tidal forcing leads to a decrease of the basal melting and the energy dissipation, and hence influences the whole energy cycle in the AIS cavity. Additionally, the radar surveys and in situ measurements revealed a mass of sinuous subglacial channels typically 500 m to 3 km wide, and up to 200 m high, in the ice-shelf base Stanton et al., 2013). These channels in the ice shelf base enhance the ice shelf basal melting of Pine Island Glacier as large as 0.06 m/day at the channel apex (Stanton et al., 2013). While large channels in the AIS base were found many decades ago (Mellor and McKinnon, 1960), the impact of channels on AIS basal melting has not been investigated. The channels would enhance the AIS basal melting and then lead to a much more buoyant melt water and increased the APE of the circulation under the ice shelf cavity, and therefore strengthen the whole energy cycle. In addition, the presence of these channels are likely to contribute to significant form drag that will increase the ocean/ice drag coefficient in the cavity part of the model domain, and strengthen mesoscale eddy dissipation rates.
Future studies should thus be devoted to the following aspects. Firstly, an understanding of the impacts of frazil ice processes on the energy pathway, and a more complete understanding of the energy cycle in a sub-ice-shelf cavity is needed. Secondly, the effects of tidal forcing on the energy pathway in the ice shelf cavity need systematic study. Thirdly, the 3-D structure of mesoscale eddy and the energetics of the eddy activity during its whole life cycle within the ice shelf cavity need detailed investigation. Additionally, the detailed study of the seasonal variability of energy budget in the AIS cavity is underway.

DATA AVAILABILITY STATEMENT
The data used to reproduce these figures in this manuscript can be found below: https://pan.baidu.com/s/1T4pLmvCb5bLG7mw5_ iGBXw (code: 1234). The model outputs used in this study are available upon request (yang.wu@njxzc.edu.cn).

AUTHOR CONTRIBUTIONS
YW performed the methodology, ran the numerical model, data analyses, and wrote the original manuscript draft. YW, ZW, CL, and LY participated in the manuscript revision and improvement. All authors contributed to the article and approved the submitted version.

ACKNOWLEDGMENTS
The MITgcm code was obtained freely from http://mitgcm. org/. The atmospheric forcing data were obtained freely from the NCAR's research data archive (JRA-55: https://rda.ucar.edu/ datasets/ds625.0/). We thank the editor and two reviewers for their positive and constructive comments and suggestions.