## 1. Introduction

The melting at the margins of the Greenland Ice Sheet (GrIS) and Antarctic Ice Sheet (AIS) has accelerated in recent years. Near many marine-terminating glaciers in Greenland, the submarine melt rate outweighs the contribution from surface runoff (Straneo and Heimbach 2013). The postulated main cause of the recent accelerated melting of the GrIS is the warming of the East and West Greenland currents that influence the water mass properties at the termini of tidewater glaciers (Wood et al. 2018). Similar accelerated melting of the AIS is likely due to greater heat fluxes supplied to the ice shelf cavities by the Circumpolar Deep Water currents (Rignot et al. 2013; Cook et al. 2016).

In recent decades, the melting of the GrIS contributed 1 mm yr^{−1} in global sea level rise on average and this contribution is accelerating and has the potential to contribute over 7 m total (IPCC 2019). The West Antarctic Peninsula, which is a small sector of the AIS with glaciers that terminate in fjords, contributes approximately 0.2 mm yr^{−1} in global sea level rise (Pritchard and Vaughan 2007). A major implication of the accelerated ocean-driven melting of marine-terminating glaciers in these two regions is the retreat of ice sheets, which along with calving and other ice sheet processes may lead to thinning of the outward-flowing GrIS and AIS (Seroussi et al. 2011).

Fjords abutting marine terminating glaciers have also been studied in regions other than the GrIS and West Antarctic Peninsula: the Canadian Arctic Archipelago, which is occasionally grouped with the GrIS and accounts for 9% of the freshwater flux anomaly in Baffin Bay (Bamber et al. 2018); the Patagonia Ice Field (Moffat 2014); Alaska (Sutherland et al. 2019); and Svalbard (Jakacki et al. 2017). In these regions, the fjord circulation has implications for physical and biogeochemical ocean properties and potentially regional ice sheet cover and albedo, but are not important contributors to sea level rise due to the smaller ice sheet volumes.

The oceanic exchange flows between fjords and the continental shelf constrains the ocean-driven melting of the GrIS and West Antarctic Peninsula glaciers. Although progress has been made in understanding the overall sensitivity of ice sheet melt to atmospheric and oceanic forcing (see Straneo and Cenedese 2015, and references therein), the translation of open ocean and fjord conditions to glacial melt rates is not well understood.

To better understand how fjords connect the open ocean to marine-terminating glaciers, recent idealized and regional modeling investigations have explored the constraints of the fjord-to-shelf circulation. Previous studies either use 2D simulations and do not account for rotational effects (e.g., Gladish et al. 2015; Sciascia et al. 2013; Xu et al. 2012), or use 3D simulations but focus on specific processes such as winds (Spall et al. 2017), coastally trapped waves (Fraser et al. 2018), and the wave-influenced fjord response to shelf forcing (Jackson et al. 2018). The effect of varying multiple parameters in a 3D fjord setup, e.g., sill height, tides, and wind forcing, was examined in Carroll et al. (2017). This study concluded that sill depth compared to the grounding line depth is a primary control on fjord overturning and renewal, which is amplified by both winds and tides. Carroll et al. (2017) also finds that horizontal recirculation is stronger for wider fjords, which influences the fjord stratification. However, in general, there remains a lack of theoretical constraints to predict the leading-order dynamics of fjord circulation: the fjord-to-shelf overturning circulation and the horizontal recirculation. Many of these previous numerical studies examine the sensitivity of the overturning circulation to various fjord parameters, but horizontal recirculation is rarely discussed and there are no existing theories to predict its strength.

To fill this gap, in this study we present numerical solutions of an idealized model, supported by dynamical theories of fjord-to-shelf overturning circulation and the horizontal recirculation in the fjord interior. The main difference between this study and its closest predecessor (Carroll et al. 2017) are that it allows the development of a freely evolving shelf circulation and coastal current which interact with the fjord circulation. Including this requires our model experiments to be run for 5 years to fully equilibrate the shelf circulation and coastal current adjacent to the fjord. We also test additional parameters in the sensitivity experiments to include more parameters of leading-order importance to the fjord overturning and recirculation. This expanded exploration of parameter space allows us to develop and test simple, but comprehensive dynamical theories for the overall fjord-to-shelf overturning circulation and the horizontal recirculation in the fjord interior.

In section 2, we present the model configuration and describe the setup and phenomenology of a reference simulation. In section 3, we explore the dependencies of the overturning circulation and horizontal recirculation on six key geometric fjord and forcing parameters. In section 4, we develop theoretical constraints for the overturning circulation/warm-water inflow in three regions of the shelf-to-glacier-face domain: the continental shelf, the fjord mouth sill, and the fjord head. Piecing together the theories of these three regions yields an overall overturning prediction in terms of the parameters explored in section 3, which is supported by the simulation results. In section 5, we present a theory for the recirculation strength within the fjord using the vorticity budget, which is also supported by simulations diagnostics. In section 6, we discuss additional fjord phenomena observed in our simulations: the onset and effect of hydraulic control at the sill and fjord mouth, low-frequency variability manifesting as periodic fjord flushing, and high-frequency submesoscale variability in the fjord and coastal current. In section 7, we discuss the major implications of including the fjord circulation in glacial melt rate estimates, summarize our findings, and provide concluding remarks.

## 2. Idealized fjord-to-shelf model

The design of our model setup is primarily motivated by Greenland’s fjords and continental shelf, but the results from the simulations are likely useful toward understanding fjord circulation in the West Antarctic Peninsula and other regions. Figure 1 shows the bathymetry around Greenland with zoomed-in panels of bathymetry and ice-sheet extent near six major Greenlandic fjords that are among the most observed (Morlighem et al. 2017).

We aim to capture only a few salient geometric features in our idealized model configuration. They are often long, narrow, deep submerged glacial valleys that connect to continental shelves hundreds of meters shallower. Some fjords have a shallow sill either near the mouth of the fjord or between the fjord interior and the open ocean (Ilulissat, Godthabsfjord, and Petermann are notable examples) and fjord width is typically between 2 and 20 km. The coastal winds vary significantly during the year from katabatic to alongshore winds between 0 and 14 m s^{−1} (monthly averages) and between 0 and 9 m s^{−1} (annual averages), which correspond to monthly averaged wind stresses up to 0.25 N m^{−2} and annual-averaged wind stress up to approximately 0.1 N m^{−2} (Lee et al. 2013). Subglacial discharge exits at the base of the glacier and is generally hundreds of cubic meters per second in the summer and nearly zero in winter (Straneo and Cenedese 2015; Chu 2014). Areas of elevated mixing within the fjord-to-shelf region are primarily forced by subglacial and ambient melt plumes as they are a dominant mode of mixing for the majority of Greenland’s fjords (Carroll et al. 2017; Gladish et al. 2015; Magorrian and Wells 2016). However, wind-driven mixing in the surface mixed layer (often shallow enough to be ignored for the interior fjord dynamics) and tidal/gravity-overflow mixing near shallow sills (which is outside the scope of this study) are also potentially important contributors.

We note that this simplified fjord–shelf configuration is not intended to fully represent the geometric complexity of Greenland’s fjords, but rather to capture a few geometric features that are representative of a number of these fjords. The lack of floating ice shelves, sea ice, melange, icebergs, canyons, enclosed bays, narrow straits, remotely generated coastal currents, etc., likely play important roles in individual fjord–shelf systems but are not captured in our simple model configuration. We do not anticipate these factors to qualitatively change our findings although they may be separately important as drivers or controls on fjord–shelf exchange (see section 7 for further discussion).

### a. Model configuration

To capture key aspects of the fjord-to-shelf dynamics, we implement a three-layer isopycnal model. With this simple model, we aim to include important elements of fjord–shelf dynamics with minimal complexity and computational cost, allowing us to conduct fully equilibrated (5-yr) simulations over a wide parameter space. The model uses three density layers for simplicity to represent the Atlantic layer, polar layer, and surface waters that are observed in many Greenlandic fjords (e.g., Gladish et al. 2015; Bartholomaus et al. 2016) and a three-layer isopycnal model describes the barotropic and first two baroclinic modes while prohibiting spurious diabatic mixing. By comparison, it has been shown that 75 well-positioned vertical layers are typically necessary to adequately resolve the first two baroclinic modes in *z* coordinates (Stewart et al. 2017). Observed temperature and salinity profiles within fjords often shown three relatively unstratified layers (compared to the shelf waters), which is captured in the thicknesses of our three layers in a discretized and simplified way (Carroll et al. 2016).

In our model configuration, the three layers represent Atlantic Water (AW), which is warmer, saltier, and denser; Polar Water, which is colder, fresher, and lighter; and Surface Water, which is the coldest, freshest, and lightest (see Fig. 2). The fjord overturning circulation is modeled as bottom-layer AW entering the fjord, which is then converted into either the middle-layer Polar Water or the Surface Water layer.

Configuration and geometry of our fjord-to-shelf isopycnal model. Snapshots of middle- and bottom-layer vorticity are displayed on surfaces of interface depth (*η*_{1}, *η*_{2}) for the reference case with *H*_{S} = 100 m and *W*_{fj} = 8 km above a surface of bathymetry. The parameters in blue vary between simulations while the parameters in red are fixed. The eastern glacial boundary is coupled to a dynamic plume model which transforms water masses from denser (*ρ*_{3} = red, Atlantic Water) to lighter water masses (*ρ*_{2} = yellow, Polar Water, and *ρ*_{1} = blue, Surface Water) of discrete densities.

Citation: Journal of Physical Oceanography 51, 4; 10.1175/JPO-D-20-0091.1

Configuration and geometry of our fjord-to-shelf isopycnal model. Snapshots of middle- and bottom-layer vorticity are displayed on surfaces of interface depth (*η*_{1}, *η*_{2}) for the reference case with *H*_{S} = 100 m and *W*_{fj} = 8 km above a surface of bathymetry. The parameters in blue vary between simulations while the parameters in red are fixed. The eastern glacial boundary is coupled to a dynamic plume model which transforms water masses from denser (*ρ*_{3} = red, Atlantic Water) to lighter water masses (*ρ*_{2} = yellow, Polar Water, and *ρ*_{1} = blue, Surface Water) of discrete densities.

Citation: Journal of Physical Oceanography 51, 4; 10.1175/JPO-D-20-0091.1

Configuration and geometry of our fjord-to-shelf isopycnal model. Snapshots of middle- and bottom-layer vorticity are displayed on surfaces of interface depth (*η*_{1}, *η*_{2}) for the reference case with *H*_{S} = 100 m and *W*_{fj} = 8 km above a surface of bathymetry. The parameters in blue vary between simulations while the parameters in red are fixed. The eastern glacial boundary is coupled to a dynamic plume model which transforms water masses from denser (*ρ*_{3} = red, Atlantic Water) to lighter water masses (*ρ*_{2} = yellow, Polar Water, and *ρ*_{1} = blue, Surface Water) of discrete densities.

Citation: Journal of Physical Oceanography 51, 4; 10.1175/JPO-D-20-0091.1

*W*×

*L*×

*H*= 150 km × 150 km × 800 m. The simple model geometry consists of a flat shelf connected to a deep fjord with a Gaussian sill in the

*x*direction with the maximum at

*x*

_{S}= 107.5 km, which is near the fjord mouth located at

*x*

_{f}= 100 km. The bathymetric depth is

*L*

_{S}= 12.5 km and sill height

*H*

_{S}(the amplitude of the sill above the shelf). The shelf has a depth of

*H*

_{Sh}= 400 m, and is connected to a fjord of depth

*H*

_{fj}= 800 m, length

*L*

_{fj}= 50 km, and width

*W*

_{fj}. Note that

*H*

_{S}and

*W*

_{fj}are the only geometric parameters varied between our experiments with ranges shown in Table 1 (see parameter sensitivity discussion in section 3). The shelf is 100 km wide in the across-shelf direction (

*x*) and 150 km long in the along-shelf direction (

*y*) and is periodic in

*y*. The across and along-shelf length scales are fixed and chosen to represent the width of a continental shelf and the choice of a periodic domain represents an average interfjord separation distance (approximately 100–150 km between one fjord and the next in Fig. 1).

Summary of key fjord parameters and test cases for the numerical simulations and their corresponding estimates for Ilulissat Icefjord in West Greenland. All variables are independently varied relative to the reference case in section 3c except fjord length and depth. The estimates of Ilulissat fjord properties are based on data from Gladish et al. (2015) and Beaird et al. (2017).

The lateral side boundaries of the fjord and coastline are represented as vertical walls due to the horizontal resolution limiting realistic coastal slopes to span only 2–3 horizontal grid points (1 km), i.e., a bathymetric steepness of 1/2 near the six fjords in Fig. 1 (Morlighem et al. 2017). At coarse resolution, such underresolved coastal slopes would produce spurious results in the boundary currents that emerge. A high-resolution near-fjord configuration with a smaller shelf was tested with both vertical walls and varying side slope steepness (from 1/4 to 1, not shown) without significant variation in boundary current transport. Using 90° corners at the fjord mouth led to large, spurious sources of vorticity, so we replaced them with quarter-circular rounded corners with radii of 3 km. We also experimented with a continental shelf slope of width 5 km and steepness 1/10 positioned at *x* = 0 (i.e., extending the Fig. 2 domain 20 km offshore to include a shelf and flat deep bathymetry), but found that this did not significantly alter our results.

### b. Model equations

We use the Back of Envelope Ocean Model (BEOM), which is a publicly available code (St-Laurent 2018). BEOM is a hydrostatic shallow-water isopycnal model with a nonlinear free surface that simulates rotating basins and allows for layer outcropping.

*f*plane using shallow-water momentum and continuity equations

*n*= 1, 2, 3. Here,

*u*is the zonal velocity (in the

*x*direction),

*υ*is the meridional velocity (in the

*y*direction), and the top-, middle-, and bottom-layer thicknesses are

*h*

_{1},

*h*

_{2}, and

*h*

_{3}. We parameterize the water mass transformation as

*ϖ*, surface and bottom boundary stresses as

**F**

_{wind}and

**F**

_{fric}, and eddy viscosity as

*υ*

**S**. We use an

*f*-plane approximation with a representative Coriolis parameter of

*f*= 1.31 × 10

^{−4}s

^{−1}corresponding to latitudes in central Greenland.

*A*

_{W}, each layer

*n*is restored to

*τ*

_{h}= 1 day that decreases linearly to zero in the interior edge of the nudging zone. At the eastern boundary, we parameterize the time-evolving plume-driven water mass transformation as

*ϖ*

_{n,p}using a point plume model (Turner 1979) applied to the three-layer density stratification (see appendix A for the details). We also implemented a line plume parameterization of variable width in our model, which exhibits negligible differences in the water mass transformation for small plume source widths (further discussed in appendix A). We assume in our model setup that all of the diabatic forcing occurs at the western and eastern boundary and exclude mixing within the domain due to tides and sill overflows.

**F**

_{wind,n}=

**/(**

*τ**ρ*

_{h}

*h*

_{n}) is imposed in the highest layer

*n*with nonnegligible thickness (

*h*

_{n}> 0.5 m) with

**= (**

*τ**τ*

_{x},

*τ*

_{y}). Bottom friction is parameterized by a quadratic drag

*C*

_{d}= 2.5 × 10

^{−3}and only acts in the lowest layer

*n*with nonnegligible thickness (

*h*

_{n}> 0.5 m). In numerical calculations, we control gridscale energy and enstrophy using a thickness-weighted biharmonic eddy viscosity term

*υ*

**S**

_{n}, for which

*η*

_{0}is the free surface elevation. The reduced gravity at the two interfaces

*η*

_{1}and

*η*

_{2}are

*η*

_{1}=

*η*

_{0}−

*h*

_{1}and

*η*

_{2}=

*η*

_{0}−

*h*

_{1}−

*h*

_{2}. The reference densities [chosen based on Ilulissat conditions from Gladish et al. (2015)] for the three layers are (

*ρ*

_{1},

*ρ*

_{2},

*ρ*

_{3}) = (1025.5, 1026.5, 1027.0) kg m

^{−3}, but varying stratification is also explored in section 3. This choice of stratification corresponds to a reduced gravity at the two interfaces of

*L*

_{d}-wide boundary current, we use a horizontal resolution of

*dl*= 400 m all of the runs discussed (except for the

*dl*= 68 m experiment discussed in section 6). We find that

*L*

_{d}based on

*h*

_{2}and

*h*

_{3}evaluated at the sill maximum (

*x*= 107.5 km) is a useful approximation for the boundary current width due to the sill’s role in establishing the boundary current width. We use a time step of 100 s and simulations are run for 1600 days to reach a statistically steady state, which is measured by the domain-integrated available potential energy. This duration is required to fully spin up the shelf circulation and coastal current, which influences the dynamics within the fjord.

The three-layer setup is more advantageous than a two-layer setup primarily because the plume parameterization in three layers allows a partition of the exiting water masses between the top two layers, which serves as a proxy for exiting plume depth. By comparison, the two-layer setup has no way of specifying the exiting plume depth since all of the bottom-layer inflow must exit as outflow in the top layer. Moreover, this degree of freedom provided by a three-layer setup is critical for the implementation of the plume parameterization since the overturning circulation in a two-layer setup can be determined entirely by the density of the two layers and the rate of subglacial discharge via the Knudsen relations. However, a three-layer setup must take into account the plume density and its level of neutral buoyancy, which provides a more physical control of the plume on the overturning circulation (see appendix A for further discussion). Another implication for using three layers is that the overturning between the bottom two layers is allowed to realistically transition to an overturning between the top two layers, which results in a substantially decreased heat transport since the middle layer has significantly less available heat content than the bottom layer. Although our isopycnal model does not carry a temperature variable, onshore heat transport inferences can be made by assigning realistic potential temperatures to each of the three density classes. In Greenland’s fjords, the lower layer has typical potential temperatures of 2°–4°C, while the middle and top layers are within the range from −1° to 1°C, which is why the bottom-layer inflow is particularly important.

The goal of our choice to specify the boundary conditions (wind stresses, subglacial discharge rate, and open-ocean stratification) to be constant with time is to better understand the fully equilibrated shelf-to-fjord mean circulation. While it is true that this constant forcing does not represent the full reality of Greenland’s fjords, we believe it to be a necessary step before considering the superimposed effects of variability on the system, which is further discussed in section 6.

### c. A reference case

Diagnostics from a reference case simulation with sill height *H*_{S} = 100 m and fjord width *W*_{fj} = 8 km are shown in Fig. 2. Snapshots of the middle- and bottom-layer vorticities are mapped onto the isopycnal interface depths *η*_{1} and *η*_{2}. The reference case parameters are selected based on conditions in Ilullisat Icefjord in West Greenland (Gladish et al. 2015) and are intermediate values for the parameter space explored in the sensitivity experiments in section 3.

In Fig. 3, we present a series of diagnostic fields that capture the dynamics of this reference case including the time-mean zonal transport and isopycnal gradients, instantaneous vorticity fields showing mesoscale eddies on the shelf, and the fjord-focused circulation using time-averaged potential vorticity (PV) and transport streamfunction. Nearly all of the features in this reference case are observed in the series of parameter sensitivity simulations discussed in section 3.

Reference case simulation with *H*_{S} = 100 m and *W*_{fj} = 8 km. (a) Zonal transport decomposed into mean and eddy components based on Eq. (7) for the bottom and middle layer (top-layer zonal transport is negligible and is therefore not shown). (b) Isopycnal interface depths *η*_{1}, *η*_{2} and bathymetry along the midline, *y* = 75 km. (c)–(e) Snapshots of vorticity for each layer (at day 1600). (f)–(h) Zoomed-in PV (in color) and transport streamfunction for each layer with 30-mSv contours in the top and middle layers and 20-mSv contours in the bottom layer (dotted line contours are negative values). The fields in the top and bottom rows are time-averaged over days 1300–1600.

Citation: Journal of Physical Oceanography 51, 4; 10.1175/JPO-D-20-0091.1

Reference case simulation with *H*_{S} = 100 m and *W*_{fj} = 8 km. (a) Zonal transport decomposed into mean and eddy components based on Eq. (7) for the bottom and middle layer (top-layer zonal transport is negligible and is therefore not shown). (b) Isopycnal interface depths *η*_{1}, *η*_{2} and bathymetry along the midline, *y* = 75 km. (c)–(e) Snapshots of vorticity for each layer (at day 1600). (f)–(h) Zoomed-in PV (in color) and transport streamfunction for each layer with 30-mSv contours in the top and middle layers and 20-mSv contours in the bottom layer (dotted line contours are negative values). The fields in the top and bottom rows are time-averaged over days 1300–1600.

Citation: Journal of Physical Oceanography 51, 4; 10.1175/JPO-D-20-0091.1

Reference case simulation with *H*_{S} = 100 m and *W*_{fj} = 8 km. (a) Zonal transport decomposed into mean and eddy components based on Eq. (7) for the bottom and middle layer (top-layer zonal transport is negligible and is therefore not shown). (b) Isopycnal interface depths *η*_{1}, *η*_{2} and bathymetry along the midline, *y* = 75 km. (c)–(e) Snapshots of vorticity for each layer (at day 1600). (f)–(h) Zoomed-in PV (in color) and transport streamfunction for each layer with 30-mSv contours in the top and middle layers and 20-mSv contours in the bottom layer (dotted line contours are negative values). The fields in the top and bottom rows are time-averaged over days 1300–1600.

Citation: Journal of Physical Oceanography 51, 4; 10.1175/JPO-D-20-0091.1

*y*= 75 km) isopycnal interface depths highlight the across-shelf pressure gradient in the middle and bottom layer (particularly those near the fjord mouth), which drives a baroclinic coastal current that is weaker/southward in the bottom layer and stronger/northward in the middle layer.

The vorticity snapshots for each layer in Figs. 3c–e show eddies shedding from the fjord mouth and coastal current primarily via baroclinic instability, which depends on the zonal isopycnal/pressure gradient. This is diagnosed using the same analysis as Zhao et al. (2019), which in this case shows the eddy energy production is dominated by conversion from potential energy rather than kinetic energy (not shown). These eddies are the dominant mode of transport across the *y*-periodic shelf. However, the peak in vorticity is located in the middle- and bottom-layer steady recirculation within the fjord and is connected to the overturning circulation via the bottom-layer boundary current and middle-layer coastal current.

*ψ*= 0 at the northern fjord wall and integrate meridionally across the fjord to determine

*ψ*throughout the fjord, and then integrate zonally across the shelf (i.e., using a path of integration that avoids diabatic regions for the nondiabatic interior).

In the bottom-layer streamfunction, the southward coastal current enters the fjord via a boundary current, which initially flows retrograde (with the boundary to the left of the flow) along the northern boundary of the fjord. The flow crosses the channel on the eastern side of the sill maximum since topographic beta, *β*_{topo} = −*f*∂_{y}(*z*_{B})/*h*_{3}, changes sign due to a reversal of the bathymetric slope. In this case, the flow crossing occurs at *x* = 114 km, which is diagnosed using the 20-mSv contour in Fig. 3h. The sill establishes a PV barrier in the bottom layer, which appears as a red patch in the PV field. The 20-mSv (1 Sv ≡ 10^{6} m^{3} s^{−1}) transport streamfunction illustrates the transport pathway approximately following PV isolines, which serve as barriers that guide the flow. The boundary current then feeds the gyre-like recirculation in the deeper fjord interior, where it is converted into the middle-layer water mass by the diabatic plume-driven water mass transformation. The flow in the middle layer recirculates with a small fraction flowing back out toward the open ocean via eddy transport across the shelf.

## 3. Parameter dependencies

The reference case motivates us to seek an understanding for the parameter dependencies of the two bulk fjord circulation properties: the overturning circulation and horizontal recirculation. The overturning and recirculation control parameters can be classified into geometric, forcing-related, and stratification, and our goal is to test the sensitivity of a few simple parameters that to first order capture the parameter variations among Greenland’s fjords.

Although various complex geometric controls can exist (bends in the fjord, nonuniform fjord width, shelf troughs, multiple sills, etc.), we anticipate that the features of first-order importance to the overturning are sill height *H*_{S} and fjord width *W*_{fj}, which act to horizontally and vertically constrict the exchange flow at the fjord mouth. Forcing parameters that are of first-order importance to the fjord–shelf exchange are wind direction and strength, subglacial discharge strength, and open ocean boundary conditions, which we quantify as AW depth

Therefore, we choose to vary the following six parameters: sill height, fjord width, wind direction/strength, subglacial discharge strength, AW depth, and stratification. The key parameters and test cases are listed in Table 1, with parameter variations selected to span the range of existing glacial fjord measurements.

### a. Summary of dependencies

Figures 4a–f shows the sensitivity of the overturning circulation and its root-mean-square deviation (RMSD) to each of the six parameters. Relative to the reference case, the overturning circulation varies most significantly with sill height, AW depth, winds, and subglacial discharge over realistic parameter variations.

(a)–(f) Time-averaged (days 1300–1600) mean transport for each layer *n* based on Eq. (7) diagnosed at the sill maximum (*x* = 107.5 km) and its root-mean-square deviation (color shading denotes positive and negative values calculated from the time series for each respective case). Transports are positive toward the glacier. The purple dotted line shows the parameter choice of the reference case (from Figs. 2 and 3). The wind magnitude is (N1, N2, N3) = (0.015, 0.03, 0.1) N m^{−2} and similarly for the other wind directions in (c).

Citation: Journal of Physical Oceanography 51, 4; 10.1175/JPO-D-20-0091.1

(a)–(f) Time-averaged (days 1300–1600) mean transport for each layer *n* based on Eq. (7) diagnosed at the sill maximum (*x* = 107.5 km) and its root-mean-square deviation (color shading denotes positive and negative values calculated from the time series for each respective case). Transports are positive toward the glacier. The purple dotted line shows the parameter choice of the reference case (from Figs. 2 and 3). The wind magnitude is (N1, N2, N3) = (0.015, 0.03, 0.1) N m^{−2} and similarly for the other wind directions in (c).

Citation: Journal of Physical Oceanography 51, 4; 10.1175/JPO-D-20-0091.1

(a)–(f) Time-averaged (days 1300–1600) mean transport for each layer *n* based on Eq. (7) diagnosed at the sill maximum (*x* = 107.5 km) and its root-mean-square deviation (color shading denotes positive and negative values calculated from the time series for each respective case). Transports are positive toward the glacier. The purple dotted line shows the parameter choice of the reference case (from Figs. 2 and 3). The wind magnitude is (N1, N2, N3) = (0.015, 0.03, 0.1) N m^{−2} and similarly for the other wind directions in (c).

Citation: Journal of Physical Oceanography 51, 4; 10.1175/JPO-D-20-0091.1

For tall enough sills (*H*_{S} above 150 m), deeper AW (*H*_{S}, deeper AW, and small *Q*_{0} can each lead to the complete shutoff of warm AW (bottom layer) transport toward the fjord, denoted as *Q*_{3}, it is also plausible that a weak enough stratification between the bottom and middle layers (~0.1 kg m^{−3} or less) or a small enough fjord width (1 km or less) may also lead to weakened heat transport into the fjord.

Similarly to Fig. 4 for overturning sensitivity, Figs. 5a–f shows the dependency of horizontal recirculation on the same six parameters. The recirculation in the bottom/AW layer, which is the most important due to its lateral heat transport to the glacier face, varies primarily with sill height, fjord width, subglacial discharge, and AW depth. If we compare the recirculation sensitivity in each layer with the overturning circulation sensitivity in Fig. 4, we see approximately the same trends for sill height, winds, *Q*_{0} = 500 m^{3} s^{−1} while the overturning continues to linearly increase in Fig. 4d. The middle-layer recirculation approximately opposes the recirculation in the bottom layer, except for cases of wide fjords or nonzero top-layer recirculation (especially in tall sill cases).

(a)–(f) Time-averaged (days 1300–1600) recirculation strength *ψ*_{r} [diagnosed as the streamfunction extremum using Eq. (9a)] within the fjord in each layer, where positive values correspond to cyclonic circulation. Recirculation root-mean-square deviation is shown with color shading (calculated from the time series for each respective case). The purple dotted line shows the parameter choice of the reference case (from Figs. 2 and 3). The wind magnitude is (N1, N2, N3) = (0.015, 0.03, 0.1) N m^{−2} and similarly for the other wind directions in (c).

Citation: Journal of Physical Oceanography 51, 4; 10.1175/JPO-D-20-0091.1

(a)–(f) Time-averaged (days 1300–1600) recirculation strength *ψ*_{r} [diagnosed as the streamfunction extremum using Eq. (9a)] within the fjord in each layer, where positive values correspond to cyclonic circulation. Recirculation root-mean-square deviation is shown with color shading (calculated from the time series for each respective case). The purple dotted line shows the parameter choice of the reference case (from Figs. 2 and 3). The wind magnitude is (N1, N2, N3) = (0.015, 0.03, 0.1) N m^{−2} and similarly for the other wind directions in (c).

Citation: Journal of Physical Oceanography 51, 4; 10.1175/JPO-D-20-0091.1

(a)–(f) Time-averaged (days 1300–1600) recirculation strength *ψ*_{r} [diagnosed as the streamfunction extremum using Eq. (9a)] within the fjord in each layer, where positive values correspond to cyclonic circulation. Recirculation root-mean-square deviation is shown with color shading (calculated from the time series for each respective case). The purple dotted line shows the parameter choice of the reference case (from Figs. 2 and 3). The wind magnitude is (N1, N2, N3) = (0.015, 0.03, 0.1) N m^{−2} and similarly for the other wind directions in (c).

Citation: Journal of Physical Oceanography 51, 4; 10.1175/JPO-D-20-0091.1

We now discuss the parameter sensitivity of the fjord dynamics in greater detail and describe the flow behavior in response to these key parameter variations.

### b. Sill height and fjord width

Figures 6a–d show the time-averaged isopycnal depth variations along the *y* midline and Figs. 6e–h shows the bottom-layer PV and transport streamfunction for four cases of varying sill height. As discussed in the reference case (Fig. 3), a common feature is the coastal current, which flows southward in the bottom layer and enters the fjord as a narrow *L*_{d}-wide boundary current. The boundary current flows along and across the fjord and subsequently feeds a cyclonic recirculation gyre in the fjord interior.

(a)–(d) Side profile depths of *η*_{1}, *η*_{2}, and bathymetry along midline (*y* = 75 km); and (e)–(h) zoomed-in bottom-layer PV (in color) and transport streamfunction using 20-mSv contours for four cases of varying sill height. The dotted line contours show negative values and additional pink contours in (g) and (h) highlight the fjord–shelf connectivity. All fields are time-averaged over days 1300–1600.

Citation: Journal of Physical Oceanography 51, 4; 10.1175/JPO-D-20-0091.1

(a)–(d) Side profile depths of *η*_{1}, *η*_{2}, and bathymetry along midline (*y* = 75 km); and (e)–(h) zoomed-in bottom-layer PV (in color) and transport streamfunction using 20-mSv contours for four cases of varying sill height. The dotted line contours show negative values and additional pink contours in (g) and (h) highlight the fjord–shelf connectivity. All fields are time-averaged over days 1300–1600.

Citation: Journal of Physical Oceanography 51, 4; 10.1175/JPO-D-20-0091.1

(a)–(d) Side profile depths of *η*_{1}, *η*_{2}, and bathymetry along midline (*y* = 75 km); and (e)–(h) zoomed-in bottom-layer PV (in color) and transport streamfunction using 20-mSv contours for four cases of varying sill height. The dotted line contours show negative values and additional pink contours in (g) and (h) highlight the fjord–shelf connectivity. All fields are time-averaged over days 1300–1600.

Citation: Journal of Physical Oceanography 51, 4; 10.1175/JPO-D-20-0091.1

The interface depth in Figs. 6a–d shows a transition from a 100-m zonal *η*_{2} difference (*H*_{S} = 200 m case in Fig. 6d, with the bottom layer nearly grounded on the sill bathymetry. The bottom-layer thickness inside the fjord decreases by approximately a factor of 2 between the *H*_{S} = 0 and the *H*_{S} = 200 m cases.

In Figs. 6e–h, the time-averaged PV and transport streamfunction also show a noticeable change in the fjord–shelf connectivity via the coastal and boundary current for taller sills. The sill establishes a PV barrier in the bottom layer, which appears as a red patch in the PV for cases with sills. In the *H*_{S} = 0 case, the PV barrier is weak and the streamfunction shows that the flow from the coastal current crosses the fjord gradually to join the boundary current and recirculation gyre. For taller sills, the boundary current enters as a narrower boundary current with weaker transport (outlined in a pink contour in Figs. 6g and 6h). In these cases, the isopycnal structure is suggestive of a hydraulically controlled exchange flow (Pratt and Whitehead 2007), discussed further in section 4. An anticyclonic recirculation develops on the downstream side of tall sills and the bottom-layer recirculation weakens for the tallest sills due to decreased overturning strength and the bottom friction acting on a decreased bottom-layer thickness (further discussed in section 5).

Figures 7a–h shows plots of isopycnal depth, bottom-layer PV, and transport streamfunction for varying fjord width. In Figs. 7a–d, interfacial depths of *η*_{2} along the *y* midline show a ~10% increase in zonal *η*_{2} differences between *W*_{fj} = 4 and 24 km, which is consistent with the minimal increase in overturning circulation for wider fjords shown in Fig. 4b. However, the bottom-layer recirculation strengthens by ~80% between the 4- and 8-km case and ~60% between the 8- and 24-km cases. Depressions in the isopycnal depths due to the strength of the opposing gyre recirculation in the bottom and middle layers are more clearly observed for the wider fjords, e.g., Figs. 7c and 7d. In these cases, a weak recirculation of ~20–40 mSv also develops over the sill. Regardless of fjord width, we see the flow consistently entering the fjord through an *L*_{d}-wide current in the northern boundary, which appears visually in the PV field as a small trough in the near-sill PV barrier in Figs. 7e–h. Although the narrow fjord widths cases are limited by horizontal resolution, fjord-only test cases (not shown) suggest a reduction in overturning and larger zonal isopycnal gradients for fjords narrower than *L*_{d}.

As in Fig. 6, but for four cases of varying fjord width.

Citation: Journal of Physical Oceanography 51, 4; 10.1175/JPO-D-20-0091.1

As in Fig. 6, but for four cases of varying fjord width.

Citation: Journal of Physical Oceanography 51, 4; 10.1175/JPO-D-20-0091.1

As in Fig. 6, but for four cases of varying fjord width.

Citation: Journal of Physical Oceanography 51, 4; 10.1175/JPO-D-20-0091.1

### c. Wind strength and direction

Wind stress magnitudes of *τ* = (0.015, 0.03, 0.1) N m^{−2} were tested (corresponding approximately to a range of 3.5–9 m s^{−1} wind velocities), which are fairly representative of the annual average winds along the Greenland coast and not of shorter-term extremes (Lee et al. 2013). The resulting *y*-midline depths of *η*_{1} and *η*_{2} are shown in Figs. 8a–d and time-averaged bottom-layer PV and transport streamfunction in Figs. 8e–h for four cases of varying wind direction and wind stress magnitude *τ* = 0.03 N m^{−2}.

As in Fig. 6, for four cases of varying wind direction with constant magnitude 0.03 N m^{−2}. An additional pink contour in panel e highlights the fjord–shelf connectivity for the northward wind case.

Citation: Journal of Physical Oceanography 51, 4; 10.1175/JPO-D-20-0091.1

As in Fig. 6, for four cases of varying wind direction with constant magnitude 0.03 N m^{−2}. An additional pink contour in panel e highlights the fjord–shelf connectivity for the northward wind case.

Citation: Journal of Physical Oceanography 51, 4; 10.1175/JPO-D-20-0091.1

As in Fig. 6, for four cases of varying wind direction with constant magnitude 0.03 N m^{−2}. An additional pink contour in panel e highlights the fjord–shelf connectivity for the northward wind case.

Citation: Journal of Physical Oceanography 51, 4; 10.1175/JPO-D-20-0091.1

The eastward and westward wind cases did not change the mean state appreciably, but the northward and southward cases visibly tilt both isopycnals (through Ekman transport) in Figs. 8a and 8b leading to a zonal isopycnal depth change of Δ*η*_{2} = −190 m and Δ*η*_{1} = −150 m for northward winds, and Δ*η*_{2} = 50 m and Δ*η*_{1} = 50 m for southward winds. For the northward wind case shown, 80% of the zonal gradient in *η*_{2} occurs on the shelf and the bottom layer is ~200 m thinner in the fjord interior than the eastward/westward wind cases.

The streamfunctions in Figs. 8e and 8f show an inflow that is significantly weaker in the northward wind case and slightly stronger in the southward wind case compared to the eastward and westward wind cases (Figs. 8g,h). This is influenced by the bottom-layer coastal current supplying the fjord overturning, which changes from a weak southward transport of ~40 mSv in the reference case (in Fig. 3) to a ~200-mSv northward transport for northward winds, ~500-mSv southward transport for the southward wind case, ~40 mSv (no change) for the eastward wind case, and ~20-mSv northward transport for the westward wind case. Due to the thin bottom-layer thickness above the sill for the northward wind case, there is a strong PV barrier (similar to the tall sill cases) for the northward wind case and a reduced barrier for the southward winds.

The sensitivity of the fjord dynamics to northward winds via differences in the isopycnal depths, coastal current strength, and meridional profile of the inflow lead to a 45% reduction in overturning and 40% reduction in recirculation for the intermediate wind case (0.03 N m^{−2}) and a complete shutoff of both the overturning and recirculation for the highest northward wind case (0.1 N m^{−2}). These results show northward winds are the most important in reducing the overturning and recirculation and is likely to be even more significant for fjords with weaker plume-driven overturning where the Ekman transport contribution is comparatively larger.

Our wind tests use time-constant winds that are uniform over the whole domain and are intended to capture the influence of steady winds on shelf circulation (upwelling and downwelling) and its influence on mean fjord circulation. We use annual-mean winds since the shelf circulation and across-shelf transport requires years to spin up, while seasonal winds may likely lead to strong, but transient controls on fjord–shelf exchange. In this setup, a strong northward wind (0.1 N m^{−2}) was sufficient to completely shut off the warm AW transport due to a vanishing bottom layer near the fjord mouth with strong eddies dominating the shelf, which in practice may be dampened by bathymetric features on the shelf.

Although it is likely that time-varying winds are equally or more important than the annual-mean winds, we have only included the annual-mean wind effects as a starting point for assessing the role of winds on fjord–shelf exchange in this study. A more realistic time-varying wind forcing including shorter time scale extreme events are likely to excite coastally trapped waves and other modes of variability as well as nonequilibrium rapid flushing events (e.g., Spall et al. 2017), which are not considered in this study and require further exploration.

### d. Subglacial discharge, AW depth, and stratification

Of the parameters tested, subglacial discharge has the most predictable effect on overturning strength (as shown in Fig. 4d), which increases linearly with discharge along with a moderate increase in transport RMSD. This is unsurprising given the theory of diabatic plume forcing in appendix A implemented in the eastern boundary condition. Increasing the overturning circulation via subglacial discharge from 0 to 100 mSv strengthens the boundary current from ~0 to 100 mSv, coastal current from ~0 to 300 mSv, and recirculation in the middle and bottom layers from ~0 to 300 mSv (which saturates near *Q*_{0} = 500 m^{3} s^{−1}). The strength of the recirculation and overturning is likely dependent on the grounding line depth (level of subglacial discharge), which is a parameter we do not vary.

Varying the AW depth at the western boundary *H*_{S} from 0 to 250 m (see Figs. 4a,e). This is unsurprising since the nondimensionalized sill height,

Increasing stratification (i.e., increasing *ρ*_{3} − *ρ*_{2} from 0.3 to 0.7 kg m^{−3}) has the effect of decreasing the overturning circulation from 30 to 17 mSv (shown in Fig. 4f), which, similarly to the other parameters, led to increases in the recirculation and coastal current albeit with much weaker trends (10% increase over the range of stratification). The effect of stratification on fjord dynamics in the context of plume theory is further discussed in appendix A.

## 4. Overturning circulation

Following the results from the sensitivity studies, we develop theories to predict the overturning circulation as a function of the parameters explored in section 3. For simplicity, we focus on the AW inflow in the bottom layer, *Q*_{3}, since it is nearly proportional to the heat flux toward the glacial face and the most important transport for melt rate estimates (e.g., Inall et al. 2014).

We present and assess theories for the transport across each of the three regions: the continental shelf, the fjord mouth sill, and the fjord head. We first discuss the continental shelf region with an across-shelf transport *Q*_{shelf}, primarily driven by eddies and Ekman transport. We then discuss the fjord mouth sill region with a sill-overflow transport *Q*_{fjord}, which admits both geostrophic and hydraulically controlled transport predictions (based on the theory from Zhao et al. 2019). Following this, we discuss the fjord head region with a diabatic water mass transformation *Q*_{plume}, driven by plume entrainment at the glacier face. This diabatic water mass transformation in the steady state is balanced by the diabatic transport at the western boundary and due to the restoring, this transport must match the other transports and is not included in the theory.

In the following subsections, we use diagnosed bottom-layer thicknesses at the fjord mouth

(a) Schematic for the overturning circulation showing the three components of the shelf-to-glacier-face overturning. A comparison between the strength of the simulated overturning circulation diagnosed in the model and the predictions for onshore transport from (b) *Q*_{shelf}, the sum of eddy and Ekman transports across the shelf given by Eqs. (10) and (13); (c) *Q*_{fjord}, the minimum of the geostrophic and hydraulically controlled transports given by Eqs. (14), (15), and (17); and (d) *Q*_{plume}, the diabatic water mass transformation given by Eq. (18). Increasing marker sizes correspond to increasing values of each parameter with letter labels for varying wind direction.

Citation: Journal of Physical Oceanography 51, 4; 10.1175/JPO-D-20-0091.1

(a) Schematic for the overturning circulation showing the three components of the shelf-to-glacier-face overturning. A comparison between the strength of the simulated overturning circulation diagnosed in the model and the predictions for onshore transport from (b) *Q*_{shelf}, the sum of eddy and Ekman transports across the shelf given by Eqs. (10) and (13); (c) *Q*_{fjord}, the minimum of the geostrophic and hydraulically controlled transports given by Eqs. (14), (15), and (17); and (d) *Q*_{plume}, the diabatic water mass transformation given by Eq. (18). Increasing marker sizes correspond to increasing values of each parameter with letter labels for varying wind direction.

Citation: Journal of Physical Oceanography 51, 4; 10.1175/JPO-D-20-0091.1

(a) Schematic for the overturning circulation showing the three components of the shelf-to-glacier-face overturning. A comparison between the strength of the simulated overturning circulation diagnosed in the model and the predictions for onshore transport from (b) *Q*_{shelf}, the sum of eddy and Ekman transports across the shelf given by Eqs. (10) and (13); (c) *Q*_{fjord}, the minimum of the geostrophic and hydraulically controlled transports given by Eqs. (14), (15), and (17); and (d) *Q*_{plume}, the diabatic water mass transformation given by Eq. (18). Increasing marker sizes correspond to increasing values of each parameter with letter labels for varying wind direction.

Citation: Journal of Physical Oceanography 51, 4; 10.1175/JPO-D-20-0091.1

### a. Across-shelf transport

The bottom-layer across-shelf transport *Q*_{shelf} is the sum of both eddy and mean contributions. We first discuss the eddy transport in the absence of winds and then discuss the mean Ekman transport.

*W*= 150 km is the meridional domain size,

*L*

_{Sh}= 100 km is the zonal shelf length. There are many ways of specifying the eddy diffusivity

*κ*(e.g., Gent and McWilliams 1990; Visbeck et al. 1997; Gent 2011). In the interest of simplicity, we use an empirically selected constant

*κ*= 234 m

^{2}s

^{−1}since this yields a good agreement with the across-shelf transport.

*L*= 150 km is the meridional domain length, and

*τ*

_{y}is the northward wind stress.

For the scenario where offshore Ekman return flow in the bottom layer exceeds the onshore eddy transport in the bottom layer, the bottom-layer thickness vanishes at the fjord mouth, which results in a bottom-layer transport, *Q*_{shelf} = 0, where the Ekman return flow transitions from the bottom layer to the intermediate layer such that *H*_{S} = 100 m, the theoretical prediction for a shutoff of AW access is achieved by a northward wind stress *τ*_{y} = 0.05 N m^{−2}; for the case of no sill, this is achieved by a northward wind stress *τ*_{y} = 0.08 N m^{−2}.

In Fig. 9b, we plot the overturning circulation strength diagnosed from the simulations compared to our theory for the bottom-layer across-shelf transport *Q*_{shelf} = max(*Q*_{eddy} − *Q*_{Ek}, 0) using Eqs. (10) and (13). This theory predicts the across-shelf transport with a coefficient of determination of *r*^{2} = 0.88.

### b. Sill-overflow transport

The sill-overflow transport into the fjord is driven by the zonal isopycnal gradients in the AW depth *η*_{2} outside the fjord relative to inside, which establishes a zonal pressure gradient along the fjord. This pressure gradient drives a (meridional) geostrophic flow within the fjord for smaller sill-overflow velocities, and becomes hydraulically controlled for larger velocities (due to either taller sills or other parameters).

#### 1) Geostrophic transport

The across-sill (defined here as the zonal direction) geostrophic sill-overflow transport can be estimated based on the along-sill (meridional) geostrophic transport. This is based on the assumption that boundary currents in the fjord interior establish a zonal/along-fjord pressure head that is similar to the meridional/across-fjord pressure head within the fjord (Zhao et al. 2019). This is suggested in the bottom-layer streamfunction from Fig. 3h, which shows a boundary current entering the fjord and flowing coherently across and along the sill. The pressure head across the boundary current in the along-sill and across-sill directions are thus similar values and representative of the geostrophic transport into the fjord.

*n*(see Zhao et al. 2019 for further details).

#### 2) Hydraulically controlled transport

*G*= 1, which may be defined as (Pratt and Whitehead 2007)

*g*′ defined in Eqs. (5a) and (5b) and

*W*

_{BC}= min(

*L*

_{d},

*W*

_{fj}), which is supported by our simulation results. We find that applying Eq. (17) is valid if Fr

_{3}dominates the Froude number in Eq. (16a). For tall enough sills, hydraulic control can occur in the top and middle layers (not shown), but this does not influence

*Q*

_{3}.

In Fig. 9c, we plot the overturning circulation strength diagnosed from the simulations versus our theory for the sill-overflow transport *Q*_{fjord} as the minimum of the geostrophic and hydraulic transport [using Eqs. (14), (15), and (17)]. This theory predicts the sill-overflow transport for each of the parameter variations with a coefficient of determination of *r*^{2} = 0.81.

Although hydraulic control has been applied to many sill overflows in the open ocean (Pratt and Whitehead 2007), hydrographic measurements and numerical simulations support the existence of hydraulically controlled flow within the Pine Island Glacier ice shelf cavity (Zhao et al. 2019; De Rydt et al. 2014), and outside the 79 North Glacier ice tongue cavity (Lindeman et al. 2020; Schaffer et al. 2020).

### c. Diabatic water mass transformation

*ϵ*= 1.3 (Linden 2000). The buoyancy flux

*B*=

*g*′

*Q*varies with depth, but is constant in the uniform density bottom layer

In Fig. 9d, we plot the overturning circulation strength diagnosed from the simulations compared to our theory for *Q*_{plume} in Eq. (18). This theory predicts the diabatic water mass transformation for each of the parameter variations with a coefficient of determination of *r*^{2} = 0.95.

### d. Piecing together the overturning circulation

*Q*

_{3}without knowledge of the zonal isopycnal gradients in the continental shelf and fjord mouth sill regions. The bottom-layer transport across the shelf-to-glacier-face domain can be summarized as

*Q*

_{plume}=

*Q*

_{shelf}assuming that the bottom-layer transport does not vanish, we can express

*Q*

_{hyd}=

*Q*

_{shelf}separately [Eq. (19)],

*Q*

_{hyd}) overturning. Unfortunately, these relationships do not lend themselves easily to closed form solutions.

*Q*

_{fjord}corresponds to the maximum

*Q*

_{eddy}, and thus

*Q*

_{shelf}, monotonically decreasing with increasing

*r*

^{2}= 0.86. In Fig. 10b, the simulation-diagnosed

*r*

^{2}= 0.84.

A comparison between (a) the simulated *x* = 100 km, as labeled in Fig. 9a) and the predicted *x* = 150 km, as labeled in Fig. 9a) and the predicted

Citation: Journal of Physical Oceanography 51, 4; 10.1175/JPO-D-20-0091.1

A comparison between (a) the simulated *x* = 100 km, as labeled in Fig. 9a) and the predicted *x* = 150 km, as labeled in Fig. 9a) and the predicted

Citation: Journal of Physical Oceanography 51, 4; 10.1175/JPO-D-20-0091.1

A comparison between (a) the simulated *x* = 100 km, as labeled in Fig. 9a) and the predicted *x* = 150 km, as labeled in Fig. 9a) and the predicted

Citation: Journal of Physical Oceanography 51, 4; 10.1175/JPO-D-20-0091.1

*Q*

_{plume}predicts the warm AW inflow as an explicit function of the input parameters in section 3,

In Fig. 11, the simulation-diagnosed AW inflow is shown versus the solution to Eq. (23). Even though predictions in each of the three individual regions are accurate separately (as shown in Figs. 9b–d), this comparison demonstrates that the overall prediction for the entire shelf-to-glacier-face theory predicts the AW inflow transport with a coefficient of determination *r*^{2} = 0.89 and may be calculated a priori without knowledge of

A comparison between the strength of the simulation inflow of Atlantic Water diagnosed in the model to the predicted inflow (equivalent to bottom-layer overturning circulation *Q*_{3}) calculated from Eq. (23). Increasing marker sizes correspond to increasing values of each parameter with letter labels for varying wind direction.

Citation: Journal of Physical Oceanography 51, 4; 10.1175/JPO-D-20-0091.1

A comparison between the strength of the simulation inflow of Atlantic Water diagnosed in the model to the predicted inflow (equivalent to bottom-layer overturning circulation *Q*_{3}) calculated from Eq. (23). Increasing marker sizes correspond to increasing values of each parameter with letter labels for varying wind direction.

Citation: Journal of Physical Oceanography 51, 4; 10.1175/JPO-D-20-0091.1

A comparison between the strength of the simulation inflow of Atlantic Water diagnosed in the model to the predicted inflow (equivalent to bottom-layer overturning circulation *Q*_{3}) calculated from Eq. (23). Increasing marker sizes correspond to increasing values of each parameter with letter labels for varying wind direction.

Citation: Journal of Physical Oceanography 51, 4; 10.1175/JPO-D-20-0091.1

## 5. Recirculation and vorticity balance

Although the overturning circulation is a critical component of the renewal of fjords and has received more attention in existing literature, the horizontal recirculation may play an equally important role in fjord dynamics and glacial melt rates. Specifically, recent work suggests that the near-glacier horizontal velocity, which owes its magnitude to the horizontal recirculation within the fjord, plays an important role in driving ambient front-wide glacial melt and may be comparable to the subglacial discharge-driven melt (Slater et al. 2018; Jackson et al. 2020). The importance of this contribution to melt rate is further discussed in appendix B.

*h*, and take the curl of the result to express the vorticity budget within each layer as

We can simplify this relationship with a scaling argument for Eq. (25) in terms of the recirculation strength *ψ*_{r} and bottom-layer transport *Q*_{3}. The time-average of Eq. (2b) implies that *Q*_{3} equals the area integral of *fQ*_{3}.

*∂A*and

*s*is the corresponding along-contour coordinate over the gyre recirculation region

*A*, and

*C*

_{A}is the circumference of the region

*A*. Here, the mean velocity acted on by bottom friction scales as

*ψ*

_{r}/(

*L*

_{r}

*H*

^{E}) for bottom friction concentrated in a boundary layer width

*L*

_{r}and near-glacier bottom-layer thickness

*H*

^{E}.

*L*

_{r}falls empirically between the boundary current width and the fjord half-width,

*L*

_{r}~ (

*L*

_{d}+

*W*

_{fj}/2)/2. For narrow fjords

*W*

_{fj}<

*L*

_{d}, this empirical relationship fails and the recirculation boundary width likely fills the entire fjord half-width,

*L*

_{r}~

*W*

_{fj}/2, which is untested due to resolution limitations. Therefore, for fjords of width

*L*

_{d}or larger, the scaling for recirculation strength is

*r*

^{2}= 0.87. Additionally, if we assume a constant vorticity in the recirculation gyre, the maximum velocity

*υ*

_{max}is approximately

A comparison between the strength of the simulation recirculation diagnosed in the model to the prediction for recirculation based on Eq. (28). Increasing marker sizes correspond to increasing values of each parameter with letter labels for simulations of varying wind direction.

Citation: Journal of Physical Oceanography 51, 4; 10.1175/JPO-D-20-0091.1

A comparison between the strength of the simulation recirculation diagnosed in the model to the prediction for recirculation based on Eq. (28). Increasing marker sizes correspond to increasing values of each parameter with letter labels for simulations of varying wind direction.

Citation: Journal of Physical Oceanography 51, 4; 10.1175/JPO-D-20-0091.1

A comparison between the strength of the simulation recirculation diagnosed in the model to the prediction for recirculation based on Eq. (28). Increasing marker sizes correspond to increasing values of each parameter with letter labels for simulations of varying wind direction.

Citation: Journal of Physical Oceanography 51, 4; 10.1175/JPO-D-20-0091.1

## 6. Roles of fjord geometry and variability

The simulation results in section 3 and overturning and recirculation theory in sections 4 and 5 aim to capture many factors controlling fjord circulation. However, there are additional fjord circulation characteristics and phenomena that are potentially also important and deserve further investigation. First, we present an expanded discussion on the role of vertical and horizontal hydraulic control in fjord circulation and as a driver of intrafjord variability. Following this, we diagnose the existence and role of low-frequency variability within the fjord and coastal currents in our simulations and subsequently, its high-frequency counterpart including submesoscale variability.

### a. Transition to hydraulic control

In section 4b, we applied simple theories for the transport in geostrophic and hydraulically controlled flows. Although these simple theories fit our simulation results, the onset of hydraulic control in a complex fjord-to-shelf geometry (with both horizontal and vertical constrictions) in the presence of variability is not adequately addressed in the hydraulic-control theory literature and requires further discussion.

In Fig. 13, we highlight the transition from the geostrophic to the hydraulically controlled regimes in our simulation results. We compare the diagnosed nondimensionalized transport (*x–y*-plane maps of the composite Froude number *G* for three cases of varying sill height, with the hydraulically controlled case exhibiting critical values of *G* ≈ 1.

Critical transport prediction using rotating one-layer theory from Eq. (17) and simulation results both nondimensionalized by the geostrophic transport for varying nondimensionalized sill height. The solid lines show where the geostrophic and hydraulic-control theories set the bound on transport while the dashed lines do not [in accordance with Eq. (14)], which shows a transition to hydraulic control theory for *G* over a zoomed-in domain (*x* and *y* axes in km) centered on the fjord. Increasing marker sizes correspond to increasing values of each parameter with letter labels for simulations of varying wind direction. Experiments where *G* exceeds 0.8 at any location within the domain are marked with an “x.”

Citation: Journal of Physical Oceanography 51, 4; 10.1175/JPO-D-20-0091.1

Critical transport prediction using rotating one-layer theory from Eq. (17) and simulation results both nondimensionalized by the geostrophic transport for varying nondimensionalized sill height. The solid lines show where the geostrophic and hydraulic-control theories set the bound on transport while the dashed lines do not [in accordance with Eq. (14)], which shows a transition to hydraulic control theory for *G* over a zoomed-in domain (*x* and *y* axes in km) centered on the fjord. Increasing marker sizes correspond to increasing values of each parameter with letter labels for simulations of varying wind direction. Experiments where *G* exceeds 0.8 at any location within the domain are marked with an “x.”

Citation: Journal of Physical Oceanography 51, 4; 10.1175/JPO-D-20-0091.1

Critical transport prediction using rotating one-layer theory from Eq. (17) and simulation results both nondimensionalized by the geostrophic transport for varying nondimensionalized sill height. The solid lines show where the geostrophic and hydraulic-control theories set the bound on transport while the dashed lines do not [in accordance with Eq. (14)], which shows a transition to hydraulic control theory for *G* over a zoomed-in domain (*x* and *y* axes in km) centered on the fjord. Increasing marker sizes correspond to increasing values of each parameter with letter labels for simulations of varying wind direction. Experiments where *G* exceeds 0.8 at any location within the domain are marked with an “x.”

Citation: Journal of Physical Oceanography 51, 4; 10.1175/JPO-D-20-0091.1

In the sensitivity experiments (section 3), we varied each parameter individually relative to the reference case, but further regimes are possible when we covary parameters. Figure 14 shows the nondimensionalized mean bottom-layer transport (*W*_{fj}/*L*_{d}) for one such combination of parameters: covarying sill height and fjord width. In this figure, for nondimensionalized sill heights above 0.5, the overturning circulation weakens, but for higher fjord widths, this critical sill height threshold increases to 0.9. Although the fjord widths tested in section 3 are not narrow enough to permit hydraulically controlled solutions, fjord width does lead to hydraulic control for taller sills, which is only apparent after covarying sill height and fjord width.

Time-averaged (days 1300–1600) nondimensionalized bottom-layer transport (*W*_{fj}/*L*_{d}). All other parameters are fixed to the reference case. Pink markers represent the geometric parameter combinations tested.

Citation: Journal of Physical Oceanography 51, 4; 10.1175/JPO-D-20-0091.1

Time-averaged (days 1300–1600) nondimensionalized bottom-layer transport (*W*_{fj}/*L*_{d}). All other parameters are fixed to the reference case. Pink markers represent the geometric parameter combinations tested.

Citation: Journal of Physical Oceanography 51, 4; 10.1175/JPO-D-20-0091.1

Time-averaged (days 1300–1600) nondimensionalized bottom-layer transport (*W*_{fj}/*L*_{d}). All other parameters are fixed to the reference case. Pink markers represent the geometric parameter combinations tested.

Citation: Journal of Physical Oceanography 51, 4; 10.1175/JPO-D-20-0091.1

The hydraulic control theory quantitatively captures the transport reduction in Fig. 14 for tall sills *W*_{fj}/*L*_{d} < 1. This also suggests the possibility that fjord width may lead to hydraulic control for sill heights

The right panel in Fig. 14 shows that RMSD is greater for wider fjords, where shelf eddies can more easily penetrate into the fjord, and cases near hydraulic control, which reflects the observation that the regions of critical flow (*G* ≈ 1) are also important sources of variability. This is due to the formation of isopycnal jumps/shocks with the same properties observed in Zhao et al. (2019), which have been shown to convert mean baroclinic and barotropic energy into eddy kinetic energy, and may be characterized as Kelvin-wave hydraulic shocks (Hogg et al. 2011).

### b. Long-term variability and periodic flushing events

In addition to variability on the shelf maintained by baroclinic instability (which leads to the across-shelf exchange *Q*_{eddy}) and the variability generated at hydraulic shocks near vertical/horizontal constrictions, we observe additional modes of variability.

Longer-term variability of the overturning circulation exists in our simulations and occurs simultaneously with perturbations in horizontal recirculation strength. To illustrate this, we show a time series of zonal transport using a Hovmöller plot for a *H*_{S} = 0 m case in Figs. 15a,d,g in comparison with a *H*_{S} = 100 m case in Figs. 15b,e,h (with columns corresponding to layer). In these Hovmöller plots, the transport is integrated from *y* = 75 to 79 km (inflow into northern half of the fjord) as a function of *x* and time, since it visually highlights the variability of recirculation and exchange near the fjord mouth. The right column (Figs. 15c,f,i) shows this northern half zonal transport profiled at the sill maximum (*x* = 107.5 km).

(a)–(c) Top-, (d)–(f) middle-, and (g)–(i) bottom-layer zonal transport *Q* calculated by integrating from *y* = 75 km (midline) to *y* = 79 km (northern fjord boundary). This is shown as a function of *x* for two cases of sill height: (left) *H*_{S} = 0 m and (center) *H*_{S} = 150 m for the same fjord width *W*_{fj} = 8 km. (right) The time series of this half-fjord transport at *x* = 107.5 km (sill maximum). Periodic flushing events on time scales of 60 days (long-term variability) are more apparent for the *H*_{S} = 150 m case in the middle and bottom layers, while high-frequency variability on time scales of 1–2 days is more apparent in the *H*_{S} = 0 m case, but also exists in the *H*_{S} = 150 m case.

Citation: Journal of Physical Oceanography 51, 4; 10.1175/JPO-D-20-0091.1

(a)–(c) Top-, (d)–(f) middle-, and (g)–(i) bottom-layer zonal transport *Q* calculated by integrating from *y* = 75 km (midline) to *y* = 79 km (northern fjord boundary). This is shown as a function of *x* for two cases of sill height: (left) *H*_{S} = 0 m and (center) *H*_{S} = 150 m for the same fjord width *W*_{fj} = 8 km. (right) The time series of this half-fjord transport at *x* = 107.5 km (sill maximum). Periodic flushing events on time scales of 60 days (long-term variability) are more apparent for the *H*_{S} = 150 m case in the middle and bottom layers, while high-frequency variability on time scales of 1–2 days is more apparent in the *H*_{S} = 0 m case, but also exists in the *H*_{S} = 150 m case.

Citation: Journal of Physical Oceanography 51, 4; 10.1175/JPO-D-20-0091.1

(a)–(c) Top-, (d)–(f) middle-, and (g)–(i) bottom-layer zonal transport *Q* calculated by integrating from *y* = 75 km (midline) to *y* = 79 km (northern fjord boundary). This is shown as a function of *x* for two cases of sill height: (left) *H*_{S} = 0 m and (center) *H*_{S} = 150 m for the same fjord width *W*_{fj} = 8 km. (right) The time series of this half-fjord transport at *x* = 107.5 km (sill maximum). Periodic flushing events on time scales of 60 days (long-term variability) are more apparent for the *H*_{S} = 150 m case in the middle and bottom layers, while high-frequency variability on time scales of 1–2 days is more apparent in the *H*_{S} = 0 m case, but also exists in the *H*_{S} = 150 m case.

Citation: Journal of Physical Oceanography 51, 4; 10.1175/JPO-D-20-0091.1

In the middle layer (Figs. 15d–f), we observe three cycles of a periodic flushing event on time scales of 60 days, which is approximately the residence time scale of the fjord, *τ*_{r} = *W* × *L* × *H*/*Q*. This is particularly clear in the *H*_{S} = 150 m case, where the sill is tall enough to influence the middle layer. In Fig. 15e, this periodic flushing appears as blue streaks representing westward outflow, which originate in the middle-layer fjord interior as a disruption to the anticyclonic recirculation and propagate across the shelf over a period of 20 days. The trend of the half-fjord transport at the sill overflow in Fig. 15f also clearly shows a periodic signal on 60 day cycles for the *H*_{S} = 150 m case (blue line). The bottom layer exhibits the same periodic signal, but is approximately 3 times weaker due to weak recirculation near the sill maximum since the main region of recirculation extends from *x* = 130 to 150 km (whose magnitude also observably varies over a 60 day cycle in Fig. 15h). Figure 15c,f,i shows that in general, the *H*_{S} = 0 m case has more short-term variability and the *H*_{S} = 150 m exhibits greater long-term variability.

The long-term variability (compared to the short-term) has a smaller contribution to the overall RMSD of the overturning circulation in the *H*_{S} = 0 case, but becomes increasingly important to consider for fjords with limited overturning and renewal.

The short-term variability accounts for most of the RMSD in Fig. 4 except for a few cases of weak overturning. This variability is apparent in Fig. 15 as the 1–2-day fluctuations in transport, although it is diagnosed differently. Within the fjord, this variability is dominated by coastally trapped waves, which are generated either at the fjord mouth/sill maximum (due to the horizontal and vertical constriction) or the coastal current. The amplitude of the observed waves is larger for wider fjords, intermediate sill heights, stronger winds, stronger subglacial discharge, and larger zonal pressure gradients. The daily and monthly time scales of short-term and long-term variability, respectively, coexist in Figs. 15c,f,i.

### c. Submesoscale fjord dynamics

The simulations presented so far span the noneddying to weakly eddying regime within the fjord. Although the eddy kinetic energy within the fjord is weak due to our choice of resolution, it does increase substantially for high-resolution simulations of fjord-only domains. We find that although the total overturning strength and recirculation strength do not depend strongly on resolution (~20% increase for both from *dl* = 1000 m to *dl* = 68 m), submesoscale eddies do appear within fjords and the eddy contribution accounts for a significant proportion of the overturning circulation (up to 40% in the highest resolution cases).

In Fig. 16, a reference run at high resolution (*dl* = 68 m) with *H*_{S} = 0 m and *W*_{fj} = 8 km shows evidence of submesoscale activity. The submesoscale eddies in Figs. 16d–f have a peak vorticity of *ζ*/*f* ≈ 4 and diameters on the order of 1 km, which are small compared to the deformation radius. They are found to be primarily generated near the curved sidewall regions near the mouth of the fjord. These eddies influence both the mean along-fjord and across-fjord isopycnal gradients, as can be seen in Figs. 16a–c in comparison with the corresponding low-resolution case in Fig. 6a.

High-resolution (*dl* = 68 m) simulation of a case with no sill and *W*_{fj} = 8 km with snapshots of (a),(b) surfaces of interface depth *η*_{1} and *η*_{2} and (c) their along-midline depth (*y* = 75 km). (d)–(f) Snapshot of vorticity with velocity quivers for each layer (at day 200). The maximum velocities are 0.42, 0.66, and 0.37 m s^{−1} in the top, middle, and bottom layers, respectively. See the online supplemental material for a movie of this figure.

Citation: Journal of Physical Oceanography 51, 4; 10.1175/JPO-D-20-0091.1

High-resolution (*dl* = 68 m) simulation of a case with no sill and *W*_{fj} = 8 km with snapshots of (a),(b) surfaces of interface depth *η*_{1} and *η*_{2} and (c) their along-midline depth (*y* = 75 km). (d)–(f) Snapshot of vorticity with velocity quivers for each layer (at day 200). The maximum velocities are 0.42, 0.66, and 0.37 m s^{−1} in the top, middle, and bottom layers, respectively. See the online supplemental material for a movie of this figure.

Citation: Journal of Physical Oceanography 51, 4; 10.1175/JPO-D-20-0091.1

High-resolution (*dl* = 68 m) simulation of a case with no sill and *W*_{fj} = 8 km with snapshots of (a),(b) surfaces of interface depth *η*_{1} and *η*_{2} and (c) their along-midline depth (*y* = 75 km). (d)–(f) Snapshot of vorticity with velocity quivers for each layer (at day 200). The maximum velocities are 0.42, 0.66, and 0.37 m s^{−1} in the top, middle, and bottom layers, respectively. See the online supplemental material for a movie of this figure.

Citation: Journal of Physical Oceanography 51, 4; 10.1175/JPO-D-20-0091.1

In the surfaces of interface depth (Figs. 16a,b), coastally trapped waves (as previously discussed) appear to form isopycnal shocks within the fjord and near the coastal current. These waves and shocks propagate in the prograde direction and have shock amplitudes that decay from the coast with a width *L*_{d} and are similar in behavior to the Kelvin wave hydraulic shocks discussed in Hogg et al. (2011). Interestingly, the bottom-layer coastal eddies in Fig. 16f propagate in the same direction as the waves and shocks (northward/prograde), while the background coastal mean flow in this layer is southward. In both reality and models that permit such effects, these sources of variability may lead to elevated mixing in the fjord interior and variability of the recirculation and boundary current transport, which may be explored in a future study.

## 7. Discussion and concluding remarks

### a. Summary

In glacial fjords, there is a complex interaction of dynamics in the shelf, fjord, and discharge/melt plumes, with multiple controls of the overturning circulation and horizontal recirculation (Straneo and Cenedese 2015; Carroll et al. 2017; Jackson et al. 2018). In this study, we examine the influence of key geometric controls (sill height, external stratification, and fjord width) on overturning in the shelf-to-glacial face system and horizontal recirculation in the fjord interior.

In section 2, we discuss the idealized three-layer numerical model setup to simulate the full shelf-to-glacial-face system. We examine the sensitivity of overturning and recirculation to six important parameters in section 3 that capture variations in geometry (fjord width and sill height), boundary forcing (AW depth, winds, and subglacial discharge), and stratification. We find that the overturning and recirculation increase more significantly with decreasing sill height, deeper AW, and increasing subglacial discharge (shown in Figs. 4 and 5). Additionally, the horizontal recirculation significantly increases with fjord width.

We develop and test comprehensive theories that provide clarity on the role of each control in section 4. The theory for the overturning circulation is pieced together using theories for the continental shelf, the fjord mouth sill, and the fjord head regions. The theory accurately predicts the simulated overturning over realistic ranges of each control parameter for each of the three regions and provides predictions for the AW layer thickness at the fjord mouth and fjord head (Fig. 10), which can be used to predict the overall AW transport (Fig. 11). In section 5, we develop a theory for the bottom-layer horizontal recirculation based on a vorticity balance between bottom friction and the diabatic vorticity generation of the water mass transformation, which accurately predicts the recirculation over realistic ranges of each control parameter (Fig. 12).

In section 6 we discuss the modes of external and internal variability of the system. We further discuss hydraulic control at the fjord mouth and the role of both low-frequency and high-frequency variability on the shelf and within the fjord. The sill overflow can transition from geostrophic to hydraulically controlled regimes with varying sill height, AW depth, and fjord width, and can explain the reduction in warm water inflow over realistic fjord parameters similarly to results from Zhao et al. (2019). Submesoscale variability was also observed in a configuration with a fjord attached to a smaller coastal shelf region shown in Fig. 16, and potentially plays an important role in the overturning and recirculation.

### b. Glacial melt rate implications

Using the theories we presented in sections 4 and 5 supported by the numerical simulations presented in section 3, we can estimate glacial melt rates taking into account fjord circulation. The melt rate is predominantly dependent on two parameters: the vertical velocity of the discharge plume, which depends on discharge strength and the stratification set by open ocean and the overturning circulation, and the near-glacier horizontal velocity (the main driver of ambient melt), which depends on the strength of the horizontal recirculation.

Although we do not expect accurate estimates given the possible range of the empirical coefficients *C*_{d} and *γ*_{T}, it is still useful to provide melt rate estimates based on Eqs. (39a)–(40b) with horizontal and vertical velocities from our simulation results and theory, which we hope will guide future circulation-aware glacial melt rate parameterizations (further discussed in appendix B). Our predicted maximum discharge plume-driven melt rate (or rate of undercutting) for Jakobshavn parameters is 8.7 m day^{−1} and the predicted ambient melt rate over the rest of the terminus in contact with the bottom-layer AW is 1.1 m day^{−1}. However, due to its much larger area, the ambient melt accounts for 80% of the total volume melt and is ~1.0 km^{3} yr^{−1} based on a bottom-layer thickness of 400 m and fjord width of 8 km (see appendix B for further details). However, the freshwater input is still dominated by the discharge plume rather than meltwater, which supports our model assumptions of excluding the meltwater contribution to the buoyancy forcing in the fjord and supports recent findings that ambient melt driven by horizontal recirculation may be as important or more than the subglacial discharge-driven melt (Slater et al. 2018; Jackson et al. 2020).

The connection between overturning circulation, horizontal recirculation, and melt rates raises the possibility of a dynamical feedback, which is not simulated in our model and can be described as follows: stronger horizontal recirculation leads to stronger ice front velocities, which leads to higher melt rates by increasing turbulent transfer of heat to the ice face, which leads to stronger buoyancy forcing and thus, water mass transformation and overturning, which induces stronger horizontal recirculation to balance the vorticity budget. However, additional modeling and observations are needed to assess the importance of the melt–circulation feedback.

### c. Caveats and future directions

Due to the limitations of a simplified model configuration, there are a number of caveats. These include the simplicity of geometry on the shelf, the lack of sea ice/melange/icebergs and surface buoyancy forcing in the fjord, the low-order representation of vertical structure in the ocean, and a lack of time-dependent buoyancy forcing (both the plume and open-ocean conditions). In general, the across-shelf transport is likely to be much more complicated than presented in this study, with canyons and remotely generated coastal currents playing important roles (e.g., St-Laurent et al. 2013; Moffat et al. 2009), such that a more realistic across-shelf transport component of the theory is likely more complex. Also, tests of the interfjord separation distance (not shown) suggest that the strength of the coastal current is influenced by this parameter. Furthermore, since we only consider mixing due to the entrainment of the ambient and discharge plume, our theories assume that tides and sill overflows/bottom boundary layer processes are small contributions to the overall mixing. To account for this, the theory from section 4 can be modified to include such contributions by replacing *Q*_{plume} with a total diabatic mixing term, *Q*_{diab} = *Q*_{plume} + *Q*_{tide} + *Q*_{BBL}. The overall overturning prediction in Eq. (23) can therefore by modified to include realistic parameterizations of *Q*_{tide} and *Q*_{BBL}. However, the potential importance of vertical mixing throughout the fjord to the overall overturning circulation remains an open question.

Following this study, there are a number of open questions that require further study. Additional work is needed to investigate the intrafjord submesoscale phenomenology and the distribution of mixing within the shelf-to-glacial-face system and how it influences fjord renewal through both observations and modeling. Another future avenue would be to investigate the interaction between adjacent fjords. Also, a potential feedback exists between the fjord circulation and glacial melt rates (which controls the strength of the diabatic forcing and thus, the circulation), but requires testing in a model that permits such a feedback. Finally, the boundary layer processes responsible for the melting at the glacial face requires both careful observational and modeling work in order to close the gap between our simple melt rate parameterizations based on plume and ice–ocean boundary layer theory and in situ melt rate observations. In addition, measurements of fjord recirculation and spatial density variations at depth are lacking and are critically needed to compare with our findings in the hopes of improve our understanding of fjord circulation and their influence on glacial melt rates.

## Acknowledgments

The authors thank Pierre St-Laurent for allowing the use of the open source code BEOM and two anonymous reviewers for helpful comments. The model source code is available at www.nordet.net/beom.html. This material is based in part upon work supported by the National Science Foundation under Grants PLR-1543388 and OCE-1751386.

## APPENDIX A

### Plume Parameterization in an Isopycnal Model

We derive a point-source plume solution with a piecewise-constant background density, which can be used in isopycnal models such as BEOM. This is a special case of plume theory in a continuously stratified fluid (see, e.g., Turner 1979).

The traditional theory of plumes with uniform background density predicts that buoyant plumes are largely controlled by the buoyancy forcing, which sets the entrainment and mixing of the plume with the ambient fluid (Morton et al. 1956). An axisymmetric turbulent plume can be defined based on the parameters *B* (buoyancy flux), *z* (height above the source), and *R* (radial length scale). For a constant background density, it is often assumed that the profiles are self-similar and dimensional analysis can be used to find the vertical velocity *w*, reduced gravity *g*′, and *R* as a function of *z*.

*m*=

*R*

^{2}

*w*. For the specific case of piecewise-uniform density, the buoyancy flux (third equation) can be simplified to

*B*is constant within each layer, but changes at each interface according to this definition, which is discontinuous since

*g*′ is discontinuous. We implement a simple first-order Euler scheme for

*R*and

*w*, which converges for small interval size Δ

*z*~ 0.1 m. We use this to solve for

*R*(

*z*) and

*g*′(

*z*) =

*B*/

*m*at each step in

*z*. At interfaces, we solve for the jump in

*B*and

*g*′ as Δ

*B*=

*B*

_{+}−

*B*

_{−}and

*m*and

*w*. These jumps are defined as

*Q*

_{3}and −

*Q*

_{1}. The volume transport in the middle layer must compensate the transport in the other two layers. The density flux conservation equations for the overall system can then be written as (with mass conservation already applied)

A diagram of bulk water mass transport in our BEOM plume parameterization.

Citation: Journal of Physical Oceanography 51, 4; 10.1175/JPO-D-20-0091.1

A diagram of bulk water mass transport in our BEOM plume parameterization.

Citation: Journal of Physical Oceanography 51, 4; 10.1175/JPO-D-20-0091.1

A diagram of bulk water mass transport in our BEOM plume parameterization.

Citation: Journal of Physical Oceanography 51, 4; 10.1175/JPO-D-20-0091.1

*W*

_{nudg}and

*L*

_{nudg}to be the width and length of the nudging region. In our simulations, we define our nudging region to be 5 km so the diabatic flux is resolved and distributed over at least 10 grid points, whereas the plume radius would be subgrid scale. The thickness nudging due to plume entrainment corresponding to the diagram in Fig. A1 can be expressed as

For the second case in Eq. (A9a), where

Similarly, a line plume parameterization with varying source width is implemented and test in our model. For a small discharge width (50 m or less), there is a negligible difference between the terminal volume flux of a point plume and line plume parameterization. For larger plume source widths, the overturning strength and neutral depth of the plume are quantitatively different, but the overturning still varies proportionally to the discharge.

## APPENDIX B

### Circulation-Aware Glacial Melt Rate Estimates

An important implication for fjord overturning and horizontal recirculation is the submarine melt rate implied by our simulation results and theory. The purpose of this appendix is to draw a posteriori melt rate inferences from the overturning and recirculation strengths diagnosed from our simulations. We calculate the melt rates using a front-wide ambient melt (including a line plume) in the warm bottom-layer area in contact with the glacial face as well as the melt rate within the half-cone subglacial discharge-driven plume over a much smaller area of the glacial face.

*q*(m s

^{−1}) is the glacial melt rate per unit area,

*L*

_{i}= 3.35 × 10

^{5}J kg

^{−1}is the latent heat of fusion of ice,

*c*

_{w}= 3.974 × 10

^{3}J kg

^{−1}K

^{−1}is the specific heat capacity of water,

*c*

_{i}= 2 × 10

^{3}J kg

^{−1}K

^{−1}is the specific heat capacity of ice,

*T*

_{p}and

*S*

_{p}are the plume temperature and salinity,

*T*

_{i}= −10°C is the ice temperature,

*T*

_{b}and

*S*

_{b}are the boundary layer temperature and salinity,

*γ*

_{T}and

*γ*

_{S}are the turbulent thermal and salt transfer coefficient, and

*λ*

_{1}= −5.73 × 10

^{−2}°C psu

^{−1},

*λ*

_{2}= 8.32 × 10

^{−2}°C, and

*λ*

_{3}= 7.61 ×10

^{−4}°C m

^{−1}are the freezing point slope, offset, and depth. These empirical values are consistent with those used in previous studies (Sciascia et al. 2013; Cowton et al. 2015).

*C*

_{d}= 2.5 × 10

^{−3}is the drag coefficient, Γ

_{T}= 2.2 × 10

^{−2}and Γ

_{S}= 6.2 × 10

^{−4}are the thermal and salt transfer constants, and

*υ*and

*w*are the tangential horizontal and vertical velocities at the glacier boundary. For our simulations, the plume vertical velocity (at 100 m above the discharge source) ranges from 0 m s

^{−1}(no subglacial discharge) to 3.7 m s

^{−1}(greatest discharge) and the horizontal velocity

*υ*=

*υ*

_{max}in the grid point adjacent to the glacier face in the lower layer ranges from 0.05 to 0.3 m s

^{−1}.

Although the vertical velocities (in the plume) are much larger than the horizontal velocities near the glacial face, recent work suggests that the ambient melt dynamics driven by the horizontal recirculation may be as important as the subglacial discharge-driven melt (Slater et al. 2018, Jackson et al. 2020). This is partly due to the fact that ambient melt affects a much larger area of the glacial face. Studies have also noted that ambient melt rates from the plume melt parameterizations are unrealistically low compared to the total ice flux at the terminus, but have not determined which melt processes produce these high melt rates (Jackson et al. 2020; Straneo and Cenedese 2015; Carroll et al. 2016; Fried et al. 2015).

For Jakobshavn Glacier, using a subglacial discharge of 1700 m^{3} s^{−1} (based on assuming all runoff enters as subglacial discharge in Beaird et al. 2017), our theory predicts an overturning circulation of 85 mSv and horizontal recirculation strength of 300 mSv. Here, the plume vertical velocity at the middepth point of the bottom layer (predicted to be 440 m thick) is 2.7 m s^{−1} and the horizontal velocity at the glacier boundary is 0.34 m s^{−1}. Using a bottom-layer ambient temperature and salinity of 4°C and 34.0 psu (Gladish et al. 2015), we can calculate the plume temperature and salinity at middepth in the bottom layer. Using Eqs. (39a)–(39c), this allows us to find the boundary layer temperature and salinity and the melt rates. Our predicted maximum discharge plume-driven melt rate (or rate of undercutting) is 8.7 m day^{−1} and the predicted ambient melt rate over the rest of the terminus in contact with the bottom-layer AW is 1.1 m day^{−1}. However, due to its much larger area, the ambient melt accounts for 80% of the total volume melt and is ~1.0 km^{3} yr^{−1} based on a bottom-layer thickness of 400 m and fjord width of 8 km.

We note that the ambient melt rate is dictated by *υ*_{max}, since it is ~30 times larger than the vertical velocity of the distributed line plume predicted by plume theory (Jenkins 2011). Thus, the ambient melt rate including the horizontal velocity is approximately 30 times larger than the one using only the vertical line plume velocity. Since our melt rate estimate uses the discrete density profile from our three-layer model and is only an approximation to the realistic vertical structure of temperature and salinity, it is only able to capture an approximate bulk melt rate estimate.

The ratio of areas covered by the discharge plume and ambient melt plume depends on the mean width of the discharge plume source and the vertical rise distance. The mean width for a point/cone plume is half of its radius at neutral buoyancy (*R*_{max}/2) and for a truncated-line plume with a finite width *w*_{s} discharge source, the mean width is the mean of *w*_{s} and *R*_{max}/2 (Cowton et al. 2015; Jackson et al. 2017). For a truncated-line plume of *w*_{s} = 200 m at the source (which best fits Greenland’s fjords, as shown in Jackson et al. 2017), our simulation results for the Jakobshavn test case suggests a subglacial discharge plume that occupies ~3% of the surface area of the face, but accounts for 20% of the meltwater supply. However, the buoyancy forcing is likely dominated by the freshwater from subglacial discharge rather than the meltwater production.

Recently, it has been argued that the empirical coefficients *C*_{d} and Γ_{T} are untested in tidewater glaciers and larger values are more consistent with observations, i.e., *C*_{d} = 1 × 10^{−2}, Γ_{T} = 4.4 × 10^{−2}, which would result in melt rate estimates that are 4 times greater (Jackson et al. 2020).

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