© 2021

Chuning Wang

ALL RIGHTS RESERVED

MODELING BUOYANCY-DRIVEN CIRCULATION IN AN IDEALIZED TIDEWATER

GLACIR FJORD

By

CHUNING WANG

A dissertation submitted to the

School of Graduate Studies

Rutgers, The State University of New Jersey

In partial fulfillment of the requirements

For the degree of

Doctor of Philosophy

Graduate Program in Oceanography

Written under the direction of

Robert J. Chant

and approved by

_____________________________________

_____________________________________

_____________________________________

_____________________________________

New Brunswick, New Jersey

May 2021

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ABSTRACT OF THE DISSERTATION

MODELING BUOYANCY-DRIVEN CIRCULATION IN AN IDEALIZED TIDEWATER

GLACIR FJORD

by CHUNING WANG

Dissertation Director:

Robert J. Chant

Tidewater glacier fjords are narrow, deep inlets connecting marine-terminating glaciers with

the continental shelf sea. Over the recent years, the rapid mass loss of the Greenland Ice Sheet has

sparked interest in these fjords due to their impacts on submarine glacier melting as well as

coastal ocean circulation. In tidewater glacier fjords, subglacial discharge is an important source

of buoyancy, which drives the exchange flow and controls salt and heat transports between fjord

and the coastal sea. However, due to the difficulty in acquiring field data in the ice-packed

environments, direct observations of the oceanographic condition are rare, and the general

circulation in these fjords are poorly understood.

Numerical model is an alternative approach to explore the buoyancy driven exchange in

tidewater glacier fjords. This dissertation uses a coupled ocean/plume model to study the

circulation and heat transports in idealized fjord basins and discusses the impacts on submarine

melting of the glacier front.

Firstly, in order to better represent the subglacial discharge, we developed a new model

framework, ROMS-ICEPLUME, to parameterize the rising and initial outflowing stage of the

subglacial discharge plume in vicinity of the glacier/ocean boundary. The coupled model is tested

for sensitivities over various model configurations and validated with field observations using a

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quasi-realistic setup. We find that the modeled circulation is very sensitive to the choices of

outflow parameterization and coupling method. This model is able to reproduce the strong

outflowing plume and its spatial structure observed in a Greenlandic fjord. Results from this part

provides an important tool in advancing our understanding of the exchange processes in tidewater

glacier fjords.

Secondly, using the coupled model framework, the circulation and heat exchange driven by

subglacial discharge are simulated in a series of idealized conditions. The buoyancy-driven plume

travels along the southside of the fjord channel, and to a great extent resembles the structure of a

coastal current. The physical properties of outflowing plume are estimated using model fields and

compared with that derived from a shallow water model. We find that the outflow structure is

dependent on the location of subglacial river, and plume properties are better predicted by the

shallow water model when discharge enters from the southside of the channel. Heat budgets

analysis highlights the role of discharge-driven exchange flow in removing submarine melt water

from glacier front. This mechanism in theory enhances submarine melting, but it is not well

reflected in the modeled melt rates due to limitation of the melting parameterization.

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ACKNOWLEDGEMENTS

I would like to thank the Department of Marine and Coastal Sciences at Rutgers University

for support through graduate and teaching assistantships and also for the opportunities to attend

conferences and professional development events. The last five years have forged me into a better

person, for which I will forever cherish. I would like to thank Bob Chant for his mentoring and

encouragement throughout the journey. You took me into the ocean modeling group and fed me

with the most delicious pizza I have ever had. Even when the world turned upside down you still

managed to find ways to encourage me oceans away and to help me get through the most difficult

time. I am fortunate to have had you as an advisor.

I would also like to thank John Wilkin and Becca Jackson for providing guidance whenever I

hit a bottleneck. John, your ROMS wizardry has made modeling so enjoyable. Thanks to you I

can call myself a modeler with the skills you taught me. Becca, I have learned so much about the

glacial world from you. I appreciate the time you spent reviewing and revising my manuscripts.

You are a role model for young scientists. I also want to acknowledge Carlos Moffat for

providing valuable guidance as an external committee member. My thanks extend to the members

of Ocean Modeling Group (Fernando, Alex, Jack, Piero, Julia, Eli, Jesse), who provided

friendship and technical help over the years, and Nick Beaird for sharing his wisdom on fjord

over the last two years.

Along this journey I received help from many good people. I would like to thank Enrique

Curchitser who first introduced me to the department. Thanks to the GPO students and members

of Earth System Modeling Lab for spending quality time together. Many thanks to Yair Rosenthal

for being a responsible program director and being supportive when I met tough choices. I cannot

be more grateful. Outside of work, I would like to thank my friends on basketball court for

blowing off steam after stressful days.

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Last but not least, I want to thank my mother, father, other family members, friends and loved

ones who have always believed in me. I would never have made it this far without your support

and patience. I dedicate this work to my father, who encouraged me to pursue this path and taught

me to look forward. May you rest in peace.

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TABLE OF CONTENTS

ABSTRACT OF THE DISSERTATION .....................................................................................ii

ACKNOWLEDGEMENTS .......................................................................................................... iv

TABLE OF CONTENTS .............................................................................................................. vi

LIST OF TABELS ...................................................................................................................... viii

LIST OF FIGURES ....................................................................................................................... ix

Chapter 1 - Introduction ................................................................................................................ 1

1.1 Greenland Fjords Hydrology ....................................................................................... 1

1.2 Subglacial Discharge and Buoyant Plume .................................................................. 3

1.3 Thesis Goals ................................................................................................................... 5

Chapter 2 - Modeling Subglacial Discharge Driven Flow in Tidewater Glacier Fjords.......... 7

2.1 Abstract .......................................................................................................................... 7

2.2 Introduction ................................................................................................................... 7

2.3 Methods ........................................................................................................................ 12

2.3.1 Outflow Parameterization ............................................................................................. 14

2.3.2 Coupling BPT with GCM ............................................................................................. 17

2.3.3 ROMS Implementation ................................................................................................. 20

2.3.4 Numerical Experiments ................................................................................................ 23

2.4 Results .......................................................................................................................... 25

2.4.1 2-D Simulation .............................................................................................................. 25

2.4.2 3-D Small Grid Simulation ........................................................................................... 29

2.4.3 KS Glacier Simulation .................................................................................................. 32

2.5 Discussion ..................................................................................................................... 36

2.5.1 Outflow Parameterization Methods .............................................................................. 36

2.5.2 Model Uncertainties ...................................................................................................... 39

2.6 Conclusion .................................................................................................................... 42

Chapter 3 - Buoyancy Driven Circulation and Heat Budgets in Tidewater Glacier Fjords . 44

3.1 Abstract ........................................................................................................................ 44

3.2 Introduction ................................................................................................................. 44

3.3 Methods ........................................................................................................................ 47

3.3.1 Buoyancy Driven Circulation in Tidewater Glacier Fjords .......................................... 48

3.3.2 Heat Budgets ................................................................................................................. 49

3.3.3 Numerical Experiment Configuration........................................................................... 51

3.4 Results .......................................................................................................................... 53

3.4.1 Circulation and Plume Geometry.................................................................................. 53

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3.4.2 Heat Flux ....................................................................................................................... 67

3.5 Discussion ..................................................................................................................... 70

3.5.1 Geometry of SGD Plume .............................................................................................. 70

3.5.2 Heat Budgets in Fjord ................................................................................................... 77

3.6 Conclusion .................................................................................................................... 83

Chapter 4 - Conclusions ............................................................................................................... 84

Appendix A .................................................................................................................................... 87

Appendix B .................................................................................................................................... 93

Appendix C .................................................................................................................................... 97

References .................................................................................................................................... 108

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LIST OF TABELS

Table 3.1. Mean tracer concentration (𝐶) and residence time (𝑇) of SGD, SMWP and SMWA. . 62

Table 3.2. Submarine melt rate 𝑄𝑚, subglacial buoyant plume entrainment rate 𝑄𝑒, and heat

budget terms in Equation (3.5). The residual term 𝐻𝑟 = 𝐻𝑠𝑡 − 𝐻0 −𝐻1 −𝐻2 − 𝐻𝑚. .............. 68

Table C.1. Constants and parameter values of the ICEPLUME model. ...................................... 101

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LIST OF FIGURES

Figure 2.1. Schematic diagram of the development of a subglacial buoyant plume. (a) Side view

of buoyant plume. The tidewater glacier is located at the left end; 𝑞0 is the subglacial discharge

rate at the base of glacier. The buoyant plume rises against the wall and terminates at depth 𝑧𝑇,

which marks the end of the overshooting stage. The buoyant plume then travels as a gravity

current, subducts to the neutral buoyant depth 𝑧𝑃 and outflows into the fjord. (b) 3-D schematic

of the development of buoyant plume. ........................................................................................... 12

Figure 2.2. Schematics of the Mix coupling method. .................................................................... 19

Figure 2.3. Development of the outflowing plume during the first 10.5 hours, modeled with

method HMass/OP. Upper panels are subglacial discharge dye concentrations (color) and

velocities (vector); lower panels are density (contour) and its anomality from initial condition

(color). Initially the contour lines are evenly spaced of 25 m intervals. Triangles and dash lines

mark the theoretical position of signal travels with internal wave speed. ...................................... 27

Figure 2.4. Subglacial discharge dye concentration (color) and velocity (vector) of all

experiments at hour 12.................................................................................................................... 28

Figure 2.5. Modeled velocity field (vector: 𝑢/𝑣, color: 𝑤) at 50 m. Horizontally the plots are

aligned with respect to baroclinic time step lengths (𝑑𝑡); vertically the plots are aligned with

respect to outflow parameterization and coupling methods. .......................................................... 31

Figure 2.6. Modeled velocity fields (upper, vector: 𝑢/𝑣, color: 𝑤) and subglacial discharge dye

concentration (lower) at 50 m. Contours in the lower panels are dye concentration of 0.022 kg/m3.

Horizontally the plots are aligned with respect to grid resolution in along channel direction (𝑑𝑥);

vertically the plots are aligned with respect to outflow parameterization and coupling methods. . 32

Figure 2.7. Velocity field modeled by HMass/OP (a, b) and Mix/NOP (c, d) and from observation

(e) in the near field. (a, c) Velocity averaged over the top 50 m in the near field (0-10 km from

glacier). White vectors are Sect 7 of the KS-1 observations. The observations are rotated to align

with modeled velocity direction. (b, d) Cross channel sections at 3 km from the glacier, showing

component of velocities along principal axis from model (color) and observation (contour).

Contour intervals are 0.05 m/s; the thick line marks zero and solid/dashed lines are

x

positive/negative values. (e) Observations of cross channel velocity (Sect 7 of KS-1), same as the

contour lines in (b, d). ..................................................................................................................... 33

Figure 2.8. Velocity and tracer fields modeled with HMass/OP (a, b, c, d, e) and Mix/NOP (f, g, h,

i, j) over the entire fjord domain. (a, f) Velocity averaged over the top 50 m, similar to Figure

2.7, from glacier front to fjord mouth. (b, g) Velocity profiles in along (U) and cross channel (V)

direction, averaged over the middle fjord (bounded by dashed lines in a, f). (c, h) Cross channel

sections averaged over the middle fjord region, showing component of velocities in along channel

direction. (d, i) Temperature profiles averaged over the middle fjord, of initial and modeled

states. (e, j) Cross channel sections averaged over the middle fjord, showing temperature anomaly

from initial condition (color) and subglacial discharge dye concentration (contour). ................... 35

Figure 2.9. Evolution of buoyant plume near a vertical (left) and calved (right) glacier front. .... 39

∂𝑉 ∂𝑡 = 𝑄𝑠𝑔 + 𝑄𝑚 − 𝐴𝑢𝑑𝐴 ∂ ∂𝑡𝑉𝜃𝑑𝑉 = 𝜌𝑐𝑝𝑄𝑠𝑔𝜃𝑠𝑔 + 𝜌𝑐𝑝𝑄𝑚𝜃𝑚− 𝐴𝜌𝑐𝑝𝑢𝜃𝑑𝐴 −

𝜌𝑄𝑚𝐿 + 𝑐𝑖𝜃𝑚 − 𝜃𝑖 ........................................................................................................................ 50

Figure 3.1. Numerical model configuration. (a) Grid structure of the entire domain. (b) Grid near

the glacier ocean boundary. The blue tiles mark the location of the glacier front; dark blue tiles

mark the location of subglacial discharge channels. (c, d, e) initial conditions of temperature,

salinity and density, averaged from in-situ observations................................................................ 51

Figure 3.2. Modeled velocity (a), tracer (b) and density (c) fields from the numerical experiment

𝑄𝑠𝑔 = 50 m3s-1 and 𝐿𝑟1 = 100 m. All fields are temporally averaged from model day 100-200,

and spatially averaged in along fjord direction from 𝑥 = 20 to 40 km. The black triangle in (a) is

the location of discharge point; the white dot is the coordinate of maximum velocity 𝑈𝑝, which is

referred to as the ‘plume core’. Positive velocity represents flow in downstream direction. The

solid line marks the plume boundary, and the dash lines are confidence intervals. The length

scales ℎ𝑝, 𝐿𝑝, and 𝑍𝑝 in (b) are the thickness, width and depth of the plume, extracted from the

velocity and tracer fields. (d) schematic diagram of an outflowing subsurface plume. ................. 54

Figure 3.3. Velocity fields in the mid-fjord from 9 experiments. Colors are along-fjord velocity

while vectors are cross-fjord velocity. Horizontally the subplots are aligned with respect to

subglacial river location; vertically the subplots are aligned with SGD. ....................................... 55

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Figure 3.4. Geostrophic reconstructions of along fjord velocity. Subplots are arranged similar to

Figure 3.3. ...................................................................................................................................... 56

Figure 3.5. Normalized SGD tracer. Subplots are arranged similar to Figure 3.3........................ 57

Figure 3.6. Physical properties (a-c: length (width, thickness, depth) scales; d: maximum

velocity; e-g: density, potential temperature, salinity; h: total volume flux) of the plume, with

respect to discharge rate. Line color represents simulations of different discharge locations, while

the black line is averaged over all 𝐿𝑟 values. Error bars are confidence interval determined by 𝛼

value. The gray lines in (f, g) are mean temperature/salinity of the initial profile, averaged over

the same depth of the outflowing plume......................................................................................... 58

Figure 3.7. Decomposition terms of volume flux (a, b, c) and temperature (d, e, f), with respect to

discharge rate. Line color represents simulations of different discharge locations, while the black

line is averaged over all 𝐿𝑟 values. The black dot marks the simulation with no SGD, only SMW.

........................................................................................................................................................ 60

Figure 3.8. Mean tracer concentration (a-c) and residence time (d-f) of SGD, SMWP and

SMWA. ........................................................................................................................................... 62

Figure 3.9. Top view of temperature and streamline at five time-slices, averaged over the depth

of plume. From left to right are experiments with different discharge locations (𝐿𝑟1, 𝐿𝑟10, 𝐿𝑟19);

from top to bottom are days after SGD is initiated. Discharge rate of all three experiments is 50

m3s-1. Blue triangle is the discharge location. Linewidth of streamline represents magnitude of

velocity. .......................................................................................................................................... 64

Figure 3.10. Top view of SGD tracer concentration and streamline, similar to Figure 3.9.......... 65

Figure 3.11. Similar to Figure 3.9, of 𝑄𝑠𝑔 = 200 m3s-1. ............................................................. 66

Figure 3.12. Similar to Figure 3.10, of 𝑄𝑠𝑔 = 200 m3s-1. ........................................................... 67

Figure 3.13. Heat budgets from the glacier front to a mid-fjord section (30 km), with respect to

𝑄𝑠𝑔 and 𝐿𝑟. Lines and markers are denoted similar to Figure 3.7. The residual term 𝐻𝑟 = 𝐻𝑠𝑡 −

𝐻0 − 𝐻1 − 𝐻2 −𝐻𝑚. ................................................................................................................... 69

Figure 3.14. Comparison of plume length (a, b, d, e) and velocity (c, f) scales from the analytical

shallow water model and modeled velocity fields. In the top panels (a-c) the averaged ambient

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densities (𝜌1 and 𝜌1) are determined using the whole water column, while in the lower panels (d-

f) only the layers of the plume are used. Colors represent experiments of different discharge

locations; blue is the wall plume scenario. Dots are data points, while error bars are confidence

intervals. The black dashed lines are the ratio benchmarks of 1:2, 1:1, and 2:1. Other dashed lines

are linear regression fits of each discharge location group. The blue dots and lines are highlighted

to distinguish the wall plume from other cases. ............................................................................. 72

Figure 3.15. Velocity fields in the mid-fjord with respect to discharge location and Coriolis

parameter, similar to Figure 3.3. .................................................................................................... 75

Figure 3.16. Velocity fields in the mid-fjord with respect to discharge location and fjord width,

similar to Figure 3.3. ...................................................................................................................... 75

Figure 3.17. Normalized SGD tracer in the mid-fjord with respect to discharge location and fjord

width, similar to Figure 3.5. ........................................................................................................... 76

Figure 3.18. Normalized SGD tracer in the mid-fjord with respect to discharge location and fjord

width, similar to Figure 3.5. ........................................................................................................... 76

Figure 3.19. Plume (𝑄𝑚𝑝) and ambient (𝑄𝑚𝑎) melt rate, with respect to 𝑄𝑠𝑔 and 𝐿𝑟. .............. 79

Figure 3.20. Velocity and temperature profiles of glacier near field from all experiments. Solid

lines are profiles are acquired by averaging all profiles from grid adjacent to the glacier front;

dashed lines in the upper panel are maximum velocity at each depth. The dash black line in upper

panels is 3 cm∙s-1 mark, which is the minimum background velocity of the ambient melting

parameterization. The black solid line in lower panels is the initial temperature profile; back dash

line is the temperature profile without subglacial discharge, only with submarine melting. ......... 81

Figure A.1. Density profiles of plume (black line) and ambient water (colored lines), of the

whole water column (a) and zoomed to the top 100 m (b). The gray line is the initial stratification.

The line colors represent the density profiles averaged over serval grid cells; the span of

averaging is marked in (c). ............................................................................................................. 88

Figure A.2. Hovmöller Diagram of the detrainment velocity in the top 100 m from 7

experiments. The first five experiments use ambient density profiles spatially averaged over

certain spans; the last two experiments use exponential smoothing to further average the profiles

over time. ........................................................................................................................................ 89

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Figure A.3. Hovmöller Diagram of the density deviation from initial condtion in the top 100 m

from 7 experiments. ........................................................................................................................ 90

Figure A.4. Modeled velocity field (color: along channel, vector: cross channel), 2 km

downstream from glacier front. Plots are organized similar to Figure 2.5. ................................... 92

Figure C.1. ROMS-ICEPLUME model schematics. ..................................................................... 98

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Chapter 1 - Introduction

1.1 Greenland Fjords Hydrology

Greenland fjords are narrow, deep channels that connects the Greenland ice sheet with the

large-scale ocean. Over the last two decades, the Greenland Ice Sheet (GrIS) has been losing

mass at an accelerated rate (Shepherd et al., 2012). The mass loss produces ~9.1 mm increases in

sea level from 1978 to 2018 and roughly half of this sea level increase occurred over the last 8

years (Mouginot et al., 2019). This has sparked interest in these fjords since a large portion of

mass loss is triggered by the interaction of glacier margins and the ocean via circulation through

Greenland fjords (see review by Vieli & Nick, 2011; Straneo & Heimbach, 2013).

The fundamental modes of circulation in fjords are reviewed by Farmer and Freeland (1983);

in this section we briefly summarize the established theories of fjord physics, which inspire the

ideas of this work. To a large extent the general circulation in fjords resembles the estuarine

circulation, in which the brackish outflow is on top of the deep compensation current. The

volume and salt fluxes of this process is referred to as the exchange flow, and is described by the

Knudsen’s relation (Knudsen, 1900) and reviewed by MacCready and Geyer (2010). In short,

freshwater influx from surface runoff mixes with fjord water, transporting salt outwards; the salt

lost is balanced by the inflow to maintain continuity and salt conversation. In a steady state, the

simple two-layer expression of the Knudsen’s relation is

𝑄1 =𝑠2∆𝑠𝑄𝑅 , 𝑄2 =

𝑠1∆𝑠𝑄𝑅 , 𝑄𝑅 = 𝑄1 − 𝑄2 (1.1)

where 𝑄𝑅 is the freshwater discharge, 𝑄1 and 𝑄2 are volume transports of exchange flow of the

surface and bottom layers, 𝑠1 and 𝑠2 are salinities of each layer, and ∆𝑠 = 𝑠2 − 𝑠1 is the density

stratification. In estuarine systems mixing is accomplished by tides, which reduces ∆𝑠 and in

accordance to the Knudson relationship increases the exchange. In addition, exchange can be

2

driven directly by tidal fluctuations in velocity and salinity - a term referred to as “tidal diffusive

salt flux” in the estuarine literature (MacCready & Geyer, 2010). In Fjord systems mixing will

also increase exchange flow but due to their great depths, tidal flow, and thus tidal diffusive salt

flux is weak in Fjords. Rather velocity and tracer fluctuations driven by shelf forcing has been

shown to drive significant salt, heat and freshwater flux (Jackson & Straneo, 2016).

Due to the impact of glacier calving, Greenland fjord basins tend to be deeper than the

surrounding continental shelf; the maximum depth can exceed 800 m in some fjords (e.g.,

Sermilik Fjord, Jackson & Straneo, 2016). The deep basin preserves dense water in fjords,

forming the deep layer and limiting the exchange between deep water and continental shelf water.

Some fjords possess one or more submarine sills, which separates the fjord basin from the

continental shelf and further limits the exchange of deep water.

On the other hand, the shallower sills increase the kinetic energy of tides near the sill,

powering additional mixing and enhancing the exchange flow near the constriction. The

maximum mixing (and exchange) criterion is the hydraulic control

𝑢12

𝑔′ℎ1+

𝑢22

𝑔′ℎ2= 1, 𝐻 = ℎ1 + ℎ2 (1.2)

where 𝑢1, 𝑢2 are velocities and ℎ1, ℎ2 are thicknesses of the upper and lower layers at

constriction, respectively; 𝑔′ is the reduced gravity of the two-layer. Equations (1.1) and (1.2)

combined determine the maximal volume and salt exchange in and out of fjord basin, which is

referred to as the overmixing limit (Hetland, 2010).

Density variations on the continental shelf, in the form of internal shelf waves also drives

baroclinic exchange between fjord and shelf water, which has been termed the ‘intermediary

circulation’ in the literature. It is this heaving of the isopycnal’s by shelf forcing that drives the

velocity and tracer fluctuations and net transport described by Jackson et al. (2018). Density

anomalies on the continental shelf are generated by along-shore winds or density advection. Early

3

studies of the shelf-forced circulation include Klinck et al. (1981) and Stigebrandt (1990); recent

observations have highlighted the importance of shelf forcing in various Greenland fjords (D.

Sutherland & R. S. Pickart, 2008; Straneo et al., 2010; Sutherland & Straneo, 2012). In fjords

with sills, density anomaly only drives circulation above the sill depth, thus the term

‘intermediary’ refers to the intermediate layer formed in fjord that separates from the bottom

layer; the deep layer is relatively stagnant and is not influenced by intermediary circulation.

However, in many of Greenland’s fjords these shelf-forced flows extend to the bottom (Jackson

& Straneo, 2016).

1.2 Subglacial Discharge and Buoyant Plume

In addition to the processes described above, there is another mechanism that may dominate

mixing and drives exchange in the system. Fjords in Greenland are often in direct contact with

marine-terminating (tidewater) glaciers, and these fjords termed tidewater glacier fjords. The

presence of an ice/water interface provides additional source of buoyancy: glacier melting and

calving produce freshwater flux directly at the glacier face; more importantly, subglacial rivers

discharge into the basin from the base of glaciers and drives upwelling along the glacier front and

generating strong mixing in the vicinity.

The upwelling of subglacial discharge or submarine melt water plays a key role in regulating

salt and heat transport in fjords. When injected from the base of marine terminating glacier, the

freshwater discharge has strong positive buoyancy, which drives it to rise along the glacier front

and entrains ambient water. As a result, the large volume of entrainment drives the discharge to

develop into the subglacial discharge plume. As it rises the plume’s buoyancy and acceleration

decrease, then begin to decelerate when buoyancy becomes negative. The buoyant plume

continues to decelerate until its vertical momentum becomes zero or it reaches the air-sea

boundary, which marks the termination of a vertical buoyant plume.

4

After termination, baroclinic pressure gradient forces the buoyant plume to travel away from

the glacier, which is referred to as the ‘outflowing plume’ or ‘detrainment plume’. During this

stage the plume oscillates vertically while flowing downstream, and eventually stabilizes at a

neutral buoyant depth and propagates downstream as gravity currents (Baines, 2008). The plume

forms a unique water mass in fjords and the coastal ocean and ultimately incorporated into the

coastal current.

The dynamics of subglacial buoyant plume is well reviewed by Hewitt (2020), and the

numerical implementation is discussed thoroughly in Chapter 2. The upwelling plume is an

important mechanism in mixing the isolated deep basin water to the surface (Beaird et al., 2018),

as well as providing kinetic energy to melt the glacier front. By the time the outflowing plume

detaches from the glacier front, its volume can grow by more than one order of magnitude; field

observations (Jackson et al., 2017) show that subglacial discharge of 200 m3s-1 can develop into a

strong outflow of up to 8000 m3s-1 near glacier front. Since most Greenland fjords are deep and

are lack of shallow sills, the buoyant plume can be the most significant mixing and exchange

contributor in these fjords compared with tide or wind.

In addition, since the deep basins are isolated from the continental shelf, renewal of deep

water is often slow and rare when forced by surface runoff alone. The frequency of deep water

renewal is critically dependent on the availability of dense water from outside the fjord and the

strength of mixing (Farmer & Freeland, 1983). The upwelling plume, not only being an efficient

mixing mechanism, overturns the deep water directly from the grounding line depth and flushes it

out at surface, greatly boosting the deep layer ventilation process. This also has a very strong

biological impact since the deep water upwelling is an efficient nutrient pump for biological

production (Meire et al., 2017; Hopwood et al., 2018).

The strong upwelling plume also drives circulation in the vicinity of glacier front, energizes

the ice/ocean interface and enhances submarine melting. Firstly, directly behind the upwelling

5

plume, melt rate increases due to the large upwelling velocity, and the rate of increase is scaled

with velocity (Mcconnochie & Kerr, 2017). Secondly, the entrainment and upwelling process

drives flow towards the subglacial river, providing another mechanism that potentially increases

melt rate. Direct melt of the glacier ice itself also drives weak convective upwelling (Jackson et

al., 2020), but laboratory works (e.g., Kerr & Mcconnochie, 2015) show that the self-driven

convection is not strong enough to drive additional melting beyond the velocity-independent melt

rate. Lastly, entrainment of buoyant plume quickly removes melt water from the ice/ocean

boundary and drives the relatively warm deep water towards the glacier, supplying heat that is

critical for glacier melting. This overall has a larger impact on the heat budgets and fluxes in the

fjord, which eventually determines the evolution of oceanographic conditions in fjords.

1.3 Thesis Goals

The overarching goal of this study is to explore the general circulation driven by subglacial

discharge, and its impact on submarine glacier melt rate in an idealized fjord basin. The model

framework used in this work is a custom-built version of the Regional Ocean Modeling System

(ROMS, Haidvogel et al., 2008), which couples the buoyant plume theory and submarine melt

parameterization. The principal objectives are:

i. To build and test the coupled ocean/plume model framework and evaluate the

performance of each parameterization/coupling scheme.

ii. To study the near-field circulation driven by subglacial discharge plume.

iii. To study the general circulation in the mid fjord region and evaluate heat budgets and

transports driven by subglacial discharge plume.

The above scientific objectives motivate the following dissertation chapters:

• Chapter 2: Modeling Subglacial Discharge Driven Flow in Tidewater Glacier Fjords.

6

• Chapter 3: Buoyancy Driven Circulation and Heat Budgets in Tidewater Glacier

Fjords.

7

Chapter 2 - Modeling Subglacial Discharge Driven Flow in

Tidewater Glacier Fjords

2.1 Abstract

In tidewater glacier fjords, subglacial discharge is a significant mixing mechanism near

glacier fronts and drives strong exchange flow. In this study, a new model framework, ROMS-

ICEPLUME, is developed to parameterize the rising and initial outflowing stage of the subglacial

discharge plume in the Regional Ocean Modeling System (ROMS). The coupled model is

validated with field observations using a quasi-realistic setup. We find that the modeled

circulation is very sensitive to the choices of outflow parameterization and coupling method. The

model is able to reproduce the strong outflowing plume and its spatial structure observed in a

Greenlandic fjord. This model framework is a promising tool towards advancing our

understanding of circulation in tidewater glacier fjords.

2.2 Introduction

The melting of Greenland’s tidewater glaciers is one of the major contributors of the

Greenland Ice Sheet mass loss (Straneo & Heimbach, 2013; Mouginot et al., 2019). Near the

glacier/ocean boundary, freshwater from subglacial discharge drives strong convection near the

glacier front, which develops into a turbulent buoyant upwelling plume that accelerates glacier

melting during summer seasons (Straneo & Cenedese, 2015). The development of an upwelling

plume is described by the buoyant plume theory (BPT, Holland & Jenkins, 1999; Jenkins, 2011;

Cowton et al., 2015), which uses a coupled plume-melt formulation to determine the physical

properties of the buoyant plume.

When injected from the base of marine terminating glacier, subglacial freshwater discharge

has strong positive buoyancy, which drives it to rise along the glacier front and entrains ambient

8

water. As a result, its buoyancy and acceleration decrease, and begin to decelerate when

buoyancy becomes negative. The buoyant plume continues to decelerate until its vertical

momentum becomes zero or it reaches the air-sea boundary, which marks the termination of a

vertical buoyant plume. After termination, baroclinic pressure gradient forces the buoyant plume

to travel downstream away from the glacier, which is referred to as the ‘outflowing plume’

(Figure 2.1) or ‘detrainment plume’. During this stage the plume oscillates vertically while

flowing downstream, and eventually stabilizes at a neutral buoyant depth and propagates

downstream as gravity currents (Baines, 2008). The plume forms a unique water mass in fjords

and the coastal ocean (Beaird et al., 2018) and ultimately incorporated into the coastal current.

The initial outflowing stage of a buoyant plume near glacier front is difficult to observe and

until recently has only been studied with numerical models and theory (Jenkins, 1991, 2011; Xu

et al., 2012; Xu et al., 2013; Slater et al., 2018). Conventional hydrographic surveys are difficult

to obtain near glacier fronts as a result of the extremely harsh environment; in addition, the

buoyant plume sometimes emerges at subsurface, making it difficult to be detected by remote

sensing techniques such as satellites or radars that are used to characterize conventional river

plumes (Kilcher & Nash, 2010). Recently, however, a series of observational programs have

quantified aspects of the discharge plume in both Greenlandic and Alaskan fjords (e.g., Mankoff

et al., 2016; Jackson et al., 2017) that can be used to test the efficacy of plume parameterizations.

In addition to field observations, attempts have also been made to directly resolve or

parameterize the buoyant plume in numerical models. Slater et al. (2018) first used near glacier

survey data from the Sarqardleq fjord to drive circulation close to a glacier front in a numerical

simulation. In the simulation, observed near glacial velocity profiles are used as boundary

condition to force an exchange flow driven by the upwelling plume, and the modeled circulation

agrees well with simultaneous field observation. While this method produces an accurate

reconstruction of the near field velocity, one limitation is that it requires high resolution data near

9

the glacier front, which is rare for the reasons stated above. Xu et al. (2012) used the non-

hydrostatic version of Massachusetts Institute of Technology General Circulation Model

(MITgcm, Marshall, Adcroft, et al., 1997; Marshall, Hill, et al., 1997) to simulate the

development of subglacial buoyant plume in 2-D (X-Z) and later extended it to 3-D scenario (Xu

et al., 2013). However, this method requires a non-hydrostatic setup as well as very high

horizontal and vertical resolution (1~10 m), which is computationally expensive and currently

impossible for fjord scale (~100 km) simulations.

A third approach, which we follow here, is to drive the ocean model with a parameterization

of an upwelling buoyant plume, which is frequently used in recent modeling works (Cowton et

al., 2015; Carroll et al., 2017; Oliver et al., 2020). It avoids the need to fully resolve the

nonhydrostatic processes, requires only intermediate temporal and spatial resolution and allows

for fjord scale simulations. The overall model performance is highly dependent on the

parameterization itself; recent studies using this approach include model studies of Cowton et al.

(2015); Carroll et al. (2017), in which the MITgcm is coupled with BPT to drive the subglacial

discharge plume. However, the model framework of Cowton et al. (2015) only considers the

rising stage of the buoyant plume, and the initial development of the outflowing stage is over-

simplified. The outgoing flux of plume is approximated by adding additional divergence in a

single grid cell near neutral buoyant depth; considering the large flux rate of discharge plume

(~1000 m3/s) this scheme requires a small timestep to avoid violating the CFL condition. In

addition, these numerical experiments are idealized and lack verification due to lack of field

observation and thus it is unclear if these simulations accurately reproduce the circulation and

buoyancy flux advected away from the glacier.

While there are limited observations of the initial outflowing stage of buoyant plume in

glacial fjords, similar phenomena have been observed in other oceanic and atmospheric

environments. On large scale, ocean overflows (e.g., Medditerranean Overflow, Price et al., 1993)

10

bring dense water over sills and down the continental shelf, which detrains into the ocean basin

from the shelf slope. Legg et al. (2006) investigated the detrainment of overflows over sloped

topography with several model frameworks, and concluded that the entrainment and detrainment

processes are sensitive to both model coordinate system and resolution. McConnochie et al.

(2020) used model and laboratory experiments to demonstrate the initial outflowing stage of a

wall fountain on centimeter scale, and applied the results to observations of the surface

expression from a Greenlandic subglacial discharge plume. Between these two scales,

observations of several other similar phenomena, i.e., volcanic eruptions, bubble plume in lakes

and coastal seas, and convection from hydrothermal vents also contribute to our knowledge of the

development of buoyant plume and are reviewed by Woods (2010).

Over the past few decades, systematic studies of the development of outflowing plumes are

largely limited to laboratory experiments (e.g., Ching et al., 1993; Baines, 2002; Monaghan,

2007). The experiments are usually carried out in a long, narrow tank, where the buoyancy source

is introduced from the edge of a narrow side to initiate an upwelling plume. The buoyancy source

can be either from surface (saline water injection) or from bottom (bottom heating or freshwater

injection). Since in most experiments the tank is narrow, movement in cross tank direction is

usually neglected and the system is considered 2-D in X-Z direction.

In these experiments, the buoyant plume is marked by dyes and captured with a fixed camera.

Depending on the background stratification, the outflowing plume is either a surface or

subsurface gravity current, with a ‘nose’ traveling downstream followed by relatively steady

current. Ching et al. (1993) observed the near field velocity of subsurface outflow in a two-layer

stratified fluid, and reported an asymmetric Gaussian velocity profile in the outflow. Similarly,

Baines (2002) reported a bell-shaped outflow velocity profile in a continuously stratified

environment. In both experiments, the observed outflow is strong, the velocity of which is much

greater than the inflow driven by deep entrainment.

11

In other laboratory experiments (e.g., Hogg et al., 2017; Bonnebaigt et al., 2018), the

upwelling plume is not fully mixed with the entrained ambient water, and a density gradient is

formed from the core of the plume to its periphery; as a result, the plume does not detrain into the

ambient environment as a single pulse. Instead, the buoyant plume ‘peels off’ at several depths

where the density of the peeled water equals to the ambient environment, and each peeled layer

detrains at its own depth.

In glacial fjords, the development of a subglacial discharge plume can be divided into three

stages (Figure 2.1): (1) the upwelling stage, in which buoyancy is positive and plume accelerates

in vertical; (2) the overshooting stage, in which buoyancy is negative and plume decelerates; and

(3) the outflowing stage, in which the plume gains horizontal momentum and flows downstream.

In this study, a new model framework, ROMS-ICEPLUME, is developed to parameterize these

stages in a widely used general circulation model (GCM), the Regional Ocean Modeling System

(ROMS, Haidvogel et al., 2008). The model framework includes (1) the general circulation model

ROMS, (2) the upwelling and overshooting parameterization BPT, (3) the outflow

parameterization, and (4) three coupler options to integrate BPT, outflow parameterization and

ROMS. In Section 2.3, the outflow parameterization, coupling methods, and three testing

experiments are described in detail; in Section 2.4, results of the experiments are reported and

compared with in-situ observations to fully demonstrate the model performance; in Section 2.5,

the combinations of outflow parameterization and coupling methods are discussed, and some

known uncertainties associated with the model framework are presented.

12

Figure 2.1. Schematic diagram of the development of a subglacial buoyant plume. (a) Side view

of buoyant plume. The tidewater glacier is located at the left end; 𝑞0 is the subglacial discharge

rate at the base of glacier. The buoyant plume rises against the wall and terminates at depth 𝑧𝑇,

which marks the end of the overshooting stage. The buoyant plume then travels as a gravity

current, subducts to the neutral buoyant depth 𝑧𝑃 and outflows into the fjord. (b) 3-D schematic of

the development of buoyant plume.

2.3 Methods

We coupled a new module named ICEPLUME with ROMS, which uses BPT to approximate

the upwelling and overshooting stages, and an outflow parameterization for the outflowing stage.

BPT computes the plume volume flux, tracer concentrations, and entrainment volume fluxes at

various depths, but it does not provide enough information to prescribe the outflow characteristics

such as thickness and velocity; the outflow parameterization uses the information provided by

BPT to calculate outflow rates at various depth and reports them back to ROMS to drive the

circulation.

13

In previous numerical applications, Cowton et al. (2015) prescribed the outflowing volume

flux as vertical mass transport into a single grid cell of 200 × 200 × 10 m. This is implemented

by applying an increment to vertical convergence/divergence at the boundary of each grid cell,

which is equivalently vertical velocity for a fixed grid cell surface. The size of the grid cell is

arbitrary determined in the z-coordinate GCM. Thus, the outflow velocity is determined by grid

resolution rather than physical reasoning. This very simplified method is convenient to

implement, but with the same convergence/divergence, the velocity increases with increased grid

resolution, which can generate instability in tracer fluxes. In addition, mixing, circulation and

other critical properties will potentially be a function of model resolution if the initial outflow

velocity is sensitive to grid size, which a source of uncertainty in model performance. Therefore,

in high resolution simulations the outflowing plume should be distributed in several grid cells to

prescribe realistic outflow velocities and avoid numerical instabilities.

In addition, it is not clear if vertical convergence/divergence, as used by Cowton et al. (2015),

is the optimal method to prescribe the plume driven flow. In particular, since the horizontal scale

of a subglacial discharge plume tends to be smaller than the grid cell itself, distributing the mass

transport of a small convective plume to a large grid cell may be inaccurate. Rather, we prescribe

the entrainment/detrainment as a horizontal advective fluxes some distance downstream (Figure

2.1, gray line) of the glacier. This approach forces the exchange flow without introducing large

vertical convergence/divergence, and prescribes a more widely distributed outflow away from the

glacier front that, due to its broader distribution, is better resolved by the model grid.

We next describe a parameterization and three different model implementations to represent

the initial outflowing of the plume. In Section 2.3.1, we describe a parameterization algorithm

that estimates the velocity and vertical extent of outflowing plume; in Section 2.3.2, we present

the strategies to couple the plume-driven volume fluxes with GCM; in Section 2.3.3, we discuss

14

the ROMS implementation of outflow parameterization and coupler; in Section 2.3.4, we present

three numerical experiments used to test the ROMS implementation.

2.3.1 Outflow Parameterization

To simplify the problem, the ambient stratification is represented with a two-layer setup,

where 𝜌1 < 𝜌2 are average densities of the upper and lower layer, respectively; 𝑔′ =

𝑔(𝜌2 − 𝜌1)/𝜌𝑟𝑒𝑓 is defined as the reduced gravity between two layers; 𝜌𝑟𝑒𝑓 is a reference density

and 𝑔 is the gravitational acceleration. Subglacial discharge plume rises along the glacier wall,

during which the plume properties are predicted by the BPT. The plume density is 𝜌𝑝 when the

rising stage terminates. If the value of 𝜌𝑝 falls between the densities of the two layers ( 𝜌1 <

𝜌𝑝 < 𝜌2), the outflowing plume forms near the density jump; otherwise the plume outflows at

surface and travels downstream as gravity current. Assuming that the plume detrains as one

uniform water mass, the nose speed 𝑈𝐷 of the outflowing current is estimated with an empirical

parameterization developed by Noh et al. (1992); Ching et al. (1993). For outflow in two-layer

fluid, the nose velocity is dependent on a modified Richardson number

𝑅𝑖 =𝑔′𝑙𝑃

𝑊𝑃2 (2.1)

where 𝑙𝑃 is a length scale of the buoyant plume (roughly the along fjord width of the upwelling

plume, Figure 2.1), and 𝑊𝑃 is the scale of vertical velocity of the plume during rising stage. Both

𝑙𝑃 and 𝑊𝑃 are predicted by the BPT. The outflow velocity is then calculated using a piecewise

function

𝑈𝐷 = {0.7𝑅𝑖0.17𝑊𝑃 , 𝑅𝑖 ≤ 6

0.95𝑊𝑃 , 𝑅𝑖 > 6 (2.2)

Even though the Ching et al. (1993) parameterization is empirically derived, the dependence

on Richardson Number suggests it is tightly related to the laws of stability in sheared flow. When

15

stratification is weak (𝑅𝑖 < 6), the upwelling plume overshoots deeply into the top layer, and the

transition from an upwelling plume to a horizontal outflow is very slow. Therefore, the outflow

adjusts on a Richardson Number bases, coinciding with the behavior of a gravity current. When

stratification is strong (𝑅𝑖 > 6), the overshooting phase is short, and the density interface acts like

a solid boundary (McConnochie et al., 2020), redirects the momentum from vertical to horizontal

direction (with a 5% loss), thus the dependence on Richardson Number is weak and nose velocity

is scaled with 𝑊𝑃. When stratification is very wake and the plume density is smaller than the

surface layer, the outflow emerges at surface, and the expression of 𝑅𝑖 > 6 is adopted to

determine the nose velocity since air-sea interface also acts as a large density interface, similar to

the strongly stratified scenario.

The next step is to determine the velocity structure of the outflow which we based on the

velocity profiles reported by Ching et al. (1993); Baines (2002). We use an asymmetric Gaussian

function to approximate the velocity profile; the Gaussian shape is able to generate a smooth

transition from the core of outflow to the ambient water, and reduces the artificial mixing caused

by unrealistic shear.

The assumed expression of the Gaussian velocity profile is

𝑢𝐷(𝑧) =

{

𝑈𝐷exp [−0.5 (

1

𝜎

𝑧 − 𝑍𝑃ℎ2

)2

] , 𝑍𝑃 − ℎ2 < 𝑧 < 𝑍𝑃

𝑈𝐷exp [−0.5 (1

𝜎

𝑧 − 𝑍𝑃ℎ1

)2

] , 𝑍𝑃 < 𝑧 < 𝑍𝑃 + ℎ1

(2.3)

This expression is asymmetric around the outflow core depth 𝑍𝑃; on either side it is

normalized by length scales ℎ1, ℎ2. The standard deviation σ is determined arbitrarily; σ → +∞

corresponds to a uniform 𝑈𝐷 in the outflow. In general, σ = 0.5 is a reasonable value to provide a

smooth profile for numerical implementation.

16

The corresponding outflow volume flux (per unit depth) is 𝑞𝐷(𝑧) = 𝑙𝑚𝑢𝐷(𝑧), where 𝑙𝑚 is

roughly the ‘width’ of the plume (Figure 2.1) predicted by BPT. The total outflow volume flux is

then 𝑄𝐷 = ∫ 𝑞𝐷(𝑧)𝑍𝑃+ℎ1𝑍𝑃−ℎ2

𝑑𝑧, which is equivalent to the upwelling plume volume flux predicted

by BPT; since the profile is gaussian on either side, its integration can be easily computed.

Therefore, the total thickness of the outflow is

ℎ ≡ ℎ1 + ℎ2 =𝑄𝐷

𝑙𝑚𝑈𝐷 ∫ exp[−0.5(�̃�/𝜎)2]𝑑�̃�1

−1

(2.4)

When a gravity current intrudes into a two-layer stratified fluid, in most cases the intrusion is

asymmetric (Ungarish, 2010). As a result, the vertical extents of the outflowing plume below and

above the layer boundary are usually not equal (ℎ1 ≠ ℎ2), and the intrusion depths into upper and

lower layer need to be determined independently. Following Ungarish (2010), ℎ1 and ℎ2 are

determined by matching the pressure at the base of outflow with the pressure of the ambient water

of the same depth.

When the buoyant plume detrains from the glacier front, it pushes ambient water downstream

and changes the local pressure. The pressure at the bottom of the outflow (𝑍2 in Figure 2.1) is

estimated by integrating from surface

𝑃2 = 𝑃0 + 𝑔[𝜌1(𝑍𝑃 − ℎ1) + 𝜌𝑃(ℎ1 + ℎ2)] (2.5)

At the same depth, the ambient water pressure is

𝑃′2 = 𝑃0 + 𝑔(𝜌1𝑍𝑃 + 𝜌2ℎ2) (2.6)

The ‘steady’ condition for gravity current intrusion requires 𝑃2 = 𝑃′2. When 𝑃2 < 𝑃′2, the

dense ambient water (𝜌2) is forced to move upstream, lifting the buoyant plume to a shallower

depth; when 𝑃2 > 𝑃′2, the outflow water mass sinks and pushes the ambient water downstream.

Rearrange, Equations (2.4) to (2.6) give

17

ℎ1 =𝜌𝑃 − 𝜌1𝜌2 − 𝜌1

ℎ,

ℎ2 =𝜌2 − 𝜌𝑝

𝜌2 − 𝜌1ℎ

(2.7)

and at this stage the properties of outflow are fully determined.

2.3.2 Coupling BPT with GCM

Once the entrainment/outflowing rates are calculated, the fluxes are added to the ROMS grid

as point sources in each vertical level. Previously, ROMS provides two options to add point

source into a grid cell: LuvSrc or LwSrc, where the point source is prescribed as horizontal or

vertical mass fluxes, respectively. Similarly, to incorporate entrainment/outflow fluxes in ROMS,

the simplest method is to prescribe a horizontal mass flux from a nearby ocean grid (hereinafter

HMass), which takes advantage of the existing framework of LuvSrc. When HMass is activated, the

horizontal velocity from glacier to ocean is determined by

𝑢(𝑘) =1

𝑑𝑦𝑑𝑧(𝑘)[−𝑞𝐸(𝑘) + 𝑞𝐷(𝑘)], 𝑜𝑟

𝑣(𝑘) =1

𝑑𝑥𝑑𝑧(𝑘)[−𝑞𝐸(𝑘) + 𝑞𝐷(𝑘)]

(2.8)

where 𝑞𝐷 and 𝑞𝐸 are total entrainment and outflow volume fluxes into/out of a ROMS grid cell

predicted by BPT and the outflow parameterization; 𝑑𝑥, 𝑑𝑦 and 𝑑𝑧 are length dimensions of the

ocean grid; 𝑘 is the grid cell index in vertical direction.

In the second method, the point source is prescribed as vertical mass fluxes (hereinafter

VMass), which is implemented as an increment in vertical velocity

∆𝑤(𝑘) =1

𝐴∑ −𝑞𝐸(𝑖) + 𝑞𝐷(𝑖)

𝑘

𝑖=1 (2.9)

where 𝐴 = 𝑑𝑥𝑑𝑦 is the area of the grid. In ROMS this is equivalent to adding

convergence/divergence in an ocean grid. This is the default method to add point source in

18

MITgcm, and is later adopted by Cowton et al. (2015) to couple the buoyant plume model with

ocean model.

The tracer fluxes, 𝑇, on the other hand, are independent of the choices of HMass or VMass

𝑇(𝑘) =1

𝐴[−𝑞𝐸(𝑘)𝑇𝐴𝑚(𝑘) + 𝑞𝐷(𝑘)𝑇𝑃] (2.10)

where 𝑇𝐴𝑚 and 𝑇𝑃 are tracer concentrations of the ambient water and plume water, respectively.

The third method (hereinafter Mix), unlike HMass or VMass, does not add horizontal or vertical

mass fluxes to the ocean grid. This method is newly developed to process the large values of

plume driven convergence and divergence in VMass, which generates instabilities in numerical

implementations. Since subglacial discharge overturns deep, high density water to shallower

depths (Straneo & Cenedese, 2015; Beaird et al., 2018), the isopycnals near the neutral buoyant

depth 𝑍𝑃 will be distorted as a result of the overturning. Baroclinic pressure gradient then drives

the exchange flow, forming an outflowing plume similar to VMass. Unlike VMass, Mix calculates

the expanding/shrinking of isopycnals internally; then the new profiles of tracers are rewritten

back into the ocean grid before the execution of baroclinic timestep to mimic the distortion in

isopycnals.

In Mix, we consider each grid cell is a rectangular box of volume 𝑣𝑜𝑙0(𝑘) = 𝐴𝑑𝑧(𝑘) and

uniform tracer concentration 𝑇0(𝑘). Each grid cell is allowed to expand/contract and modify its

tracer concentration based on the volume and tracer flux into or out of the cell by entrainment 𝑞𝐸

and outflow 𝑞𝐷 (Figure 2.2). Integrating over one baroclinic timestep ∆𝑡, the volume and tracer

concentration for each level become

𝑣𝑜𝑙12⁄(𝑘) = 𝑣𝑜𝑙0(𝑘) + ∆𝑡[−𝑞𝐸(𝑘) + 𝑞𝐷(𝑘)]

𝑇12⁄(𝑘) =

[𝑣𝑜𝑙0(𝑘) − ∆𝑡𝑞𝐸(𝑘)]𝑇0(𝑘) + ∆𝑡𝑞𝐷(𝑘)𝑇𝑃𝑣𝑜𝑙1

2⁄(𝑘)

(2.11)

19

The subscript 1 2⁄ denotes that this is merely an intermediate step. After this step the vertical

grid spacing is distorted, and a transformation is required to project the new profile back to the

original grid space (Figure 2.2). This is achieved by weighted averaging

𝑇1(𝑘) = ∑ 𝛿(𝑖, 𝑘)𝑇12⁄(𝑖)

𝑁

𝑖=1 (2.12)

where 0 ≤ 𝛿 ≤ 1 is the weight function of each level to transform from the intermediate to

original grid space. The value of 𝛿 is 0 where the intermediate grid does not overlap with the

original grid, and is between 0 and 1 if the two partially overlap. After the transformation, tracer

concentration profile is rewritten back into the ocean model, without introducing any momentum

into the grid cells.

Figure 2.2. Schematics of the Mix coupling method.

In a hydrostatic GCM like ROMS, Mix and VMass generate very similar solutions. The

advantage of Mix over VMass is that it is numerically stable and can tolerate much larger

baroclinic timesteps. One limitation of Mix is that it cannot properly represent the barotropic

20

response of buoyant plume. However, since the circulation driven by subglacial discharge is

mostly baroclinic, the overall barotropic contribution is minimal and sometimes can be neglected.

In ROMS, the options LuvSrc and LwSrc are generally used to represent rivers in estuarine

and coastal simulations. Fluxes of freshwater are added as uniform barotropic flow from one end

of a narrow river channel, in which case LuvSrc and LwSrc produce similar results. Subglacial

discharge, on the other hand, drives strong baroclinic circulation in the near field. The barotropic

signal is relatively weak, and the prescribed fluxes are naturally associated with great shear. As a

result, HMass, VMass and Mix are expected to produce very different results due to the way that

initial shear is prescribed. Moreover, subglacial discharge drains into fjords through opening of

~100 m width, which is relatively small compared to the width of glacier front (1-10 km). As a

result, unlike a river, the outflowing plume near the glacier front is not bounded by topography

and will spread laterally. Since VMass and Mix prescribe the outflow without initial horizontal

momentum in the along channel direction, the outflow behaves more isotropic and balloons out in

all direction; on the other hand, HMass prescribes the outflow with momentum in along channel

direction, thus produces a jet-like flow in the near field. The outflow produced by HMass is

significantly different from VMass and Mix, and is more consistent with observations as we show

in the results of this study.

2.3.3 ROMS Implementation

The simulations in this study are carried out with the ROMS-ICEPLUME coupled modeling

system (https://github.com/ChuningWang/roms-iceplume). Since it is the first time the outflow

parameterization and couplers are implemented and fully tested in ROMS, here we briefly

describe the model structure and implementing procedure.

Firstly, the upwelling parameterization (BPT) calculates the physical properties of the

buoyant plume based on information of subglacial discharge rate, plume structure, tracer

21

concentrations, etc., and terminates to upwell when the plume’s vertical momentum becomes

zero, where the corresponding depth is 𝑍𝑇. At this point, the buoyant plume is slightly denser

than the ambient water. Secondly, the neutral buoyant depth 𝑍𝑃 is determined as the core depth of

outflowing plume. This is achieved by searching downward from 𝑍𝑇 until the density of plume

water is smaller than that of ambient water.

The density profile of ambient water is determined by averaging over a small ‘box’ of 𝑚× 𝑛

grids near the subglacial point source. The values of 𝑚 and 𝑛 are arbitrary and are defined in

model input files. The reason to use a regional average instead of only the adjacent grid is to

avoid strong fluctuations caused by the convective plume itself. A demonstration of the effect of

averaging is shown in Appendix A.1.

The outflow parameterization is based on studies that assumes a two-layered ambient

stratification, in contrast to a continuously stratified environment. In most Greenlandic fjords,

during high freshwater discharge periods (summer), the surface is heavily stratified throughout

the top tens of meters, and there is not a clear ‘pycnocline’ that separates the water mass into two

distinctive water masses. Water below the heavily stratified surface layer, however, is relatively

weakly stratified, despite the fact that it consists of two different water masses (Polar Water and

Atlantic Water, Straneo & Cenedese, 2015). Therefore, conventional methods to determine layer

boundaries, e.g., finding maximum stratification, may not apply in this condition, and new

methodology is required.

As pointed out in Section 2.3.1, in a two-layer stratified fluid, the buoyant plume tends to

detrain at the layer boundary due to its intermediate density; vice versa, the outflow depth 𝑍𝑃

acquired in the previous step is a reasonable approximation for the boundary between the upper

and lower layers. Hence, we use 𝑍𝑃 to define the interface between the layers, and densities of the

upper and lower layer are calculated based on the mean layer density.

22

The total thickness of the outflowing plume ℎ is determined using Equation (2.4). On the

other hand, ℎ1 and ℎ2 are not calculated directly with Equation (2.7); instead, it is determined by

adding grid cells above and below 𝑍𝑃, until the total thickness is greater than ℎ while the ‘steady’

condition 𝑃2 = 𝑃′2 is still met. This is achieved by iteratively searching layers around 𝑍𝑃; more

details are provided in Appendix A.2.

Tracer fluxes in each layer are calculate as 𝐹𝑡𝑟𝑐(𝑘) = 𝑞𝐷(𝑘)𝑡𝑃 − 𝑞𝐸(𝑘)𝑡𝐴𝑚(𝑘), where 𝑡𝑃 is

predicted by BPT, and 𝑡𝐴𝑚 is acquired from the ocean model. The plume tracer concentration 𝑡𝑃

is a single value and has no depth dependence. It is possible to modify the tracer concentration for

each level to match the local density; however, we lack the knowledge to prescribe the variability

in plume properties, and additional numerical tests show that varying tracer concentration did not

produce better results. Therefore, the tracer concentrations in buoyant plume are fixed to a single

value in all simulations.

We choose horizontal resolution to be equal or slightly larger than the maximum 𝑙𝑚 over the

course of the simulation. Since 𝑙𝑚 is either predefined as a model input or computed by BPT,

setting the grid size to the maximum 𝑙𝑚 allows the inflow and outflow to be prescribed to a single

horizontal grid. Distributing the plume-driven inflow and outflow over multiple horizontal grids

is feasible but usually not necessary; numerical tests (not shown) suggest that using horizontal

resolution greater than 𝑙𝑚 will not greatly alter the near field circulation structure.

This model framework can be reverted to the Cowton et al. (2015) style plume setup by

turning off the outflow parameterization and using a VMass or Mix style coupler. When the

outflow parameterization is turned off, the upwelling algorithm terminates when 𝑍𝑃 is determined

and all outflow volume goes into the single layer near 𝑍𝑃. To increase model flexibility, we give

the option to turn the outflow parameterization on (hereinafter OP) or off (hereinafter NOP) for

each coupler option. By combining OP/NOP with HMass/VMass/Mix, in total six outflow/coupling

methods are available. Theoretically, the outflow parameterization OP is used to describe the

23

horizontal transport of the plume, which is only consistent with HMass, and coupling OP with

VMass or Mix has no physical meaning; nevertheless, vertically distributing the plume volume flux

in VMass or Mix helps to suppress numerical instability, despite it is only an engineering fix and

lacks theoretical justification. Therefore, we keep all six outflow/coupling options available to

users, and the performance of each is tested fully in the next few sections.

2.3.4 Numerical Experiments

Three groups of experiments are carried out to test the model framework. The first group (Ex

1) uses a 2-D setup, aiming to demonstrate the outflow parameterization and to produce an

outflow best resembling the tank experiments of Ching et al. (1993). The grid is oriented in X-Z

direction; horizontally, the spatial resolution d𝑥 is uniformly 300 m, and the total length of model

domain is 30 km. Depth is uniformly 200 m with 40 vertical layers, with increased vertical

resolution in the top 100 m. Initially the salinity is stratified in two-layers; the lower layer (below

50 m) salinity 𝑆0 is fixed to 35 PSU, and a variety of surface salinities 𝑆1 = 5, 15,30, 33, 34 PSU

are used to produce variation in background stratification. Vertical salinity structure across the

halocline is represented by a hyperbolic tangent function 𝑆(𝑧) = 𝑆0 + 0.5(tanh(0.2𝜋(𝑧 +

50)) + 1)(𝑆1−𝑆0). Subglacial discharge is added from the bottom of water column (200 m) at

𝑥 = 0, and the other end (𝑥 = 30 km) an open boundary condition is used to allow the plume to

exit the domain. A range of subglacial discharge values 𝑄0 = 10, 25, 50, 100 m3/s are used as

another parameter to modify the Richardson number 𝑅𝑖. In total 24 experiments are carried out,

which corresponds to 𝑅𝑖 values ranging from 0.245 to 15.612.

The second group of experiments (Ex 2) extends the simulation from 2-D to 3-D to

demonstrate the direct influence of the outflow parameterization and coupling methods on the

near field circulation. The six outflow/coupling methods (HMass/OP, HMass/NOP, VMass/OP,

VMass/NOP, Mix/OP, Mix/NOP) are first tested with four barotropic timesteps (𝑑𝑡 = 1, 5, 30, 60

24

s) to demonstrate the numerical stability. In addition, two methods (HMass/OP, Mix/NOP) are

tested with three horizontal resolutions (𝑑𝑥 =100, 300, 600 m) in along channel direction to

demonstrate sensitivity to grid spacing.

For the stability (𝑑𝑡) tests, the glacier near field is represented with a 6000 × 4500 × 260 m

basin, with 40 vertical layers intensified in the top 100 m. The channel is oriented in east-west

direction, with glacier located on the west boundary. Horizontal resolution in along/cross channel

direction is 200/300 m, respectively. Subglacial discharge is added from a single grid near the

glacier centerline; initial discharge is kept constant (200 m3/s) during the simulation. Initially the

ambient water is stratified in two-layers similar to Ex 1, with surface salinity fixed to 33 PSU.

For the sensitivity (𝑑𝑥) tests, the model domain is extended to 12000 × 4500 × 260 m to allow

simulations of coarser resolution (up to 600 m); all other conditions are kept the same.

Lastly, a large grid, fjord scale setup (Ex 3) is designed to test the model performance in

quasi-realistic conditions. The fjord is simplified as a rectangular basin of 60 km by 4.2 km by

400 m with 40 vertical layers, intensified in the top 100 m; horizontal resolution is uniformly 600

by 300 m inside the fjord. Outside the fjord is a uniform shelf of 64 by 84 km, with resolution

linearly decreasing from 700 by 300 m to 3600 by 2800 m. To prevent the buoyant plume

recirculating around the fjord mouth, a southward coastal current of 1 cm/s is prescribed to

remove buoyant plume exiting the fjord, which is a reasonable value based on field observations

(Harden et al., 2014). To suppress instabilities generated in the shelf region, the east offshore

boundary is closed, forcing the coastal current to travel in north/south direction. Initial salinity

and temperature profiles are prescribed using hydrographic surveys from the KS Glacier in

Uummannaq fjord of west Greenland (Jackson et al., 2017, Hereinafter Jackson17), which

resembles a typical summer condition in Greenlandic fjords.

This final set of simulations is spun-up for 200 days without subglacial discharge; after 200

days subglacial discharge is slowly ramped up to 200 m3/s within 10 days. After the spin-up

25

period the simulation continues for 300 days; The last 200 days of model outputs are averaged as

the ‘steady state’ condition. Based on the test results of Ex 2, only two outflow/coupler methods,

HMass/OP and Mix/NOP, are tested in Ex 3.

For the above simulations, a finite-line style parameterization (Jenkins, 2011; Jackson et al.,

2017) is used for the entrainment parameterization. Vertical mixing is parameterized using the k-ε

closure scheme (Warner et al., 2005); horizontally a grid-scaled harmonic viscosity is used to

suppress grid-scale noise. The MPDATA (Smolarkiewicz & Margolin, 1998) advection scheme is

used to guarantee tracer positivity. To track the trajectory of outflow, dye is released into the

subglacial discharge to mark the buoyant plume.

2.4 Results

2.4.1 2-D Simulation

In this section, we focus on the simulation carried out with outflow/coupler method

HMass/OP. Figure 2.3 shows snapshots of the outflowing plume in the first 12 hours of one single

simulation (𝑄0 = 50 m3/s, 𝑆1 = 33 PSU). Once the simulation initiates, the buoyant plume forces

downstream outflow around the pycnocline (50 m), and the outflow is compensated by inflows

from both above and below. Since the plume is a uniform water mass, it generates density

anomaly of opposite signs near the pycnocline, which travels downstream with the plume. The

negative density anomaly, however, propagates faster than the positive anomaly. It travels

downstream with internal wave speed 𝐶𝑝 = √𝑔′𝐻1𝐻2

𝐻1+𝐻2 (Figure 2.3), suggesting the plume excites

a ‘bow wake’ propagating ahead of the outflow.

Since the outflow parameterization is strongly dependent on the Richardson number 𝑅𝑖

defined by Equation (2.1), the outflow structure is expected to vary with respect to both 𝑄0 and

𝑆1. Figure 2.4 shows the modeled outflow at hour 12 in all simulations. At high 𝑅𝑖 values (𝑅𝑖 >

26

1.6), the outflow is almost strictly below the pycnocline, which is compensated by a deep return

flow. The strong density jump prevents the plume water from mixing with the top layer, and very

little momentum penetrates across the pycnocline. In some cases (e.g., 𝑄 = 100 m3/s, 𝑆1 = 5

PSU), a separate circulation develops in the top layer only, but the structure is not consistent and

is not well correlated with the main outflow. At intermediate 𝑅𝑖 values (0.4 < 𝑅𝑖 < 1.6), the

outflow partly penetrates into the top layer, and return flows are generated both above and below.

One exception is 𝑆1 = 34 PSU and 𝑄0 = 25 m3/s, in which case the stratification is too weak and

the plume quickly mixes with the top layer, thus the surface return flow is not identifiable. When

𝑅𝑖 is small (𝑅𝑖 < 0.4), the upwelling plume penetrates fully into the top layer, and a surface

outflow is formed. In this case, the major pycnocline is the base of the plume instead of the

original density jump. Therefore, the actual 𝑅𝑖 is defined using the density difference between

plume and ambient water, which is higher than the background stratification.

27

Figure 2.3. Development of the outflowing plume during the first 10.5 hours, modeled with

method HMass/OP. Upper panels are subglacial discharge dye concentrations (color) and

velocities (vector); lower panels are density (contour) and its anomality from initial condition

(color). Initially the contour lines are evenly spaced of 25 m intervals. Triangles and dash lines

mark the theoretical position of signal travels with internal wave speed.

28

Figure 2.4. Subglacial discharge dye concentration (color) and velocity (vector) of all

experiments at hour 12.

The modeled outflow structure, as a function of 𝑅𝑖, agrees well with the observations of

Ching et al. (1993). The modeled subglacial dye distribution well resembles that of the tank

experiments, and so does the velocity profiles. Even though the small turbulent structures are not

directly captured by the model, the outflow is able to generate enough shear to mix with ambient

water, which to some degree accounts for the mixing caused by turbulent eddies.

When applied to a continuously stratified condition, the parameterization is also able to

generate an outflow structure well resembling the laboratory observations of Baines (2002) (not

shown), which is carried out in conditions of constant 𝑁2. This suggests that this outflow

parameterization is applicable in continuously stratified water, despite the two-layer fluid

assumption.

29

Similar experiments are also carried out using the Mix/NOP parameterization. In general,

Mix/NOP produces similar results in terms of baroclinic flow structure, thus they are not reported

here. The primary differences are shown in 3-D simulations, which will be reported in the

following two sections.

2.4.2 3-D Small Grid Simulation

When a buoyant water mass such as subglacial discharge plume emerges in a 3-D basin,

baroclinic pressure gradient drives it to spread radially while the water mass flows downstream;

in addition, the outflowing plume is deflected by Coriolis force and is not symmetric around the

centerline. In this section we focus on the outflow in the near field within ~5 km from glacier

front. In later sections we will briefly show and discuss the evolving plume as it travels further

downstream where it develops a regime similar to a coastal current.

The velocity fields at 50 m, which is the pycnocline and outflow core depth, after one model day

are shown in

Figure 2.5 to demonstrate the initial pathway of outflowing plume. In general, HMass

produces a jet-like flow with strong momentum in the along channel direction, which is quickly

diverted to the south boundary under influence of the Coriolis force. Without the outflow

parameterization, HMass/NOP drives a strong outflow that occupies the top 150 m in the near field

(Figure A.4 in Appendix A.1). This is a result of the strong shear generated in the near field

without the outflow parameterization; the large volume flux that goes into a single layer forces

unrealistically strong vertical shear, which causes the turbulent closure scheme to overestimate

mixing in downstream grids.

On the other hand, as expected, VMass and Mix produce an outflow that initially spreads

laterally and fills the width of the domain within 2km from the glacier. Further downstream the

Coriolis force diverts the flow to the south wall, forming a coastal current. Overall, the flow

30

structure is significantly different from that of HMass. However, solutions of VMass and Mix are

less sensitive to the outflow parameterization than HMass. When numerically stable, VMass/OP,

VMass/NOP, Mix/OP and Mix/NOP produce very similar results.

In terms of numerical stability, in general Mix is the most stable among the three coupling

methods, while VMass performs poorly overall. At 𝑑𝑡 = 5, even though VMass/NOP still produces

solution of a surface plume, a detailed examination shows vertical fluxes has violated the CFL

condition and the solution will soon diverge. Similarly, VMass/OP begins to diverge at 𝑑𝑡 = 10

and produces slightly different solution in the near field; the reason that a solution is still

produced is that vertical mixing partly neutralizes the instability caused by vertical advection.

However, over longer term the instability will continue to develop. Overall, the two most

promising method for further applications are HMass/OP and Mix/NOP; HMass/NOP forces

unrealistic vertical shear overestimate mixing, while VMass/OP, VMass/NOP and Mix/OP produce

similar solutions to Mix/NOP but are numerically less stable.

Lastly, the outflow/coupling methods are tested with respect to along channel grid resolution.

Since HMass/OP and Mix/NOP are the two best performing methods, only the results of these two

are shown in Figure 2.6. The comparison suggests that in general, model performance of

HMass/OP is less sensitive to 𝑑𝑥. At very high resolution (𝑑𝑥 = 100 m), the velocity fields

modeled by HMass/OP and Mix/NOP show some resemblance, and in both cases subglacial dye

does not reach the northern boundary. As 𝑑𝑥 increases, the solution begins to diverge for

HMass/OP and Mix/NOP. At the coarsest resolution (𝑑𝑥 = 600 m), HMass/OP produces solutions

very similar to the high-resolution cases in velocity fields; in addition, the main path of the

plume, which is marked by the 0.022 contour in dye concentration, does not change significantly

from high resolution cases. On the other hand, in Mix/NOP, as 𝑑𝑥 increases, the plume tends to

spread in cross channel direction. Subglacial dye reaches the north bound as opposed to the

31

highest resolution case; the plume becomes more diffusive in the initial outflowing stage, as

suggested by the distribution of subglacial dye concentration.

Figure 2.5. Modeled velocity field (vector: 𝑢/𝑣, color: 𝑤) at 50 m. Horizontally the plots are

aligned with respect to baroclinic time step lengths (𝑑𝑡); vertically the plots are aligned with

respect to outflow parameterization and coupling methods.

32

Figure 2.6. Modeled velocity fields (upper, vector: 𝑢/𝑣, color: 𝑤) and subglacial discharge dye

concentration (lower) at 50 m. Contours in the lower panels are dye concentration of 0.022 kg/m3.

Horizontally the plots are aligned with respect to grid resolution in along channel direction (𝑑𝑥);

vertically the plots are aligned with respect to outflow parameterization and coupling methods.

2.4.3 KS Glacier Simulation

In Ex 3, since the primary forcing (subglacial discharge) is constant after a 10-day ramp up

period, the response is also expected to be in steady state over longer modeling time. In the near-

field (within 10 km of glacier front), the outflow is relatively steady with little temporal

variability; however, in the far-field (~10 km away from glacier front), the downstream velocity

shear generates strong oscillations (not shown), which could be a result of baroclinic instability

developed in the fjord channel. This eddy-like motion in the far-field is beyond the scope of this

work, and the dynamics will be discussed in future works; instead, in this study we focus on the

near-field velocity structure driven by subglacial discharge. To remove temporal variability, the

modeled velocity and tracer fields are averaged over the last 200 model days to get a quasi-steady

state solution.

33

Figure 2.7. Velocity field modeled by HMass/OP (a, b) and Mix/NOP (c, d) and from observation

(e) in the near field. (a, c) Velocity averaged over the top 50 m in the near field (0-10 km from

glacier). White vectors are Sect 7 of the KS-1 observations. The observations are rotated to align

with modeled velocity direction. (b, d) Cross channel sections at 3 km from the glacier, showing

component of velocities along principal axis from model (color) and observation (contour).

Contour intervals are 0.05 m/s; the thick line marks zero and solid/dashed lines are

positive/negative values. (e) Observations of cross channel velocity (Sect 7 of KS-1), same as the

contour lines in (b, d).

A comparison between models and observation is shown in Figure 2.7. Since the geometry

of KS glacier front is very complex, and the fjord channel connected to KS glacier is curved with

a complex coastline, the ‘principal axis’ of the channel is difficult to be determined for alignment

with the idealized model topography; instead, we rotate the observed velocity vector to align with

the primary direction of outflow in the model. Overall, HMass/OP is able to reproduce the strong

34

outflow extended from near surface to depth of 50 m. The outflow in the observations and

HMass/OP is predominantly in the along channel direction and carries strong momentum. Below

the strong outflow is a compensatory inflow, extending from 50 to 150 m. On the other hand,

Mix/NOP produces a very different flow structure; the outflow bifurcates from the point source

and follows both the north and south walls of the fjord, while an inflow is formed in the middle.

The outflow extends from near surface to roughly 50 m; the inflow extends deeply to 150 m,

decaying with depth.

Jackson et al. (2017) calculated the volume flux of outflowing plume by integrating the

transport within the 0.03 m/s velocity contour line that encloses the maximum velocity, and

reports a mean outflow of 7200 ± 500 m3/s averaged over 8 sections in which the plume

structure is prominent. Using the same method (integrating within 0.03 m/s velocity contour,

enclosing the plume core for HMass/OP, or the two plume cores for Mix/NOP), modeled velocity

field yields a total outflow volume flux of 8800 and 10100 m3/s in HMass/OP and Mix/NOP,

respectively, suggesting both model setups overestimate the total flux. This overestimation from

model could be attributed to the uncertainties of the model, which are discussed in Section 2.5.2.

35

Figure 2.8. Velocity and tracer fields modeled with HMass/OP (a, b, c, d, e) and Mix/NOP (f, g, h,

i, j) over the entire fjord domain. (a, f) Velocity averaged over the top 50 m, similar to Figure

2.7, from glacier front to fjord mouth. (b, g) Velocity profiles in along (U) and cross channel (V)

direction, averaged over the middle fjord (bounded by dashed lines in a, f). (c, h) Cross channel

sections averaged over the middle fjord region, showing component of velocities in along channel

direction. (d, i) Temperature profiles averaged over the middle fjord, of initial and modeled

states. (e, j) Cross channel sections averaged over the middle fjord, showing temperature anomaly

from initial condition (color) and subglacial discharge dye concentration (contour).

In the middle fjord region, the responses driven by HMass/OP and Mix/NOP show more

similarities compared with the near field (Figure 2.8). When averaged between 20-40 km, the

strong downstream flow is focused on the south wall of the fjord, and the return flow occurs near

the north wall (Figure 2.8, c, h). The along channel velocity profile (Figure 2.8 b, g) shows a

36

three-layer exchange flow, with a strong outflow from roughly 5 to 50 m, and two return flows

above and below the outflow. The top layer (above 5 m) contributes very little to the exchange

volume flux, and the outflow is primarily compensated by the inflow at depth. When traveling

downstream, the outflow mixes with fjord water below the plume, and subglacial dye is diffused

to deeper depth (Figure 2.8 e, j), especially in HMass/OP; in both cases subglacial dye mixes to

~100 m deep, greater than the bottom of outflowing plume. The plume homogenizes the

subsurface temperature, creating a negative temperature anomaly near surface and a positive

anomaly below the plume. Since subsurface mixing is more prominent in HMass/OP, the

temperature anomaly also reaches deeper, showing a greater influence at intermediate depth.

Modeled velocity fields (Figure 2.8, b, c, g, h) suggest that the outflowing in HMass/OP

extends deeper than Mix/NOP; near the south wall, the outflow expands to 80 m in HMass/OP

(compared with 60 m in Mix/NOP). On contrary, maximum outflow speed is found at 14 m in

HMass/OP, which compared with Mix/NOP (18 m) is slightly shallower. In addition, below the

outflow, a much stronger return flow is formed in HMass/OP, which suggests that more mixing is

generated and stronger exchange flow is established. In the middle fjord section, the integrated

outflow volume fluxes are 9700 and 6900 m3/s in HMass/OP and Mix/NOP, respectively.

Overall, in the near field, the model setup HMass/OP produces outflow better resembling the

observation. In the middle fjord, the along channel flow structure show more similarity in the two

setups; HMass/OP generates more mixing at subsurface, which results in stronger exchange flow

and greater downstream volume flux.

2.5 Discussion

2.5.1 Outflow Parameterization Methods

Using in-situ observations from a tidewater glacier fjord, we have demonstrated that ROMS-

ICEPLUME is able to reproduce the structure of an outflowing plume generated by subglacial

37

discharge. In the previous section we have proposed that the two recommended model setups are

HMass/OP and Mix/NOP; both are numerically stable and produce circulation resembling

buoyancy driven exchange flow.

Even though both HMass/OP and Mix/NOP produce reasonable results, the forms of

circulation in the two simulations are very different. HMass/OP drives a strong jet-like buoyant

flow from the point source, which travels predominantly in the along channel direction and

follows the south wall; Mix/NOP produces a pure buoyancy-driven gravity current, which

spreads in cross channel direction and travels downstream along both walls. The correlation

coefficients of velocity modeled by HMass/OP and observation of each survey section range from

0.24 to 0.68 and average 0.51, which is significantly greater compared with Mix/NOP (-0.22 to

0.38, average 0.04).

While limited data prevents a detailed quantitative assessment of the efficacy of these

parameterizations, we argue that based on the available data that HMass/OP is advantageous over

Mix/NOP. Here we attempt to interpret the results and explain why HMass/OP performs better.

Near a strictly vertical glacier front, the buoyant plume initially travels only in vertical direction,

and when it is released from glacier front, the plume should spread radially in the horizontal

direction, which is better prescribed by Mix/NOP; meanwhile, if the glacier front is inclined, the

plume leaves the glacier front with horizontal momentum and tends to travel in direction

perpendicular to glacier front, which is better represented with HMass/OP. The model results

suggest that the outflowing plume carries initial momentum in the along channel direction and is

a jet-like buoyant flow, which is potentially the result of an inclined glacier front. This flow

structure is also observed in other tidewater glacier fjords (Motyka et al., 2013; Mankoff et al.,

2016; Kienholz et al., 2019; Jackson et al., 2020), suggesting a consistent pattern in outflowing

subglacial discharge plumes.

38

The geometry of glacier fronts varies constantly due to melting and calving processes when

the glacier interacts with ambient water, and the rates of glacier undercutting cannot be easily

prescribed with an analytical model. Using a 1-D idealized model, Slater et al. (2017) concluded

that subglacial discharge enhances subglacial melting behind the buoyant plume, and therefore

promotes glacier undercutting near the point source. Discharge driven undercutting is also

observed in multiple glacier systems (Fried et al., 2015; Rignot et al., 2015; Sutherland et al.,

2019). This has two consequences: (1) The new geometry redirects the buoyant plume and

transfers vertical momentum to horizontal direction, and when the plume exits the cavity it has

already primarily travels in horizontal direction. (2) The cavity restrains the plume in a narrower

space compared with the rest of glacier front, prevents the plume from spreading in the cross-

channel direction, and forces the flow to move primarily in a direction normal to the glacier front

(Figure 2.9). Over time the buoyant plume continues to melt the glacier behind, undercuts the

cavity deeper into the glacier and the outflowing plume gains more horizontal momentum. Even

when the glacier is initially vertical, the buoyant plume will change the geometry of glacier and

eventually develop into a jet-like flow as shown in the observation. This process will continue

until calving removes ice mass around the cavity.

39

Figure 2.9. Evolution of buoyant plume near a vertical (left) and calved (right) glacier front.

2.5.2 Model Uncertainties

This study highlights the strong control of subglacial buoyancy source to the near-field

circulation in tidewater glacier fjords. In this section we discuss uncertainties associated with the

ICEPLUME module, and the potential impacts on the modeled near field circulation. Due to lack

of observation and the chaotic nature of subglacial discharge plume, these uncertainties cannot be

eliminated in the current model setup. The goal of this section is to remind future researchers to

revisit these topics as our knowledge of subglacial discharge plume advances.

Firstly, the upwelling parameterization based on BPT has been previously used in several

studies, thus its uncertainties have been discussed in those works. One of the major sources of

uncertainty is the configuration of plume’s vertical structure, more specifically the choice of a

line style plume vs. axisymmetric style plume or other prescribed forms (Jackson et al., 2017).

Each plume style predicts a unique entrainment volume flux and tracer concentration, which

determines the initial exchange flow near the glacier front. The choice is rather arbitrary, but

basic in-situ surveys could provide information to inversely deduce it from several prescribed

forms. The procedure is described in detail in Jackson et al. (2017), and the result of which is

40

used in Ex 3 to force the simulation. Hence, the reliability of the model depends on the in-situ

measurements and the deduction procedure in the specific system.

There is still an ongoing investigation on the choice of optimal plume style in realistic

applications. When BPT is first applied to subglacial discharge plume (Jenkins, 1991), it is

assumed the buoyant source is uniformly released and generates a line style plume, which is

appropriate for large ice shelves or plumes driven primarily by basal melting. In Greenlandic

glacier fjords, subglacial discharge is mostly drained from channelized systems (Fried et al.,

2015; Mankoff et al., 2016), and the axisymmetric form is more commonly used. The

development of axisymmetric plume is investigated by various modeling and experimental

studies (Kimura et al., 2014; Ezhova et al., 2018; McConnochie et al., 2020), but none of them

includes the undercutting of glacier front (Fried et al., 2015; Rignot et al., 2015; Sutherland et al.,

2019). The channel formed by undercutting (Figure 2.9) could restrict the plume to spread in

cross channel direction, thus force a line style plume along the glacier front before it outflows

into the fjord. This agrees with in-situ observation of Jackson et al. (2017), where the volume flux

and tracer concentration of a buoyant plume is better predicted by the line-style plume theory.

This single case is not sufficient to prove that line-style plume is more advantageous over

other plume forms; rather, it suggests that the structure of buoyant plume is complex and cannot

be simplified with one single form. When applying ROMS-ICEPLUME in other systems, it is

advantageous if data exists to assess if the discharge is best characterized by a line or point

source.

Secondly, plume model parameters, for instance, subglacial discharge rate, are difficult to

obtain and often inferred from other type of measurements or products, thus the uncertainties in

them propagate into the model parameters and eventually influence the model performance.

Subglacial discharge rate is estimated from hydrology models (e.g., Chu, 2014; Beamer et al.,

2016; Noel et al., 2018), or hydrographic measurements by calculating ‘freshwater flux’ or other

41

mixing-based formulas (Jenkins, 1999; Jackson et al., 2017). These estimates are crude due to the

complexity of subglacial drainage system (Chu, 2014), and are considered a great source of error

other than the uncertainties of parameterization algorithm itself.

In the outflow parameterization, the key parameter of the algorithm is the advance speed 𝑈𝐷

of the outflow. ROMS-ICEPLUME provides two methods to approximate 𝑈𝐷, (1) the Ching et al.

(1993) parameterization, and (2) the internal wave speed approximation. The Ching et al. (1993)

parameterization is the default algorithm and recommended in most applications. The internal

wave speed approximation assumes outflow travels with internal wave speed 𝐶𝑝 = √𝑔′𝐻1𝐻2

𝐻1+𝐻2,

which requires a two-layer approximation of the ambient stratification, similar to the Ching et al.

(1993) parameterization. The expression suggests that 𝐶𝑝 is highly sensitive to the layer

thicknesses (𝐻1, 𝐻2). In tidewater glacier fjords, the strongest stratification is usually found near

surface, thus 𝐻1 tends to be much smaller than 𝐻2, and 𝐶𝑝 is heavily dependent on 𝐻1. A small

variation in 𝐻1 leads to great variability in 𝐶𝑝, and when the surface layer vanishes 𝐶𝑝 converges

to 0. This leads to great instability when coupling the buoyant plume model with ROMS, which is

often problematic in applications with weak stratification.

In addition, using internal wave speed suggests the signal travels downstream purely as

gravity current, and that there is no initial horizontal momentum when the buoyant plume

detrains. This assumption is only consistent with the couplers VMass and Mix, thus is incompatible

with HMass. In idealized experiments, 𝐶𝑝 can be used with VMass or Mix to be an alternative

configuration from the default one used in this work; in realistic applications with continuous

stratification, due to the instability issue we strongly suggest not to use 𝐶𝑝 to approximate 𝑈𝐷.

In addition to melting driven directly by the upwelling plume, melting of the ambient glacier,

which is sometimes referred to as the ‘ambient melting’, is another source of buoyancy in the

glacier near field. Recent attempts to parameterize ambient melting in GCM include Cowton et al.

42

(2015), where the three-equation formula is applied to the glacier/ocean boundary. The friction

velocity is approximated with modeled velocity from adjacent ocean grid; this is known to be an

underestimation since the model cannot resolve the small-scale turbulence near the ice/ocean

boundary, which is the primary driver of melting (Magorrian & Wells, 2016; Slater et al., 2018;

Jackson et al., 2020). Field observations (Sutherland et al., 2019) also shows that the background

melting is up to two order of magnitude greater than coupled plume-melt theory predicted.

Although attempts of engineering fixes have been applied to artificially increase ambient melt

rate in models (e.g., Cowton et al., 2015), they are not validated and thus are not recommended in

applications where ambient melting is considered important.

2.6 Conclusion

In order to better understand subglacial discharge driven flow in tidewater glacier fjords, a

new model framework, ROMS-ICEPLUME is developed by coupling the upwelling and outflow

parameterizations with a hydrostatic general circulation model. The model is composed of (1) the

general circulation model ROMS, (2) the BPT module adopted from Jenkins (2011); Cowton et

al. (2015), (3) outflow parameterization option OP developed from Noh et al. (1992); Ching et al.

(1993), and (4) one of three coupler options (HMass/VMass/Mix) to integrate the parameterizations

with ROMS.

The outflow parameterization OP uses an empirical function to determine the nose velocity

of outflowing plume, and distributes the outflow vertically in several model layers. The coupler

options provide different schemes to incorporate plume driven momentum/tracer fluxes in model

grids: HMass/VMass uses horizontal/vertical mass fluxes to prescribe the momentum/tracer fluxes,

respectively, while Mix computes the distortion of isopycnals internally and do not add extra

momentum fluxes. The performance of outflow parameterization and coupling methods is tested

with idealized numerical experiments.

43

To validate the module, background stratification and subglacial discharge rates measured or

inferred from Jackson et al. (2017) are used to setup and force a semi-realistic simulation.

Model/observation comparison suggests that a combination of outflow parameterization OP and

HMass coupler is able to reproduce the strong outflowing plume in the near field and the inflow at

depth. Due to the chaotic nature of buoyant plume, there are still some uncertainties associated

with the parameterizations that cannot be eliminated; these uncertainties should be addressed in

future studies as our understanding of circulation in tidewater glacier fjords advances.

44

Chapter 3 - Buoyancy Driven Circulation and Heat Budgets in

Tidewater Glacier Fjords

3.1 Abstract

Using a coupled ocean/buoyant-plume model, the development of subglacial discharge plume

in an idealized fjord is simulated. The plume travels along the southside of the fjord channel, and

to a great extent resembles the structure of a coastal current. The physical properties of

outflowing plume are estimated using model fields and compared with that derived from a

shallow water model. We find that the outflow structure is dependent on the location of subglacial

river, and plume properties are better predicted by the shallow water model when discharge enters

from the southside of the channel. Heat budgets analysis highlights the role of discharge-driven

exchange flow in removing submarine melt water from glacier front. This mechanism in theory

enhances submarine melting, but it is not well reflected in the modeled melt rates due to

limitation of the melting parameterization.

3.2 Introduction

Over the last two decades, the Greenland Ice Sheet (GrIS) has been losing mass at accelerated

rate (Shepherd et al., 2012). Recent observations suggest that 66% of the GrIS mass loss is

attributed to the dynamic change of tidewater glaciers (marine-terminating glaciers), producing a

~9.1 mm increase in sea level from 1978 to 2018 and roughly half of this sea level increase

occurring in the last 8 years (Mouginot et al., 2019).

The two dominating mechanisms of tidewater glacier mass loss are iceberg calving and direct

submarine melting. Due to the difficulty in acquiring field data near a glacier front, direct

observations of glacier mass loss are sparse. Previously iceberg discharge is indirectly inferred

from measuring calving-generated surface gravity wave (e.g., Minowa et al., 2018) or using

45

satellite images (e.g., Moyer et al., 2019); submarine melt rate is estimated using freshwater or

noble gas mixing model (Beaird et al., 2017, 2018), or boundary layer parameterizations such as

the three-equation formula that was first developed for long, extended ice shelves (Holland &

Jenkins, 1999; Cowton et al., 2015). Even though measurement of gravity wave packets is

accurate in representing the frequency of calving events, it remains challenging to estimate the

total ice volume discharge of each calving event. Recent observations also suggest that for

vertical or inclined tidewater glaciers, the three-equation parameterization tends to underestimate

submarine melt rate by up to two order of magnitude (Minowa et al., 2018; Sutherland et al.,

2019). Therefore, acquiring glacier mass loss rate is still a challenging task in the oceanography

community.

From a bulk point of view, submarine melting produces negative heat flux and lowers the

overall heat content. Over seasonal scale, the negative flux is primarily balanced by oceanic heat

sources. On the Greenland continental shelves, the relatively warm and salty subtropical Atlantic

Water (AW) lies below the cold and fresh polar-origin water (PW), fills the deep basins of most

Greenlandic fjords (D. A. Sutherland & R. S. Pickart, 2008; Straneo et al., 2011). The efficiency

of glacier melting is thus influenced by the availability of AW to tidewater glaciers, as suggested

by long-term observations of oceanic properties and glacier behavior (Holland et al., 2008;

Carroll et al., 2017).

There are still many critical but poorly understood mechanisms that control exchange

between the continental shelf and marine terminating glaciers. In Greenland, the warm AW must

transit through the deep, narrow fjord basin before it gets in contact with glaciers. Heat transport

through fjords is controlled by both oceanographic and bathymetric factors. Firstly, freshwater

discharge drives estuary-like circulation in fjords and accelerates heat transport from ocean to

glaciers (Motyka et al., 2003; Motyka et al., 2013; Carroll et al., 2017). Other atmospheric and

oceanic controls, including shelf forcing (e.g., Jackson et al., 2018), wind forcing (e.g., Spall et

46

al., 2017) and tides (e.g., Carroll et al., 2017), also influences the general circulation and heat

transport in fjords. Bathymetric sills may form near fjord mouth, limiting exchange and blocking

entrance of AW at depth (Farmer & Freeland, 1983; Schaffer et al., 2020), which makes glaciers

less vulnerable to warm water flux. This mechanism is often less important in Greenland since the

sills are usually several hundred meters deep, but is frequently found in other locations (e.g.,

Valle-Levinson et al., 2006).

Freshwater enters glacial fjords in several ways. Apart from iceberg calving and submarine

melting, freshwater discharge is one of the primary sources of buoyancy. Surface melt water of

the GrIS sinks through the porous glacier ice to the base of the glacier and enters fjords from the

glacier grounding depth, which is referred to as subglacial discharge (Bartholomaus et al., 2015).

The buoyant freshwater flux drives strong upwelling plume near the glacier front and mixes

intensively with fjord deep water and forms the subglacial discharge plume; the dynamics of

subglacial buoyant plume is comprehensively review by Hewitt (2020). The most commonly

acknowledged theoretical framework of plume evolution is developed by Jenkins (2011), which

combines the buoyancy-driven convection parameterization (Morton et al., 1956) with melting

parameterization (Holland & Jenkins, 1999) to predict the plume entrainment rate and glacier

melt rate as the plume rises.

Subglacial discharge and surface runoff both drive estuarine-like circulation and facilitate

heat exchange between fjord and continental shelf. Like typical estuaries, surface river runoff

mixes with fjord water when it flows downstream, and the exchange flow satisfies the Knudsen

relations (Knudsen, 1900). In estuarine systems mixing is broadly distributed and generated by

bottom boundary layer tidal stresses (Peters, 2003). Due to the great depth in fjord, mixing of

surface runoff is often characterized as weak (Geyer & Maccready, 2014) compared with shallow

riverine estuaries. On the other hand, in fjords, subglacial discharge generates mixing through

vigorous convective upwelling, which mostly occurs near the ice/ocean boundary and is directed

47

from grounding line depth to surface. This mechanism increases the overall kinetic energy near

glacier front and enhances submarine melting (Motyka et al., 2013).

Since subglacial discharge plume overturns deep water to shallower depth, the residence time

of deep water is expected to be greatly reduced (Gladish et al., 2015) during discharge periods.

The quick renewal of deep AW provides heat that is available for submarine melting. Since

subglacial discharge operates on seasonal scale, compared with other synoptic scale events (e.g.,

Arneborg, 2004; Spall et al., 2017) this mechanism is persistent in transporting warm AW

upstream into the system over summer months to sustain both submarine melting and glacier

calving events. Moreover, the overturing circulation in fjord upwells nutrients to the euphotic

zone, making these systems highly productive (Hopwood et al., 2018).

In this study, our primary goal is to quantify the discharge driven circulation and heat

transport in tidewater glacier fjords with numerical model. The model framework is able to

isolate the component of discharge-driven plume, submarine melt and other heat sources, thus

provides more advantages over previous modeling and observational studies. The model is

coupled with subglacial discharge plume and submarine melting parameterization developed by

Holland and Jenkins (1999); Jenkins (2011); heat budgets analysis is based on the observational

study of Jackson and Straneo (2016), which uses the decomposition method of Lerczak et al.

(2006) to isolate the barotropic and baroclinic components from modeled velocity fields. In

section 2, we describe the decomposition and heat budgets analysis methods and model

configuration; in section 3, the modeled velocity and tracer fields are shown, and the results of

decomposition and heat budgets analysis are summarized. Section 4 discusses two primary topics:

(1) the shape and geometry of outflowing plume, which are summarized with three basic

parameters and compared with a shallow water model; (2) variability of each heat budget term

and their interaction.

3.3 Methods

48

3.3.1 Buoyancy Driven Circulation in Tidewater Glacier Fjords

Buoyancy driven flow forms when subglacial discharge (SGD) enters the fjord at glacier

grounding line. From glacier front to fjord mouth, the evolution of SGD is roughly a two-stage

process. The first stage is the upwelling stage, in which the freshwater upwells near the glacier

front and entrains the ambient water, forming the SGD plume. This stage is nonhydrostatic and

cannot be resolved by hydrostatic ocean models; instead, it is often approximated using the

buoyant plume theory (BPT) parameterization. Detailed description of the BPT implementation is

reported in the previous chapter.

The second stage, which is the primary research objective of this study, is the downstream

transport of SGD plume from glacier front to fjord mouth. Since the plume entrains large amount

of deep water during the upwelling stage, when it detrains from glacier front it is already well

mixed with density similar to the ambient environment. In comparison with typical fjord-like

estuaries, the plume is relatively homogenous, and sometime emerges beneath the heavily

stratified surface layer. The plume travels downstream in the form of a coastal current, and

recirculation redistribute the plume in cross fjord direction.

In order to quantitatively describe the barotropic and baroclinic response to plume, for any

cross-section in fjord, the velocity and tracer fields are decomposed following the method of

Lerczak et al. (2006)

𝜙0 =1

𝐴0⟨∫𝜙(𝑦, 𝑧, 𝑡)𝑑𝐴𝐴

⟩ , 𝜙𝑒 = ⟨𝐴(𝑡)

𝐴0𝜙(𝑦, 𝑧, 𝑡)⟩ − 𝜙0, 𝜙𝑡 = 𝜙(𝑦, 𝑧, 𝑡) − 𝜙𝑒 −𝜙0 (3.1)

where the angled brackets indicate temporal averaging; 𝐴 is the cross-section area; 𝜙 refers to

the velocity 𝑢 or tracer fields such as 𝜃 and 𝑆. The subscripts 0, 𝑒 and 𝑡 represent fields of mean,

exchange, and time-dependent states, respectively. Tracer transport through the cross section is

easily obtained through the integration across the fjord’s cross-section

49

𝐹𝜙 = ∫𝑢𝜙𝑑𝐴𝐴

= ∫(𝑢0 + 𝑢𝑒 + 𝑢𝑡)(𝜙0 + 𝜙𝑒 +𝜙𝑡)𝑑𝐴𝐴

≈ 𝐴0𝑢0𝜙0 + 𝐴0𝑢𝑒𝜙𝑒 + 𝐴𝑢𝑡𝜙𝑡 ≡ 𝐹0 + 𝐹𝑒 + 𝐹𝑡

(3.2)

where 𝜙 refers to the tracer fields 𝑇 or 𝑆 and 𝑢 the along fjord velocity. For the exchange and

time-dependent component, it is also useful to separate the inflowing/outflowing volume

transport and tracer properties

𝑄𝑒+ = ∫ 𝑢𝑒𝑑𝐴𝐴+

, 𝑄𝑒− = ∫ 𝑢𝑒𝑑𝐴𝐴−

, 𝑄𝑡+ = ⟨∫ 𝑢𝑡𝑑𝐴𝐴+

⟩ , 𝑄𝑡− = ⟨∫ 𝑢𝑡𝑑𝐴𝐴−

⟩

𝜙𝑒+ =∫ 𝑢𝑒𝜙𝑒𝑑𝐴𝐴+

𝑄𝑒+, 𝜙𝑒− =

∫ 𝑢𝑒𝜙𝑒𝑑𝐴𝐴−

𝑄𝑒−, 𝜙𝑡+ =

⟨∫ 𝑢𝑡𝜙𝑡𝑑𝐴𝐴+⟩

𝑄𝑡+, 𝜙𝑡− =

⟨∫ 𝑢𝑡𝜙𝑡𝑑𝐴𝐴−⟩

𝑄𝑡−

(3.3)

The barotropic transport 𝐹0 corresponds to salt and heat transports moved by the mean flow,

which is equal to the total freshwater influx from SGD and submarine melt water flux (SMW).

The exchange transport 𝐹𝑒 represents the fluxes generated by steady shear in the velocity and

tracer fields. Often vertical shear dominates the total shear transport in narrow estuaries, but in

fjord-like systems where width is of the order of the Rossby radius of deformation, horizontal

shear is also significant. The time dependent transport 𝐹𝑒 mainly describes the tidally varying

component in typical estuaries; in fjords tidal current and the corresponding tracer fluxes are low

due to the greater depth; 𝐹𝑡 describes other forms of time-dependent variability. Observations

reveal that synoptic scale shelf forcing dominates the time-dependency in Greenland Fjords

(Jackson et al., 2018); in this study variability is generated by instabilities forced by a steady

SGD.

3.3.2 Heat Budgets

The previous section introduces the decomposition method, describing components of

volume and heat transports through an arbitrary cross section in the fjord. When surface fluxes

are neglected, heat budgets in the fjord are dominated by the heat storage, heat exchange between

50

glacier/ocean boundary, and heat flux downstream. From the tidewater glacier terminus to an

arbitrary mid-fjord section, the overall volume and heat budgets are (Jackson & Straneo, 2016)

∂𝑉

∂𝑡= 𝑄𝑠𝑔 + 𝑄𝑚 − ∫𝑢 𝑑𝐴

𝐴

∂

∂𝑡∫𝜃𝑑𝑉𝑉

= 𝜌𝑐𝑝𝑄𝑠𝑔𝜃𝑠𝑔 + 𝜌𝑐𝑝𝑄𝑚𝜃𝑚 −∫𝜌𝑐𝑝𝑢𝜃 𝑑𝐴𝐴

− 𝜌𝑄𝑚[𝐿 + 𝑐𝑖(𝜃𝑚 − 𝜃𝑖)]

storage discharge melt advection latent

(3.4)

where 𝑉 is the water volume from glacier front the cross section, 𝑄𝑠𝑔 is subglacial discharge

rate, 𝑄𝑚 is submarine melt rate, 𝜃𝑠𝑔, 𝜃𝑚 and 𝜃𝑖 are temperatures of SGD, SMW and glacier ice,

respectively. 𝑐𝑝 and 𝑐𝑖 are heat capacity of water and ice, respectively; 𝐿 is the latent heat of

fusion. Taking temporal average and applying the decomposition method, the budgets equations

can be rewritten into

⟨∂𝑉

∂𝑡⟩ = 𝑄𝑠𝑔 + 𝑄𝑚 − 𝑢0𝐴0

𝜌0𝑐𝑝 ⟨d

d𝑡∫𝜃𝑉

d𝑉⟩ = 𝜌0𝑐𝑝(𝑄𝑠𝑔𝜃𝑠𝑔 + 𝑄𝑚𝜃𝑚 − 𝑢0𝐴0𝜃0) − 𝜌0𝑐𝑝∫𝑢𝑒𝜃𝑒𝐴

d𝐴 − 𝜌0𝑐𝑝 ⟨∫𝑢𝑡𝜃𝑡𝐴

d𝐴⟩ − 𝜌0𝑄𝑚[𝐿 + 𝑐𝑖(𝜃𝑚 − 𝜃𝑖)]

𝐻𝑠𝑡 = 𝐻0 + 𝐻1 + 𝐻2 + 𝐻𝑚

storage barotropic exchange time-dependent melting

(3.5)

The barotropic transport is the divergence of barotropic fluxes between the glacier boundary

and downstream boundary; exchange and time-dependent transports are the contribution of steady

and time-varying shear through the downstream boundary; the melting term is the latent heat

required to melt the glacier surface. Even though 𝑄𝑠𝑔 and 𝑄𝑚 are both time dependent in reality,

in our numerical simulations 𝑄𝑠𝑔 is constant after the ramp up period, and 𝑄𝑚 is nearly constant

over the period of interest, thus these two terms are not decomposed using Equation (3.1). When

averaged over longer time periods (> weeks) the volume is approximately conserved; therefore,

the barotropic volume flux 𝑢0𝐴0 is roughly equal to discharge rate 𝑄𝑠𝑔 plus SMW flux.

51

3.3.3 Numerical Experiment Configuration

Figure 3.1. Numerical model configuration. (a) Grid structure of the entire domain. (b) Grid near

the glacier ocean boundary. The blue tiles mark the location of the glacier front; dark blue tiles

mark the location of subglacial discharge channels. (c, d, e) initial conditions of temperature,

salinity and density, averaged from in-situ observations.

The numerical model framework is modified from a preceding study, which uses an

idealized, rectangular basin to represent the fjord channel. Details of the original model

52

configuration are provided in Chapter 2. To better resolve the structure of the outflowing plume,

horizontal resolution inside the fjord is enhanced from 600 × 300 m to 300 × 220 m in along

and cross fjord direction, respectively. The total dimension of fjord channel is 60000 × 4180 m,

equivalent to 200 × 19 grids in grid space. The rectangular fjord is connected to a shelf of

uniform depth, with resolution linearly increasing from 300 × 220 m to 2800 × 1470 m. Since

bathymetry is not the focus of this study, a uniform depth of 400 m is used over the whole model

domain. The simulation is initialized using temperature and salinity profiles collected during in-

situ surveys near the KS glacier in Uummannaq fjord of west Greenland, which resembles a

typical summer condition in Greenlandic fjords. To prevent the buoyant SGD plume recirculating

around the fjord mouth, a southward coastal current of 1 cm/s is prescribed to remove the plume

water exiting the fjord, which is a reasonable value based on field observations.

This study explores two model parameters, both of which have significant influence on the

mass and heat exchange in fjords. The first parameter is discharge rate 𝑄𝑠𝑔; since SGD is the

primary driver of circulation, a wide range of 𝑄𝑠𝑔 (0, 5, 10, 20, 50, 100, 150, 200, 300 m3s-1) is

covered to test the sensitivity of SGD plume transport to buoyancy injection. the second

parameter, which is not investigated in previous studies, is the distance of discharge channel from

one side of the fjord wall. This parameter is addressed either as distance 𝐿𝑟 from the fjord south

boundary or in model grid space; where 𝐿𝑟1 = 110 m is the grid adjacent to the southside wall,

𝐿𝑟10 = 2090 m is the centerline grid, and 𝐿𝑟19 = 4070 m is the grid adjacent to the northside

wall (Figure 3.1). The number in subscript is the grid index from the southside wall. For

simplicity, we neglect the interaction between several SGD plume and only apply one discharge

location in each experiment. In addition to direct discharge, melting of the glacier front is also

parameterized using the three equation formulation (Holland & Jenkins, 1999); the total SMW

flux is composed of melt driven directly by the upwelling subglacial buoyant plume, hereinafter

plume melt (SMWP), and melt away from the plume, hereinafter ambient melt (SMWA).

53

For each experiment, the simulation is spun-up for 200 days without SGD, and after 200

days the discharge is slowly ramped up to 𝑄𝑠𝑔 over the course of 10 days. After spin-up the

simulation continues for another 200 days. To illustrate the pathway of SGD and SMW and

estimate their residence time in fjord, three types of passive tracers are used for freshwater

sources of SGD, SMWP and SMWA, respectively.

3.4 Results

3.4.1 Circulation and Plume Geometry

Figure 3.2 shows the cross-sectional velocity and SGD tracer fields in experiment 𝑄𝑠𝑔 = 50

m3s-1 and 𝐿𝑟1 = 110 m. The area enclosed by the solid black line in Figure 3.2 b marks the

outflowing plume, defined using two criteria: (1) 𝑢 > 𝛼𝑢𝑝 and (2) 𝑐𝑠𝑔 = 𝛼𝑐𝑝, in which 𝑢 and 𝑐𝑠𝑔

are velocity and SGD tracer concentration on the averaged transect; 𝑢𝑝 and 𝑐𝑝 are maximum

velocity and tracer concentration, respectively. The threshold parameter 𝛼 is arbitrary, and

repeated tests suggest that 𝛼 = 0.1 is a reasonable value to highlight the plume geometry. In

addition, 𝛼 = 0.02 and 𝛼 = 0.5 are also used to build a confidence interval for the plume

geometry, which is shown as dashed lines in Figure 3.2.

The maximum velocity of the SGD plume 𝑈𝑝 is near the southside wall, at depth of ~50 m.

We refer to this depth as the ‘core’ depth of the plume. To better describe the geometry of the

outflowing plume, we define the depth scales 𝑍𝑝 to be the distance from grounding line depth to

the core depth; 𝐿𝑝 and 𝐷𝑝 to be the width and thickness of the plume, respectively. In addition,

the total volume flux 𝑄𝑝 and mean density 𝜌𝑝 of the plume is determined by integrating over the

plume region. Figure 3.3 to Figure 3.5 show the velocity and tracer fields from 9 experiments,

and the summary of plume physical properties are shown in Figure 3.6. In all experiments the

plume forms near the southside of the channel, regardless of the discharge location. The

outflowing plume follows the southside wall due to Coriolis force, and in the cross-fjord direction

54

the flow is to a large extent geostrophic, with a geostrophic reconstruction of the along fjord flow

nearly identical to the mean flow (Figure 3.4). The plume core is in general near the southside

wall, but not necessarily directly attached to it. The depth parameter 𝑍𝑝 is mainly a function of

SGD and is nearly independent of 𝐿𝑟 (Figure 3.7); this is because the plume outflowing depth is

determined by the density of the plume, which is primarily a function of SGD buoyancy flux. The

plume width 𝐿𝑝, however, is sensitive to 𝐿𝑟. When the discharge location is near the southside

wall (𝐿𝑟1 = 110 m), the outflowing plume is narrower, and flows faster compared with other two

discharge locations shown in Figure 3.7. 𝐿𝑝, at 𝐿𝑟4 and 𝐿𝑟7 is similar to the other ‘non-wall’

plume cases.

Figure 3.2. Modeled velocity (a), tracer (b) and density (c) fields from the numerical experiment

𝑄𝑠𝑔 = 50 m3s-1 and 𝐿𝑟1 = 100 m. All fields are temporally averaged from model day 100-200,

and spatially averaged in along fjord direction from 𝑥 = 20 to 40 km. The black triangle in (a) is

the location of discharge point; the white dot is the coordinate of maximum velocity 𝑈𝑝, which is

referred to as the ‘plume core’. Positive velocity represents flow in downstream direction. The

solid line marks the plume boundary, and the dash lines are confidence intervals. The length

55

scales ℎ𝑝, 𝐿𝑝, and 𝑍𝑝 in (b) are the thickness, width and depth of the plume, extracted from the

velocity and tracer fields. (d) schematic diagram of an outflowing subsurface plume.

When discharge location is away from the southside wall (𝐿𝑟4-𝐿𝑟19), the plume widens

horizontally. 𝐿𝑝 still increases slightly with SGD, but the overall increase is small. In all

experiments 𝐿𝑝 is roughly half of the fjord width 𝐿𝑤, suggestive the fjords width influences 𝐿𝑝,

which is explored later.

Figure 3.3. Velocity fields in the mid-fjord from 9 experiments. Colors are along-fjord velocity

while vectors are cross-fjord velocity. Horizontally the subplots are aligned with respect to

subglacial river location; vertically the subplots are aligned with SGD.

56

Figure 3.4. Geostrophic reconstructions of along fjord velocity. Subplots are arranged similar to

Figure 3.3.

57

Figure 3.5. Normalized SGD tracer. Subplots are arranged similar to Figure 3.3.

58

Figure 3.6. Physical properties (a-c: length (width, thickness, depth) scales; d: maximum

velocity; e-g: density, potential temperature, salinity; h: total volume flux) of the plume, with

respect to discharge rate. Line color represents simulations of different discharge locations, while

the black line is averaged over all 𝐿𝑟 values. Error bars are confidence interval determined by 𝛼

value. The gray lines in (f, g) are mean temperature/salinity of the initial profile, averaged over

the same depth of the outflowing plume.

A summary of the plume physical properties from all simulations is shown in Figure 3.6. In

general, 𝑍𝑝 increases with 𝑄𝑠𝑔 but is insensitive to 𝐿𝑟, while 𝐿𝑝 mainly depends on 𝐿𝑟, in

agreement with the flow structure of Figure 3.3 to Figure 3.5. The simulations of 𝐿𝑟1 is

distinctive from other 𝐿𝑟 values; the plume is more compact when the discharge is directly

released from the southside wall. Therefore, we refer to the experiments of 𝐿𝑟1 as ‘wall plume’,

to distinguish them from other simulations.

59

The plume thickness ℎ𝑝, has a decreasing trend with respect to 𝑄𝑠𝑔. The wall plume is

slightly thinner compared with non-wall plume, and the overall decreasing rate is smaller. Plume

maximum velocity 𝑈𝑝 increases with 𝑄𝑠𝑔; the wall plume is constantly faster compared with

other cases, by a factor of 5.8 at lowest 𝑄𝑠𝑔 and 1.7 at highest 𝑄𝑠𝑔. Even though the velocity of

wall plume is greater, the overall volume flux is smaller due to mixing in the non-wall plumes.

Plume density and salinity decrease with 𝑄𝑠𝑔, and are not sensitive to 𝐿𝑟; only at high

discharge (𝑄𝑠𝑔 ≥ 200 m3s-1) 𝜌𝑝 and 𝑆𝑝 begin to diverge for different 𝐿𝑟 values. In general, 𝑆𝑝 is

slightly lower than the initial salinity, which is determined by averaging over the same depth from

the initial profile. Plume temperature 𝜃𝑝 is more complex as 𝑄𝑠𝑔 increases; at low discharge

(𝑄𝑠𝑔 ≤ 50 m3s-1) 𝜃𝑝 is higher than the ambient temperature, and decreases with 𝑄𝑠𝑔 as the

upwelling buoyant plume entrains cold water from the subsurface layer; at intermediate discharge

(50 ≤ 𝑄𝑠𝑔 ≤ 150 m3s-1), 𝜃𝑝 increases with 𝑄𝑠𝑔 as the plume begin to entrain warm surface

water; at high discharge, the plume reaches sea surface and cannot further entrain surface water;

instead 𝜃𝑝 decreases with 𝑄𝑠𝑔, and becomes lower than ambient temperature.

60

Figure 3.7. Decomposition terms of volume flux (a, b, c) and temperature (d, e, f), with respect to

discharge rate. Line color represents simulations of different discharge locations, while the black

line is averaged over all 𝐿𝑟 values. The black dot marks the simulation with no SGD, only SMW.

Using Equation (3.3), the decomposed volume flux and temperature are calculated, and the

results are shown in Figure 3.7. Unlike the spatially averaged fields shown in Figure 3.2 to

Figure 3.6, the decomposition is applied to a single cross-fjord slice at 𝑋 = 30 km. The total

barotropic volume flux 𝑄0 = 𝑄𝑠𝑔 +𝑄𝑚 is not shown since 𝑄𝑠𝑔 ≫ 𝑄𝑚, thus only the smaller

SMW portion is presented.

Total SMW flux 𝑄𝑚 is mainly a function of 𝑄𝑠𝑔 and shows little dependence on 𝐿𝑟. SMWA

is relatively consistent (4 m3s-1, Figure 3.7 d, black dot), and is on the same order of SMWP flux

(0-10 m3s-1). The exchange flux 𝑄𝑒 increases nonlinearly with 𝑄𝑠𝑔; even though the velocity field

61

of wall plume is distinctive from others, the overall exchange volume flux is seemingly less

influenced by 𝐿𝑟. The time dependent flux 𝑄𝑡, due to the unsteadiness of the plume, is more

complex, showing an increasing trend for both 𝑄𝑠𝑔 and 𝐿𝑟.

In the wall plume experiments, the barotropic temperature 𝜃0 is relatively constant (1.56 ℃)

over a wide range of discharge rate, which is roughly the initial mean temperature in fjord. In

other experiments, however, 𝜃0 increases rapidly with 𝑄𝑠𝑔 from 1.56 to 1.66 ℃, then decreases to

1.54 ℃. On contrary, the exchange temperature 𝜃𝑒 first decreases with 𝑄𝑠𝑔, and then increases

slightly at higher discharge. The outflowing component 𝜃𝑒+ is constantly lower than the

inflowing component 𝜃𝑒−, suggesting the exchange flow has a positive net contribution to the

heat flux into fjord. The time dependent component 𝜃𝑡 shows no clear trend over 𝑄𝑠𝑔 and is small

compared to 𝜃𝑒, suggesting it only has a minor contribution to the overall heat flux.

𝐿𝑟 [m] 𝑄𝑠𝑔 [m3s-1] 𝐶𝑠𝑔 [10-4 kg∙m-3] 𝐶𝑠𝑚𝑤𝑝 [10-4 kg∙m-3] 𝐶𝑠𝑚𝑤𝑎 [10-4 kg∙m-3] 𝑇𝑠𝑔 [Day] 𝑇𝑠𝑚𝑤𝑝 [Day] 𝑇𝑠𝑚𝑤𝑎 [Day]

110

5 0.4 0.15 5.58 4.6 4.8 116.8

10 0.6 0.16 4.64 3.46 3.73 84.33

20 1.05 0.19 3.66 3.01 3.22 58.23

50 2.38 0.24 3.26 2.72 2.83 61.65

100 4.05 0.27 2.51 2.33 2.41 38.79

150 6.64 0.34 1.82 2.55 2.63 26.83

200 8.59 0.35 1.85 2.44 2.49 27.62

300 11.16 0.34 1.44 2.13 2.15 17.96

770

5 6.23 2.22 7.33 29.83 29.71 57.93

10 5.37 1.37 4.69 12.36 12.38 32.83

20 5.4 0.94 3.89 11.59 11.68 50.29

50 7.94 0.81 2.9 7.76 7.79 33.76

100 15.81 1.02 1.96 7.24 7.28 23.5

150 18.52 0.93 1.85 6.27 6.34 22.11

200 20.75 0.84 1.76 5.35 5.39 21.75

300 30.15 0.91 1.68 4.14 4.14 16.01

1430

5 6.85 2.43 7.61 39.42 39.08 73.27

10 7.74 1.97 5.56 18.29 18.3 37.96

20 8.16 1.42 3.89 10.46 10.5 25.54

50 10.91 1.09 2.51 7.64 7.66 21.62

100 16.85 1.08 1.84 7.11 7.14 19.3

150 23.01 1.13 1.74 6.22 6.25 17.3

200 26.4 1.07 1.34 5.02 5.04 11.98

300 30.03 0.91 1.7 4.13 4.15 15.67

2090

5 7.71 2.66 8.48 52.59 52.43 90.09

10 8.93 2.26 6.2 24.27 24.24 48.12

20 9.46 1.64 4.17 14.45 14.45 32.12

50 12.25 1.23 2.53 7.85 7.87 18.39

100 18.7 1.2 2.01 6.71 6.73 16.97

150 24.7 1.21 1.63 6.31 6.33 14.76

200 29.66 1.2 1.46 5.06 5.08 11.52

300 31.19 0.94 1.34 4.3 4.31 12.2

4070 5 7.79 2.72 8.5 52.29 51.84 74.62

62

10 8.51 2.15 5.96 26.17 26.19 43.51

20 8.77 1.51 4.7 14.59 14.59 37.22

50 14.2 1.4 3.23 10.58 10.59 25.22

100 22.38 1.43 2.61 8.67 8.68 21.89

150 25.76 1.26 2.09 6.42 6.44 17.66

200 28.52 1.14 1.54 5.81 5.81 13.78

300 29.7 0.89 1.64 4.23 4.23 14.41

Table 3.1. Mean tracer concentration (𝐶) and residence time (𝑇) of SGD, SMWP and SMWA.

Figure 3.8. Mean tracer concentration (a-c) and residence time (d-f) of SGD, SMWP and

SMWA.

Figure 3.8 shows the mean concentration and residence time of the three types of freshwater

passive tracers (SGD, SMWP and SMWA). Residence time of all three tracers decreases with

𝑄𝑠𝑔 because higher discharge drives stronger exchange flow, which is more efficient at

transporting freshwater out of the fjord. Since SGD and SMWP are both transported with the

SGD plume, these two components have roughly the same residence time; SMWA, however, has

a much longer residence time. Concentration of SGD increases quasi-linearly with 𝑄𝑠𝑔, while

63

SMW tracers show a decreasing trend with 𝑄𝑠𝑔 in general. For SGD and SMWP, the tracer

concentration and residence time of wall plume is one order of magnitude smaller compared with

non-wall plume; for SMWA, tracer quantity is invariant of discharge location, while residence

time of wall plume is one order of magnitude shorter compared with non-wall plume.

The longer residence time of non-wall plume is attributed to the recirculation of SGD plume

and the slower advective speed. In Figure 3.9 to Figure 3.12, temperature, SGD tracer

concentration and horizontal circulation in fjord are compared between experiments of three

discharge locations. In addition to passive tracers, temperature is also an indicator of SGD

pathway: when discharge is low (𝑄𝑠𝑔 = 50 m3s-1, Figure 3.9 and Figure 3.10), the plume

emerges at subsurface and does not touch the surface (Figure 3.3), with temperature higher than

the ambient environment (Figure 3.6); when discharge is high (𝑄𝑠𝑔 = 200 m3s-1,Figure 3.11 and

Figure 3.12), the plume reaches surface (Figure 3.3), and its temperature is slightly higher than

the ambient environment (Figure 3.6). For the wall plume, the plume water mass is mostly found

near the southside wall. Weak recirculation forms in the fjord channel, with velocity much

smaller than the strong outflowing plume; only a very small portion of SGD is mixed to the

northside of the channel. For non-wall plume, the outflow meanders greatly as it travels

downstream, forming strong gyres of recirculation in the fjord channel. As a result, large amount

of SGD water recirculates upstream before it is transported out of the fjord mouth. The SGD

water flushes the local water out of fjord, occupies its neutral buoyant depth and alters the local

temperature over longer term. Therefore, the total heat content in fjord is greatly modified by the

recirculation of non-wall plumes.

64

Figure 3.9. Top view of temperature and streamline at five time-slices, averaged over the depth

of plume. From left to right are experiments with different discharge locations (𝐿𝑟1, 𝐿𝑟10, 𝐿𝑟19);

from top to bottom are days after SGD is initiated. Discharge rate of all three experiments is 50

m3s-1. Blue triangle is the discharge location. Linewidth of streamline represents magnitude of

velocity.

65

Figure 3.10. Top view of SGD tracer concentration and streamline, similar to Figure 3.9.

66

Figure 3.11. Similar to Figure 3.9, of 𝑄𝑠𝑔 = 200 m3s-1.

67

Figure 3.12. Similar to Figure 3.10, of 𝑄𝑠𝑔 = 200 m3s-1.

3.4.2 Heat Flux

Heat budgets from the glacier front to a mid-fjord cross section (𝑥 = 30 km) are calculated

using Equation (3.5), and the results are summarized in Table 3.2 and Figure 3.13. Since the

primary forcing (SGD) is constant after the ramp-up period, heat fluxes are relatively stable after

the adjustment period, and the budget terms are not time dependent. Hence, we use a longer

temporal averaging span (model day 100-200) to seek the ‘steady state’ heat budgets in fjord.

𝐿𝑟 𝑄𝑠𝑔 [m3s-1] 𝑄𝑚 [m3s-1] 𝑄𝑒 [m3s-1] 𝐻𝑠𝑡 [107W] 𝐻0 [107W] 𝐻1 [107W] 𝐻2 [107W] 𝐻𝑚 [107W] 𝐻𝑟 [107W]

N/A 0 3.95 0 -48.26 -2.49 98.45 0.11 -143.02 -1.32

110

5 5.56 1083.28 -20.76 -6.7 183.42 -0.74 -201.31 4.58

10 6.34 1584.43 -14.72 -10.4 220.33 -0.49 -229.5 5.35

20 7.33 2221.34 -63.57 -17.33 210.96 0.89 -265.49 7.4

50 8.91 3346.35 -1.81 -36.91 348.16 4.78 -322.64 4.79

68

100 10.58 4533.84 -30.91 -69.79 413.5 -1.1 -382.96 9.44

150 11.58 5293.15 53.31 -101.89 558.54 2.81 -419.32 13.18

200 12.32 5869.22 2.79 -133.23 569.96 3.95 -446.1 8.2

300 13.57 6773.4 61.03 -197.28 723.39 5.06 -491.52 21.37

770

5 5.91 1077.97 21.51 -7.14 246.23 -4.62 -213.94 0.98

10 6.54 1585.76 28.89 -11.1 268.51 2.34 -236.98 6.11

20 7.5 2224.06 11.93 -17.97 284.24 12.78 -271.68 4.56

50 9.18 3394.31 17.2 -38.36 384.11 -5.19 -332.47 9.11

100 10.54 4535.23 -13.67 -70.17 425.36 2.63 -381.59 10.1

150 11.65 5304.02 -21.5 -102.59 486.02 5.01 -421.8 11.87

200 12.31 5885.61 -14.35 -132.3 539.45 12.08 -445.72 12.13

300 13.38 6793.26 -11.5 -191.55 676.85 -33.76 -484.52 21.48

1430

5 5.98 1075.07 49.93 -7.19 288.73 -16.77 -216.6 1.77

10 6.68 1586.12 3.47 -11.24 251.77 2.17 -241.93 2.71

20 7.69 2223.87 11.59 -18.64 289.64 13.28 -278.48 5.79

50 9.1 3368.99 -10.88 -38.38 340.05 7.59 -329.64 9.5

100 10.49 4525.94 -12.93 -70.03 426.34 2.54 -379.8 8.02

150 11.51 5291.54 -42.11 -101.05 469.87 -5.72 -416.78 11.58

200 12.32 5892.28 -24.93 -131.56 588.96 -39.21 -445.93 2.82

300 13.34 6788.12 37.37 -191.68 731.92 -22.88 -483.21 3.21

2090

5 6.12 1054.72 57.12 -7.27 295.23 -13.91 -221.78 4.85

10 6.79 1577.52 36.97 -11.31 300.27 -10.11 -246.02 4.14

20 7.78 2227.38 12.01 -18.68 297.56 7.25 -281.88 7.77

50 9.19 3385.01 -25.37 -38.78 325.62 14.03 -332.91 6.67

100 10.52 4528.73 -63.41 -70.23 386.21 -13.14 -380.81 14.56

150 11.51 5289.85 -26.36 -100.97 497.83 -15.7 -416.89 9.37

200 12.25 5889.22 9.19 -130.95 658.97 -87.66 -443.45 12.29

300 13.26 6784.16 44.77 -190.6 736.2 -37.31 -480.28 16.76

4070

5 6.38 1071.68 109.06 -7.46 357.97 -14.47 -230.93 3.95

10 7.32 1579.5 50.07 -11.47 307.38 13.12 -264.95 5.98

20 8.11 2220.95 -26.83 -18.71 275.99 0.55 -293.8 9.15

50 9.37 3349.57 -19.91 -38.33 351.84 -1.3 -339.26 7.15

100 10.86 4525.2 -1.08 -70.42 460.09 -7.59 -393.41 10.26

150 11.63 5281.97 36.51 -101.23 567.16 -18.91 -421.23 10.71

200 12.22 5875.66 -2.81 -130.67 574.21 -18.42 -442.39 14.45

300 13.85 6786 45.35 -191.02 740.57 -6.88 -501.56 4.24

Table 3.2. Submarine melt rate 𝑄𝑚, subglacial buoyant plume entrainment rate 𝑄𝑒, and heat

budget terms in Equation (3.5). The residual term 𝐻𝑟 = 𝐻𝑠𝑡 − 𝐻0 − 𝐻1 −𝐻2 − 𝐻𝑚.

69

Figure 3.13. Heat budgets from the glacier front to a mid-fjord section (30 km), with respect to

𝑄𝑠𝑔 and 𝐿𝑟. Lines and markers are denoted similar to Figure 3.7. The residual term 𝐻𝑟 = 𝐻𝑠𝑡 −

𝐻0 −𝐻1 −𝐻2 − 𝐻𝑚.

As 𝑄𝑠𝑔 increases, the three dominant heat budget terms are barotropic, exchange and

submarine melt fluxes. The barotropic term 𝐻0 ≈ −(𝑄𝑠𝑔 + 𝑄𝑚)𝜃0 is proportional to the total

freshwater influx and is only slightly influenced by the barotropic temperature. Submarine melt

flux 𝐻𝑚 is solely determined by submarine melt rate; both 𝐻0 and 𝐻𝑚 contribute negatively to the

heat budgets. Exchange flux 𝐻1 is positive, and to a large extent balances the heat loss of

barotropic and melt flux. The maximum exchange heat flux is 740 ×107 W when 𝑄𝑠𝑔 = 300 m3s-

1. The storage term shows no clear dependence on 𝑄𝑠𝑔, and is in general smaller compared with

the three dominant terms. 𝐻𝑠𝑡 ranges from -63×107 to 109×107 W, and is largest at low

70

discharge, since the weak exchange flow delays the system from reaching steady state. 𝐻2 is

minimal at low discharge and increases slightly with 𝑄𝑠𝑔. The residual 𝐻𝑟 is also a small term

and ranges from -1.3×107 to 21.5×107 W, suggesting the decomposition method is able to

capture the primary forms of heat flux in fjord.

3.5 Discussion

3.5.1 Geometry of SGD Plume

In this section we use a three-layer, shallow water model to derive the geometry of a

geostrophic balanced, subsurface plume and compare with length scales from the modeled fields

(Figure 3.2). The analytical model is extended from a two-layer, surface gravity current model

first proposed by Stern et al. (1982), in which the gravity current is formed by releasing a

‘locked’ volume of freshwater over saltwater. Assuming the ambient environment is two-layer

stratified of density �̂�1 and �̂�2, and the outflow is formed as the third layer by releasing the

‘locked’ plume of density �̂�𝑝 into the water. The plume travels downstream between the two

layers, forming a wall attached gravity current.

The geometry and velocity of the outflowing plume is derived using potential vorticity

conservation, and the derivation is shown in Appendix B. The simple expression of plume

geometry is

ℎ̂(𝑦) = ℎ0[𝑈(𝐿 − 𝑦) − 0.5(𝐿 − 𝑦)2]

�̂�1 =𝑔′

𝑔1𝑝′ ℎ̂, �̂�2 = −

𝑔′

𝑔2𝑝′ ℎ̂

(3.6)

in which 𝑔1𝑝′ =

𝑔(𝜌𝑝−𝜌1)

𝜌0 and 𝑔2𝑝

′ =𝑔(𝜌2−𝜌𝑝)

𝜌0 are reduced gravities between the plume and

upper/lower layers, 𝑔′ ≡𝑔1𝑝′ 𝑔2𝑝

′

(𝑔1𝑝′ +𝑔2𝑝

′ ) is the ‘effective’ reduced gravity of the plume, ℎ0 =

71

(2�̂�𝑓)0.5𝑔′−0.5

is the thickness of the plume, 𝑦 = (𝑔′ℎ0)−0.5𝑓�̂� is the normalized distance in

cross plume direction, and �̂� is the total volume flux of plume. When the top layer density is very

small (𝜌2 ≫ 𝜌1, 𝜌𝑝 ≫ 𝜌1), the reduced gravity 𝑔′ ≈ 𝑔2𝑝′ and the three-layer model degenerates

into the two-layer model of Stern et al. (1982). The length and velocity parameters 𝐿 = 0.42 and

𝑈 = 2.5 are only weak functions of the initial plume potential vorticity and can be approximated

as constants. The plume width is

�̂� = (𝑔′ℎ0)0.5𝑓−1𝐿 = (2�̂�𝑔′)

0.25𝑓−0.75𝐿 (3.7)

and plume velocity is

�̂�(𝑦) = 𝑢0[𝑈 − 𝐿 + 𝑦] ≡ (𝑔′ℎ0)

0.5[𝑈 − 𝐿 + 𝑦] (3.8)

We compare the length scales �̂�, ℎ0 and velocity scale �̂�(0) = 𝑢0(𝑈 − 𝐿) from the shallow

water model with the plume dimensions 𝐿𝑝, ℎ𝑝 and 𝑈𝑝 acquired from the modeled velocity fields,

and the results are summarized in Figure 3.14. We use two different averaging strategies to

determine the ambient densities 𝜌1 and 𝜌1: (1) the averaged densities are calculated by integrating

density from the plume core depth to surface/bottom of the whole water column, excluding the

plume region; (2) the integration only extends the depth where the plume exists (bounded by the

white dashed lines in Figure 3.2), and the water column above/below the plume layer is

excluded.

Figure 3.14 a-c shows the results calculated using the first averaging method. Firstly, the

plume thickness is poorly predicted by the shallow water model. In the wall plume, the modeled

thickness ℎ𝑝 is invariant of ℎ0; for other discharge locations, ℎ𝑝 and ℎ0 are inversely correlated.

The plume thickness decreases in model due to surfacing of the buoyant plume; as the plume rises

and reaches sea surface, the upper boundary is confined by the free surface and cannot further

expand. In all simulations ℎ𝑝 is constantly greater than ℎ0. In the shallow water model, the

72

maximum thickness ℎ̂(0) = ℎ0(𝑈𝐿 − 0.5𝐿2) ≈ 1.14ℎ0 is slightly higher than ℎ0; however, even

when scaled by 1.14, the modeled thickness is still significantly higher than the analytical

prediction.

Figure 3.14. Comparison of plume length (a, b, d, e) and velocity (c, f) scales from the analytical

shallow water model and modeled velocity fields. In the top panels (a-c) the averaged ambient

densities (𝜌1 and 𝜌1) are determined using the whole water column, while in the lower panels (d-

f) only the layers of the plume are used. Colors represent experiments of different discharge

locations; blue is the wall plume scenario. Dots are data points, while error bars are confidence

intervals. The black dashed lines are the ratio benchmarks of 1:2, 1:1, and 2:1. Other dashed lines

are linear regression fits of each discharge location group. The blue dots and lines are highlighted

to distinguish the wall plume from other cases.

In terms of plume width, the shallow water model prediction is in good agreement with that

of the wall plume but is not well correlated with other cases. For the wall plume, 𝐿𝑝 is initially

73

smaller than �̂�, and then become slightly larger as 𝑄𝑠𝑔 increases. For other discharge locations,

𝐿𝑝 is significantly greater than �̂�, but seemingly limited by the fjord width, as shown in Section

3.1; regression analysis reveals no clear correlation between 𝐿𝑝 and �̂�.

Plume velocity is the best predicted variable among all three parameters. For both wall plume

and other discharge locations the correlation is strong, even though the velocity of wall plume is

greater than other cases for the same discharge. For the wall plume, the modeled velocity 𝑢𝑝 is

slightly greater than 𝑢0, while for other discharge locations 𝑢𝑝 is smaller than 𝑢0. Since in the

shallow water model, the in-situ velocity �̂� is also scaled by the factor [𝑈 − 𝐿 + 𝑦], which ranges

from 2.08 on �̂� = 0 to 2.5 on �̂� = 𝐿, thus in fact the predicted velocity �̂� is constantly higher than

𝑢𝑝. Besides, in the shallow water model �̂� increases from �̂� = 0 to �̂� = 𝐿; in the model results

velocity has a decreasing trend from the wall side to the plume boundary.

When the second averaging method is applied, the results do not significantly change the

conclusion, except that in the wall plume scenario the intercept of plume width and velocity

regression is smaller. In summary, when the SGD plume is initiated from the southside wall, the

plume thickness is poorly predicted by the shallow water model. The plume velocity is slightly

larger than the model prediction but show good correlation, while the width is well predicted.

When the plume is initiated away from the southside wall, the plume thickness and width are not

well predicted by the shallow water model. The plume velocity is even smaller than that of the

wall plume, but also show good correlation with the model prediction. The discrepancies between

the two averaging methods are likely attributed to the assumptions of shallow water model.

Firstly, the thickness scale ℎ0 = (2�̂�𝑓)0.5𝑔′−0.5

is inversely correlated with 𝑔′, while the

velocity scale 𝑢0 = (𝑔′ℎ0)0.5 = (2�̂�𝑔′𝑓)

0.25 is positively correlated with 𝑔′. The definition of

𝑔′ depends on the density stratification and is sensitive to the averaging extent in a continuously

stratified environment. Since this study does not attempt to quantitively predict the plume

74

geometry from density profiles, we do not seek the ‘better’ averaging strategy among the two.

The latter one seems to produce better regression results, but the averaging area is very small

when plume is relatively thin. In addition, other factors also contribute to uncertainties of the

shallow water model; for example, friction and diapycnal mixing are not considered in the

shallow water model, but the strong shear generated by the outflow could change the along fjord

momentum balance and mix the plume, which could explain the thicker plume in modeled fields.

To further illustrate the scaling of plume properties, additional experiments are carried out to

test the sensitivity to Coriolis parameter 𝑓 and fjord width 𝐿𝑤. We scale 𝑓 by factors of 0.5, 0.8,

1.2 and 1.5 and 𝐿𝑤 by 2, 4, respectively, and the cross-sectional velocity and SGD tracer fields

are shown in Figure 3.15 to Figure 3.18. For the wall plume, plume width 𝐿𝑝 is inversely

correlated with 𝑓, in agreement with the scaling of Equation (3.7), while the plume thickness ℎ𝑝

only increases slightly with 𝑓. This is in agreement with the conclusion that 𝐿𝑝 is better predicted

by the shallow water model. For non-wall plumes, the plume length scales 𝐿𝑝 and ℎ𝑝 are both

invariant of 𝑓, but the plume shape is evidently changed with 𝑓: it is ‘rounder’ towards the tip of

the plume as 𝑓 increases. In terms of fjord width, 𝐿𝑝 increases with 𝐿𝑤, up to a point where the

plume width is limited by 𝑓 for wide fjords. ℎ𝑝 is mostly invariant of 𝐿𝑤. In summary, these

comparisons further address the discrepancy between wall plume and non-wall plumes and

highlight the recirculation of non-wall plume water in fjords, providing additional information to

our scaling analysis.

75

Figure 3.15. Velocity fields in the mid-fjord with respect to discharge location and Coriolis

parameter, similar to Figure 3.3.

Figure 3.16. Velocity fields in the mid-fjord with respect to discharge location and fjord width,

similar to Figure 3.3.

76

Figure 3.17. Normalized SGD tracer in the mid-fjord with respect to discharge location and fjord

width, similar to Figure 3.5.

Figure 3.18. Normalized SGD tracer in the mid-fjord with respect to discharge location and fjord

width, similar to Figure 3.5.

77

3.5.2 Heat Budgets in Fjord

The heat flux analysis in Section 3.2 highlights the major contributors on heat budgets -

barotropic, exchange, and melting fluxes. In this section we discuss the mechanism of each

contributor in detail, and then unify them to form a theory about SGD driven heat exchange in

fjord.

Among the three dominant budget terms, the barotropic and melting fluxes contribute

negatively to heat budgets, while the exchange flux balances them to maintain a closed budget.

Since the SGD and MW have constant temperature (𝜃𝑠𝑔 = 𝜃𝑚 = 0℃), the barotropic term 𝐻0 =

𝜌0𝑐𝑝(𝑄𝑠𝑔𝜃𝑠𝑔 + 𝑄𝑚𝜃𝑚 − 𝑢0𝐴0𝜃0) = 𝜌0𝑐𝑝𝑄0𝜃0 is simply determined by the barotropic volume

transport and mean temperature. 𝑄0 ≈ 𝑄𝑠𝑔 + 𝑄𝑚 is primarily determined by the discharge itself;

𝜃0 is relatively constant and only varies by 0.12℃ as 𝑄𝑠𝑔 increases from 0 to 300 m3s-1, less than

10% of the initial mean temperature. Therefore, the total flux is approximately a linear function

of discharge rate 𝐻0~𝑄𝑠𝑔, as shown in Figure 3.13 b.

Small perturbations in 𝐻0 include the SMW flux 𝑄𝑚 and the variability in mean temperature

𝜃0. Model predicted 𝑄𝑚 only varies from 4 to 14 m3s-1 as 𝑄𝑠𝑔 increases from 0 to 300 m3s-1. In

the model domain this melt water flux is prescribed as advection on u/v faces of a glacier-

adjacent ocean grid, which produces barotropic pressure gradient near glacier front. In reality,

melting of glacier is accompanied by retreat of submarine glacier itself, thus in theory the melt

water takes the place of retreated glacier and should not generate barotropic pressure gradient.

This discrepancy is produced since the grid is rigid and cannot mimic the slow retreat of glacier

front location. As a result, the total freshwater transport 𝑄0 is likely slightly overestimated in the

numerical model.

Barotropic temperature 𝜃0 is the mean temperature of the outer section of the ‘box’; to some

degree it also reflects the mean heat content of the box since the along-fjord temperature gradient

78

is low. The perturbation in 𝜃0 is largely related to the exchange between fjord water and plume

water. For non-wall plumes, the perturbation in 𝜃0 is tightly related to the depth and temperature

of outflowing plume (Figure 3.6 f). At low discharge (𝑄𝑠𝑔 ≤ 100 m3s-1), the plume detrains

below 30 m, coinciding with layer of minimum temperature (Figure 3.1 c). After mixed with

ambient water, the plume quickly flushes the low temperature subsurface water out of the fjord

and upwells the warm deep water to the subsurface layer, increasing the overall heat content. At

high discharge (𝑄𝑠𝑔 > 100 m3s-1), the plume detrains at shallower depths, where ambient water is

slightly warmer, and the plume flushes the surface water out of the fjord and decreases the overall

heat content. For the wall plume, the plume also detrains at different depth as 𝑄𝑠𝑔 increases, but

the plume flows on the southside and is not able to flush water of the entire layer out of the fjord.

The stratification away from southside wall is maintained; 𝜃0 varies only by 0.02 as 𝑄𝑠𝑔

increases, and the overall heat content is not significantly changed.

Melting heat flux 𝐻𝑚 = −𝜌0𝑄𝑚[𝐿 + 𝑐𝑖(𝜃𝑚 − 𝜃𝑖)] is proportional to submarine melt rate 𝑄𝑚,

as other variables 𝜌0, 𝐿, 𝑐𝑖, 𝜃𝑚 and 𝜃𝑖 are model constants. Due to the large latent heat required to

melt the glacier ice, the process of melting is more efficient at removing heat than direct melt

water discharge - the effective temperature of SMW is -90℃ (Gade, 1979) significantly lower

than the temperature of SGD (0℃). Without subglacial discharge (𝑄𝑠𝑔 = 0), ambient melting

𝑄𝑚𝑎 produces minimum amount of melt water flux of 4 m3s-1; as 𝑄𝑠𝑔 increases from 0 to 300

m3s-1, plume melting 𝑄𝑚𝑝 produces additional melt water flux of 0~9 m3s-1 while 𝑄𝑚𝑎 is

relatively unchanged (Figure 3.19). The plume melt rate is primarily determined by the BPT

parameterization and scales with plume upwelling velocity, which scales with the cubic root of

subglacial discharge 𝑄𝑚𝑝~𝑈𝑝~𝑄𝑠𝑔13⁄ .

79

Figure 3.19. Plume (𝑄𝑚𝑝) and ambient (𝑄𝑚𝑎) melt rate, with respect to 𝑄𝑠𝑔 and 𝐿𝑟.

The purpose if distinguishing plume melt water from ambient melt water is that SMWP is

fully mixed with the subglacial plume during upwelling stage, while SMWA is generated away

from the plume and only partially mixed with the plume. Without subglacial discharge, the weak

and well-spread SMWA flux is not able to generate enough upwelling to form the exchange flow.

As a result, the melt water remains in fjord for much longer time before it is flushed out, with

mean residence time of 2650 days (Figure 3.8). This overall has a cooling effect in the glacier

near field, which is reflected in the heat storage 𝐻𝑠𝑡 in Figure 3.13; without additional forcing

this cold layer of water isolates the warm deep water from the glacier front and prevents further

melting.

80

Ambient melt rate 𝑄𝑚𝑎 is slightly different for each discharge location (Figure 3.19); in

general, 𝑄𝑚𝑎 is slightly higher when the discharge point is located on the northside (𝐿𝑟19). At low

discharge, 𝑄𝑚𝑎 of the wall plume is slightly lower than non-wall plumes; at high discharge the

discrepancy is not as significant. Figure 3.20 shows the profiles of near glacier temperature and

velocity extracted from the model fields, as 𝑄𝑚𝑎 is primarily determined by these two factors. To

maintain the minimum amount of ambient melting, a background velocity 𝑉𝑏𝑘𝑔 = 3 cm∙s-1 is

applied in the parameterization. This background flow driven melting is very weak and only

velocity greater than 𝑉𝑏𝑘𝑔 enhances ambient melting. At low discharge, velocity near glacier front

is low and rarely exceeds 𝑉𝑏𝑘𝑔, thus 𝑄𝑚𝑎 is heavily dependent on temperature profile; for the

wall plume, the lower temperature towards surface highlights the high melt water content and

weaker recirculation, thus 𝑄𝑚𝑎 is slightly reduced. At high discharge, ambient temperature

profiles show less discrepancy, and velocity becomes the primary driver of variability; the mean

velocity of northern most (𝐿𝑟19) plume is greater than other discharge locations, thus 𝑄𝑚𝑎 is

slightly higher.

Since the process of submarine melting near glacier front is complex, the three-equation

parameterization is very crude and cannot resolve the small-scale convection and turbulence in

the ice/ocean boundary layer that drives submarine melting. Previous studies suggest that the

ambient melt rate 𝑄𝑚𝑎 is very likely underestimated using the model parameterization and the

actual melt rate should be at least one order of magnitude higher (Sutherland et al., 2019). The

purpose of applying the background velocity is to represent this process without resolving the

small-scale turbulence in the boundary layer. However, we lack the information to determine the

magnitude of melt driven turbulence in the near field, thus the modeled 𝑄𝑚𝑎 is likely very crude

as well. Our estimate is likely a lower bound of the actual melt rate, and the relationship shown in

Figure 3.19 is not determinant.

81

Figure 3.20. Velocity and temperature profiles of glacier near field from all experiments. Solid

lines are profiles are acquired by averaging all profiles from grid adjacent to the glacier front;

dashed lines in the upper panel are maximum velocity at each depth. The dash black line in upper

panels is 3 cm∙s-1 mark, which is the minimum background velocity of the ambient melting

parameterization. The black solid line in lower panels is the initial temperature profile; back dash

line is the temperature profile without subglacial discharge, only with submarine melting.

The plume driven exchange flow is very efficient at transporting the remnant melt water out

of the fjord; even with the smallest discharge rate (𝑄𝑠𝑔 = 5 m3s-1) the mean residence time of

SMWA is reduced to 83 days. This highlights the role of exchange heat flux 𝐻𝑒 in balancing heat

budgets in fjord. The upwelling subglacial plume directly entrains large volume of water near the

glacier front, rapidly removes the melt-rich content away from the ice/ocean interface. In

addition, recirculation driven by non-wall plume is also able to transport melt water away from

glacier front; for the same discharge rate, the wall plume is slightly less efficient at transporting

82

melt water compared with non-wall plumes, as suggested by the greater residence time of

ambient melt water content (Figure 3.8 f).

The strong removal mechanism of upwelling plume is mostly dependent on the initial

buoyancy of discharge and is invariant of the ambient melt rate. We argue that even if the melt

rate is one order of magnitude greater than model predicted, the plume is still able to sufficiently

entrain the melt water and remove it from the glacier front. As a result, 𝐻1 increases with 𝐻𝑚, and

the barotropic flux 𝐻0 will not be significantly changed, and the overall heat content of fjord

water is maintained. Hence, the exchange flux can be scaled by the summation of three major

components 𝐻1~𝑐0𝑄𝑠𝑔 + 𝑐𝑚𝑝𝑄𝑠𝑔13⁄ + 𝑐𝑚, in which the three terms on the RHS are barotropic,

plume melt and ambient melt flux.

Other smaller heat budget terms include heat storage 𝐻𝑠𝑡 and time dependent flux 𝐻2. Over

longer terms 𝐻𝑠𝑡 should be approximately zero since the primary forcing (SGD) is constant; the

modeled heat storage is not strictly zero because the system takes slightly longer to reach steady

state compared with the averaging span. One exception is the experiment with no subglacial

discharge, in which case the lack of exchange mechanism prevents the melt water from being

transported. As a result, cold water accumulates in fjord, and the system is expected to take much

longer time to reach steady state.

The time dependent flux 𝐻2 is slightly negative on average and is dependent on the discharge

location - for wall plume 𝐻2 is nearly zero, while for non-wall plumes 𝐻2 shows a decreasing

trend over 𝑄𝑠𝑔 (Figure 3.13 c). 𝐻2 is mostly relevant to the oscillatory recirculation formed when

the buoyant plume travels downstream (Figure 3.9 to Figure 3.12). Even though the time

dependent volume transport 𝑄𝑡 is on the same order of magnitude with the exchange volume

transport 𝑄𝑒 (Figure 3.7 b, c), the overall heat flux is much smaller since the recirculation

transports the same water mass in and out of fjord.

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3.6 Conclusion

Using a coupled ocean-BPT model, plume-driven circulation and submarine melt rate in

tidewater glacier fjord is investigated. Physical properties (velocity, width and thickness) of the

downstream flowing plume are estimated using modeled outputs and are compared with those

derived using a shallow water model. Results suggest that when subglacial discharge is generated

from the southside wall (wall plume), plume width and velocity are well predicted by the shallow

water model, while the plume thickness is relatively constant and greater than the model

predicted value; when the discharge is generated away from southside (non-wall plume), the

shallow water model is not able to well predict the physical properties of plume, which appears to

be constrained by the fjord width.

The results also highlight the role of recirculation generated by non-wall plumes in fjord. The

oscillatory recirculation transports plume water away from the southside wall, mixes plume water

with local water and flushes local water out of fjord. This mechanism ventilates the local fjord

water, accelerates exchange flow and have influence on the total heat content in fjord. However,

the recirculation contributes little to the heat flux.

Decomposition analysis suggests that the heat budgets in fjord are primarily controlled by

barotropic, exchange, and melting heat fluxes when the circulation is forced by constant

subglacial discharge. The plume driven exchange flow is very efficient at flushing the cold, melt

water out of fjord, thus the negative barotropic and melting heat flux is balanced by the positive

exchange heat flux. The quick removal of melt water by exchange flow potentially increases the

near field temperature and accelerates submarine melting of the ambient glacier; however, due to

the limitation of the three-equation parameterization, this mechanism is not well reflected in the

modeled melt rates. Efforts to improve the melting parameterization is critical for advancing our

understanding of glacier mass loss in future.

84

Chapter 4 - Conclusions

Subglacial discharge enters tidewater glacier fjords from the base of marine-terminating

glaciers, which drives strong upwelling near the glacier front, entrains large volume of ambient

deep water and forms the subglacial buoyant plume. This process enhances exchange between

fjords and the continental shelf, and controls salt and heat budgets in glacial fjords. In this

dissertation we developed a new model framework, ROMS-ICEPLUME, to parameterize the

rising and initial outflowing stage of the subglacial discharge plume in vicinity of the

glacier/ocean boundary. Using this model framework, we examined the near-field circulation of

the outflow plume, and estimated heat transports and budgets using idealized simulations.

In Chapter 2, we introduce the outflow parameterization and coupling options implemented

on the model framework. The outflow parameterization OP is based on the experimental work of

Noh et al. (1992); Ching et al. (1993), which uses a modified Richardson number criterion to

determine the initial states of the outflowing plume and distributes the outflow vertically in

several model layers. The three coupling options are HMass/VMass/Mix, which use horizontal

advection, vertical convergence/divergence, and isopycnal displacement to prescribe the plume-

driven volume and tracer fluxes in the ocean grid, respectively.

The outflow parameterization and coupling options are tested with idealized and semi-

realistic numerical experiments. Comparison with field observations shows that a combination of

OP and HMass can best capture the strong outflowing plume in vicinity of the glacier front. Both

the magnitude and direction of the outflow are well reproduced in the OP/HMass simulation. We

suspect that the surface geometry of the glacier front redirects the vertical upwelling flow to

horizontal direction, thus the initial flow is better represented with horizontal momentum flux

(HMass).

85

In Chapter 3, we extended the simulations to explore the sensitivity of plume-driven

circulation and heat transports to subglacial discharge, subglacial river location, and a few other

oceanographic parameters. The model domain is an east-west orientated rectangular basin with

tidewater glacier located on the west end and coastal ocean on the other end. Physical properties

of the plume are estimated using the modeled velocity and tracer fields and compared with those

derived from a shallow water model. Our analysis shows that the subglacial plume travels

downstream along the southside of the fjord resembling a coastal current. When subglacial

discharge is generated from the southside wall (wall plume), plume width and velocity are well

predicted by the shallow water model, while the plume thickness is relatively constant and greater

than the model predicted value; when the discharge is generated away from southside (non-wall

plume), the shallow water model is not able to well predict the physical properties of plume,

which appears to be constrained by the fjord width.

Results also suggest that non-wall plumes generate strong recirculation in the fjord channel.

The oscillatory recirculation transports plume water away from the southside wall, mixes plume

water with local water and flushes local water out of fjord. This mechanism ventilates the local

fjord water, accelerates exchange flow and have influence on the total heat content in fjord.

However, the recirculation contributes very little to the overall heat flux.

Using the decomposition method of Lerczak et al. (2006), heat budgets from the glacier front

to a mid-fjord section are calculated. The analysis suggests that when forced by constant

subglacial discharge, the heat budgets in fjord are primarily determined by barotropic, exchange,

and melting heat transports. The exchange flow driven by subglacial discharge is very efficient at

flushing the cold, melt water out of fjord, thus the negative barotropic and melting heat flux is

balanced by the positive exchange heat flux. This mechanism quickly removes melt water from

the ice/ocean interface, potentially increases the near field temperature and accelerates submarine

melting of the ambient glacier; however, due to the limitation of the melting parameterization

86

applied in the model framework, the modeled melt rates are not significantly enhanced in the

simulations. Efforts to improve the melting parameterization is critical for better estimation of

subglacial melt rate and thus glacier mass loss in future.

This model framework is a powerful tool to advance our understanding of circulation in

tidewater glacier fjords. However, due to the chaotic nature of buoyant plume, there are still

uncertainties associated with the parameterizations that cannot be eliminated. This work

highlights the importance of acquiring field data to improve our knowledge of the nature of the

upwelling plume, and to reduce these uncertainties in future.

87

Appendix A

This appendix contains additional text and figures supporting the methods, results and

discussions presented in Chapter 2. Section A.1 contains additional detail in employing averaging

of ambient density used in the detrainment parameterization. Section A.2 describes the algorithm

to determine the vertical extent of detrainment outflow in a continuously stratified fluid.

A.1 Averaging Ambient Density Profile

To demonstrate how spatially averaging the ambient density profile changes the model

behavior, we run the same setup of Ex 2 for 10 model days, and the density profiles of plume and

ambient water are plotted in Figure A.1. The gray line is the initial density stratification, while

the colored lines are density profiles at the end of day 10. The plume density profile is calculated

offline using ROMS output files. The blue line is the density profile from the grid adjacent to the

buoyant plume point source, while other lines are profiles averaged over a small span. As the

plume rises, the plume density quickly converges to ambient density. Since the plume

detrainment depth is defined as where the plume density first exceeds the ambient density (from

surface to bottom), at this step the plume should detrain at approximately 20 m (where the black

and blue lines intersect). However, even after averaging over a very small span, the density

profile changes drastically, and the detrainment depth should be 60 m instead.

88

Figure A.1. Density profiles of plume (black line) and ambient water (colored lines), of the

whole water column (a) and zoomed to the top 100 m (b). The gray line is the initial stratification.

The line colors represent the density profiles averaged over serval grid cells; the span of

averaging is marked in (c).

Another step to average the ambient density profile is exponentially smoothing the profile at

each timestep

𝜌𝑡 = 𝛼𝜌0𝑡 + (1 − 𝛼)𝜌𝑡−1 (A.1)

where 𝜌0 is the area averaged density profile of current step, and 𝛼 is the exponential smoothing

factor. This prevents ambient density changes too sharply within a few time steps. In practice, it

may be not necessary to perform this step, in which case 𝛼 = 1 is used.

The averaging has a prolonged effect in detrainment outflow over modeling time. The

evolution of the detrainment outflow from a series of simulations are shown in Figure A.2. From

89

top to bottom, the averaging span increases from a single grid (1 × 1) to a 5 × 9 box; to

demonstrate the effect of temporal averaging two additional experiments with α = 0.5 are carried

out.

Without any averaging, the outflow is stable for the first two model days, but quickly shows

strong variability over time. As the averaging span increases, the outflow becomes more stable;

even with a small averaging span (2 × 3) the outflow is mostly kept at the same depth over the

course of simulation. The temporal averaging, on the other hand, is not very effective, as the

outflow still show great variability. The same response is also shown in density anomaly, which

is shown in Figure A.3.

Figure A.2. Hovmöller Diagram of the detrainment velocity in the top 100 m from 7

experiments. The first five experiments use ambient density profiles spatially averaged over

certain spans; the last two experiments use exponential smoothing to further average the profiles

over time.

90

Figure A.3. Hovmöller Diagram of the density deviation from initial condtion in the top 100 m

from 7 experiments.

A.2 Vertical Extent of Plume in Continuously Stratified Environment

In section 2.3.1, vertical extent of the detrainment outflow is determined by matching the

pressure with ambient water (Equation (2.5)-(2.7)). It is a simplified expression in a two-layer

setup, and in continuously stratified environment it cannot be direction applied. Instead, an

iterative method is used to satisfy the same condition in a discrete vertical grid of continuously

stratified water.

Assuming the background stratification for each level is ρ(𝑘), where 𝑘 is the vertical grid

index. The plume detrains at depth 𝑧 = 𝑧(𝑘𝑃), where the density of plume matches the ambient

water ρ𝐴𝑚(𝑘𝑃) ≈ 𝜌𝑃. Define the total ‘pressure anomaly’ flux introduced by the buoyant plume

91

Δ𝑃 =∑ 𝑔𝑢(𝑖)𝑑𝑧(𝑖)[ρ𝐴𝑚(𝑘𝑃) − 𝜌𝑃]𝑁

𝑖=0/𝑈0 (A.2)

where 𝑢 is the velocity of detrainment outflow in each level and 𝑈0 is a velocity scale. If all

detrained volume flux goes into one signal layer 𝑘𝑃 , then 𝑢(𝑘 = 𝑘𝑃) =𝑄𝐷

𝑑𝑧(𝑘𝑃)𝑑𝑦 and

𝑢(𝑘 ≠ 𝑘𝑃) = 0, where 𝑄𝐷 is the total volume flux of the detrainment plume. If the thickness of

this layer 𝑑𝑧(𝑘𝑃) is greater than that of the predicted outflow (ℎ, Equation 4 of the main text),

then it already satisfies the parameterization and no further action is needed. On the other hand,

when 𝑑𝑧(𝑘𝑃) is less than ℎ, another layer adjacent to 𝑘𝑃 needs to be included as part of the

detrainment outflow. The added layer can be either on top or below 𝑘𝑃, and the principle is to

make |Δ𝑃| converging to zero. Assuming density increases monotonously with depth, if Δ𝑃 is

greater/less than 0, the layer on top of 𝑘𝑃 (𝑘+) /below 𝑘𝑃 (𝑘−) is used. The total thickness of the

detrainment becomes ℎ𝑝 = 𝑑𝑧(𝑘𝑃) + 𝑑𝑧(𝑘+) or 𝑑𝑧(𝑘𝑃) + 𝑑𝑧(𝑘−), which is then compared with

ℎ again to determine if ℎ𝑝 > ℎ is satisfied. The detrainment velocity becomes 𝑢(𝑘 = 𝑘𝑃 , 𝑘+/−) =

𝑄𝐷

ℎ𝑝𝑑𝑦, and 𝑢(𝑘 ≠ 𝑘𝑃 , 𝑘+/−) = 0. Note that we used a uniform ‘bulk’ velocity here, which in

reality should be a Gaussian shaped function. However, a non-uniform velocity complicates the

algorithm, but the improvement is neglectable.

The above process is repeated until the criterion ℎ𝑝 > ℎ is met. At the last step, the velocity 𝑢

is adjusted to satisfy Δ𝑃 = 0. As a result, the density profile of detrainment outflow still satisfies

Equation (2.6), and the outflow is hydrographically stable.

92

Figure A.4. Modeled velocity field (color: along channel, vector: cross channel), 2 km

downstream from glacier front. Plots are organized similar to Figure 2.5.

93

Appendix B

The surface and bottom layers are stagnant; sea surface height �̂�0 is constant to guarantee a

stagnant surface layer. In vertical direction, we choose the depth of pycnocline as origin of

coordinate. The velocity and thickness of the plume layer are described by the shallow water

equations

𝑑

𝑑�̂��̂� + 𝑓𝒌 × �̂� = −𝑔1𝑝

′ ∇�̂�1 = 𝑔2𝑝′ ∇�̂�2

𝜕

𝜕�̂�ℎ̂ + ∇�̂�ℎ̂ = 0

(B.1)

where �̂� = �̂�𝒊 + �̂�𝒋 is the two-dimensional velocity, 𝑓 is the Coriolis frequency, �̂�1 and �̂�2 are the

slopes of layer boundaries, 𝑔1𝑝′ =

𝑔(𝜌𝑝−𝜌1)

𝜌0 and 𝑔2𝑝

′ =𝑔(𝜌2−𝜌𝑝)

𝜌0 are reduced gravities, and ℎ̂ =

ℎ̂𝑝 = ℎ̂1 − ℎ̂2 is the thickness of plume. In cross shelf �̂� direction, lateral boundary conditions are

�̂�1 = �̂�2 = 0 𝑎𝑡 �̂� → +∞

�̂� = 0 𝑎𝑡 �̂� = 0

(B.2)

Since the plume upper and lower boundary converges at �̂� → +∞, Equation (B.1) can be

rewritten into

𝑑

𝑑�̂��̂� + 𝑓𝒌 × �̂� = −

𝑔1𝑝′ 𝑔2𝑝

′

𝑔1𝑝′ + 𝑔2𝑝

′ ∇ℎ̂ ≡ −𝑔′∇ℎ̂ (B.3)

where 𝑔′ ≡ 𝑔1𝑝′ 𝑔2𝑝

′ (𝑔1𝑝′ + 𝑔2𝑝

′ )⁄ is the ‘effective’ reduced gravity.

As the buoyant plume travels downstream, potential vorticity is conserved

𝑓 + 𝜁

ℎ̂=𝑓

�̂� (B.4)

94

where 𝜁 = 𝒌 ∙ ∇ × �̂� is the two-dimensional vorticity; �̂� is the thickness of the ‘locked’ fluid

before it releases and forms the buoyant plume. Substitute Equation (B.4) into (B.3), the vorticity

form of primitive equation is

𝜕

𝜕�̂��̂� + (𝑓 + 𝜁)𝒌 × �̂� = −∇(𝑔′ℎ̂ +

1

2�̂�2) (B.5)

Nondimensionalize the primitive equations using

ℎ̂ = ℎ0ℎ(𝑥, 𝑦, 𝑡), �̂� = (𝑔′ℎ0)0.5𝑓−1𝑦, �̂� = 𝜖−1(𝑔′ℎ0)

0.5𝑓−1𝑥,

�̂� = (𝑔′ℎ0)0.5𝑢, �̂� = 𝜖(𝑔′ℎ0)

0.5𝑣, �̂� = 𝜖−1𝑓−1𝑡,

�̂� = ℎ0𝐻, �̂� = (𝑔′ℎ0)0.5𝑓−1𝐿(𝑥, 𝑡), ℎ̂(�̂�, �̂�, �̂�, ) = 0.

(B.6)

in which ℎ0 is a given depth scale, �̂� is the maximum extent of the plume (where ℎ̂ = 0), and 𝜖 is

the ratio of velocity in along/cross coast direction. By making the long-wave (𝜖 → 0) assumption,

the governing equations become

∂

∂𝑡𝑢 − (1 −

∂

∂𝑦𝑢) 𝑣 +

∂

∂𝑥(ℎ +

1

2𝑢2) = 0 (B.7)

𝑢 = −∂

∂𝑦ℎ (B.8)

∂

∂𝑡ℎ +

∂

∂𝑥ℎ𝑢 +

∂

∂𝑦ℎ𝑣 = 0 (B.9)

1 −∂

∂𝑦𝑢 =

ℎ

𝐻 (B.10)

Solution for Equations (B.8) and (B.10) is

ℎ(𝑥, 𝑦, 𝑡) = 𝐻 [1 − cosh(𝐿 − 𝑦)

𝐻12

] + 𝐻12𝑈 sinh

(𝐿 − 𝑦)

𝐻12

𝑢(𝑥, 𝑦, 𝑡) = −𝐻12 sinh

(𝐿 − 𝑦)

𝐻12

+𝑈 cosh(𝐿 − 𝑦)

𝐻12

(B.11)

95

in which 𝑈 = 𝑈(𝑥, 𝑡) is the solution of 𝑢 on 𝑦 = 𝐿. When the depth of initial ‘lock’ is greater

than the thickness of outflowing plume (𝐻 → +∞), this expression can be simplified using Taylor

expansion on the hyperbolic function in 𝐿𝐻−1

2, and the solution becomes independent of 𝐻

ℎ = 𝑈(𝐿 − 𝑦) −1

2(𝐿 − 𝑦)2

𝑢 = 𝑈 − 𝐿 + 𝑦

(B.12)

Substitute Equation (B.12) into (B.7), the along shore momentum equation becomes

∂

∂t𝑈 + 𝑈

∂

∂x𝑈 −

∂

∂t𝐿 = 0 (B.13)

To further establish the relation between 𝑈 and 𝐿, integrating Equation (B.9) from 0 to 𝐿(𝑥, 𝑡)

and applying the boundary condition

0 = 𝐻∂

∂𝑡[𝐿 − 𝑢(𝐿) − 𝑢(0)] −

1

2

∂

∂𝑥[ℎ2(𝐿) − ℎ2(0)] (B.14)

Substitute the solution (B.12) into Equation (B.14) yields the first relation between 𝐿 and 𝑈

0 = 𝑈∂

∂𝑡𝐿 + (𝑈 −

1

2𝐿) (𝑈 − 𝐿)

∂

∂𝑥𝐿 +

1

2𝐿(𝑈 − 𝐿)

∂

∂𝑥𝑈 (B.15)

The other relation requires to make certain assumptions. Assuming 𝑈 is solely a function of

𝐿(𝑥, 𝑡), thus

𝑈 = 𝑈(𝐿),𝜕

𝜕𝑡𝑈 = 𝑈′(𝐿)

𝜕

𝜕𝑡𝐿,𝜕

𝜕𝑥𝑈 = 𝑈′(𝐿)

𝜕

𝜕𝑥𝐿 (B.16)

and Equations (B.13) and (B.15) can be rewritten as function of 𝑈′. Solving for 𝑈′ yields

𝑈′ = 1 −2𝑈

𝐿 ± [𝐿2 + 2𝐿(𝑈 − 𝐿)]12

(B.17)

This expression suggests that two classes of solution (±) exist, and Stern et al. (1982)

concludes only the negative (−) solution is physically suitable.

96

This expression holds near the nose of the plume (𝐿 → 0). Note that when 𝐿 = 0 and 𝑈 ≠ 0,

𝑈′ goes to infinity, and 𝜕

𝜕𝑥𝐿 = 𝑈′

−1 𝑑

𝑑𝑥𝑈 = 0 unless

𝑑

𝑑𝑥𝑈 = 0 is also satisfied. If

𝑑

𝑑𝑥𝑈 =

𝑑

𝑑𝑥𝐿 =

0, the solution is constant in 𝑥 direction and does not describe a plume ‘nose’; therefore, only

when 𝑈(𝐿 = 0) = 0 the solution is physically suitable, and only one class in Equation (B.17) (the

one passes the coordinate origin in 𝑈/𝐿 space) satisfies it. The solution yields that 𝐿 = 0.426 and

𝑈 = 2.488. Further investigation suggests that both 𝐿 and 𝑈 is relatively stable and insensitive to

𝐻. From 𝐻 → 0 to 𝐻 → +∞, 𝐿 only ranges from 0.418 to 0.426 and 𝑈 ranges from 2.530 to

2.488, respectively.

Lastly, the depth scale ℎ0 is determined by integrating the volume flux. Away from the ‘nose’

of the plume (𝑥 → −∞), the thickness near the wall ℎ(𝑦 = 0) is the normalized thickness

(ℎ(−∞, 0, 𝑡) = 1). The total volume flux of the outflowing plume is

𝑄 = ∫ 𝑢ℎd𝑦𝐿

0

= ∫ −ℎdℎ𝐿

0

= −1

2[ℎ2(𝐿) − ℎ2(0)] =

1

2ℎ2(0) =

1

2 (B.18)

In context of subglacial plume, 𝑄 is the non-dimensional volume flux rate of the detrainment

outflow. From here the dimensional thickness ℎ0 can be estimated by �̂� = ∫ �̂�ℎ̂𝑑�̂��̂�

0=

1

2𝑔′ℎ0

2𝑓−1,

which yields ℎ0 = (2�̂�𝑓

𝑔′)

1

2.

97

Appendix C

This document is an introduction to the work in process ICEPLUME module for ROMS. It is

modified from a similar package, IcePlume for the MITgcm, first developed by Dr. Tom Cowton.

A detailed description of the MITgcm version is in Cowton et al. (2015).

C.1 Overview

In ROMS, freshwater discharge is treated as point source going into a ROMS grid, which is

activated by the compiling options LuvSrc or LwSrc. It reads in a total discharge (𝑄𝑏𝑎𝑟), a

prescribed vertical weight function (𝑄𝑠ℎ𝑎𝑝𝑒), and tracer concentrations (𝑇, 𝑆 and passive), and

volume/tracers are injected into a grid cell through horizontal advection (LuvSrc) or vertical

convergence (LwSrc).

This method works well for shallow estuaries with a barotropic freshwater discharge forcing.

In most cases, the freshwater is advected into the grid uniformly (𝑄𝑠ℎ𝑎𝑝𝑒 = 1/𝑁 or 𝑄𝑠ℎ𝑎𝑝𝑒 =

𝑑𝑧/𝐻). The expansion in volume and dilution of tracers drives a gravitational flow, which adjusts

quickly to form an estuarine circulation.

However, the freshwater discharge is not always uniform in vertical direction. At high

latitude, especially fjordic systems, summer meltwater can permeate the glacier through cracks

and pores, and releases at the bottom of glacier head. If the glacier extends into the ocean, the

freshwater discharge is injected at depth, which is called subglacial discharge.

In fjordic systems, the depth of subglacial discharge can be a few hundreds of meters. The

runoff water is cold and fresh, rises and entrains the ambient water until it reaches a neutral

buoyant layer. This process is non-hydrostatic, which cannot be accurately simulated by ROMS.

Therefore, a parameterization for this process is required to correctly model subglacial discharge.

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This document is a technical manual to introduce the ROMS-ICEPLUME coupled model,

which uses a set of parameterizations to represent the subglacial discharge driven circulation in

fjords. The parameterizations for entrainment, detrainment, background melting and coupler

options are described in Section C.2; code structure is briefly summarized in Section C.3; steps

for model usage are given in Section C.4.

Figure C.1. ROMS-ICEPLUME model schematics.

C.2 Model Parameterizations

The buoyant plume theory (BPT) is a set of equations that describes the development of

buoyant plume rising near an ocean/glacier boundary. It is first described using a one-dimensional

model by Jenkins (1991, 2011), which assumes the plume initiates from a line source and only

grows in direction normal to line source. Later this model is modified by Cowton et al. (2015),

which assumes the plume initiates from a point source instead of a line source, and grows

uniformly in horizontal direction. In reality it is likely that the development of buoyant plume

falls in between the two cases (Jackson et al., 2017). In this section we attempt to use a

99

‘generalized’ buoyant plume model to describe both cases. This generalized model gives more

flexibility in modeling different geometries of buoyant plume, and is also convenient for

numerical applications.

To summarize, the development of buoyant plume is controlled by the following budget

equations:

𝑑

𝑑𝑧[𝐴𝑢] = 𝛼𝐿𝑐𝑢 + 𝐿𝑚�̇�

𝑑

𝑑𝑧[𝐴𝑢2] = 𝑔′𝐴 − 𝐿𝑚𝑐𝑑𝑢

2

𝑑

𝑑𝑧[𝐴𝑢𝑇𝑝] = 𝛼𝐿𝑐𝑢𝑇𝑎 + 𝐿𝑚�̇�𝑇𝑏 − 𝐿𝑚Γ𝑇𝐶𝑑

12⁄ 𝑢(𝑇𝑝 − 𝑇𝑏)

𝑑

𝑑𝑧[𝐴𝑢𝑆𝑝] = 𝛼𝐿𝑐𝑢𝑆𝑎 + 𝐿𝑚�̇�𝑆𝑏 − 𝐿𝑚Γ𝑇𝐶𝑑

12⁄ 𝑢(𝑆𝑝 − 𝑆𝑏)

(C.1)

where A is the plume cross section area, u is the velocity along plume transport axis, 𝑔′ =

𝑔(𝜌𝑎 − 𝜌𝑝)/𝜌0 is the reduced gravitational acceleration. The temporal derivative of mass, �̇� is

the submarine melt rate from the glacier wall. T, S and ρ are temperature, salinity and density.

The subscripts a, p and b indicate the plume model component of ambient water, plume water,

and boundary layer, respectively. The ambient conditions are read from ROMS at each baroclinic

time step. Γ𝑇 and Γ𝑠 are non-dimensional turbulent transfer coefficients, and 𝐶𝑑 is the ice-plume

drag coefficient.

The parameter α is a constant entrainment rate. It determines the growth rate buoyant plume.

It is the key parameter controlling the model behavior. Historically α = 0.1 is the conventional

value (Jenkins, 2011; Cowton et al., 2015), which is also verified by recent numerical studies

(Ezhova et al., 2018).

𝐿𝑚 and 𝐿𝑐 are length of the cross section that is in contact with the glacier ice wall and the

ambient water, respectively. In this model the plume can take any arbitrary shape as long as 𝐿𝑚

100

and 𝐿𝑐 can be determined. For example, when 𝐿𝑚 = 2𝑏, 𝐿𝑐 = 𝜋𝑏, and 𝑏 = √2𝐴/𝜋, Equations

(C.1) degenerate into the point source model (Cowton et al., 2015), where b is the radius of the

‘half-cone’ plume. When 𝐿𝑚 = 𝐿𝑐 = 𝑐𝑜𝑛𝑠𝑡 and 𝐴 = 𝐷 ∙ 𝑐𝑜𝑛𝑠𝑡, Equations (C.1) degenerate into

the line source model (Jenkins, 2011), where D is the thickness of plume. For other plume

geometries, 𝐿𝑚 and 𝐿𝑐 can be determined analytically or numerically.

Equations (C.1) can be easily interpreted as the conservation of mass, momentum, heat and

salt. In The first equation, 𝐴𝑢 is the volume flux through a horizontal transect; the first term on

RHS is the entrainment of volume from the ambient water; the second term on RHS is the volume

of melted water from glacier wall. In the second equation, 𝐴𝑢2 is the momentum flux through a

transect; the first term on RHS is the buoyancy force; the second term on RHS is the friction

(drag) from ambient water. In Equations third and fourth equations, the first and second terms on

RHS are similar to that of the first equation; the third terms are turbulent mixing of temperature

and salinity between plume and boundary layer.

The plume-ice boundary layer is described by a set of three equations (Holland & Jenkins,

1999)

�̇�(𝑐𝑖(𝑇𝑏 − 𝑇𝑖) + 𝐿) = Γ𝑇𝐶𝑑12⁄ 𝑢𝑐𝑤(𝑇𝑝 − 𝑇𝑏)

�̇�𝑆𝑏 = Γ𝑆𝐶𝑑12⁄ 𝑢(𝑆𝑝 − 𝑆𝑏)

𝑇𝑏 = 𝜆1𝑆𝑏 + 𝜆2 + 𝜆3𝑧

(C.2)

where 𝑐𝑖 and 𝑐𝑤 are heat capacity of ice and water, 𝐿 is the latent heat of melting/freezing, 𝑇𝑖 is

the temperature of glacier ice, 𝜆1, 𝜆2, and 𝜆3 are constants of linearity. The first equation is the

heat budget during melting/freezing; LHS is the total heat absorbed by ice; RHS is the heat

transfer from plume water to the glacier wall using a bulk flux formula. The second equation is

similar to the first one but for salt budget. The last equation is the linear relationship between

melting temperature as a function of salinity and depth.

101

A list of constants and parameter values are listed in Table C.1. By solving Equations (C.1)

and (C.2), BPT gives the volume flux and tracer concentration of the buoyant plume (𝐴, 𝑢, 𝑇𝑝,

𝑆𝑝), which is then coupled with the ocean model.

Symbol Name Value Units

𝛂 Entrainment rate 0.1

𝒄𝒊 Ice heat capacity 2009 J ∙ kg−1 ∙ ℃−1

𝒄𝒘 Water heat capacity 3974 J ∙ kg−1 ∙ ℃−1

𝝆𝟎 Reference ambient water density 1020 kg ∙ m3

𝝆𝒊𝒄𝒆 Reference ice density 916.7 kg ∙ m3

𝐋 Latent heat of melting 335000 J ∙ kg−1

𝚪𝑻 Thermal turbulent transfer coefficient 0.022

𝚪𝑺 Salt turbulent transfer coefficient 0.00062

𝒄𝒅 Ice/ocean drag coefficient 0.065

𝝀𝟏 Freezing point salt slope -0.0573

𝝀𝟐 Freezing point offset 0.0832

𝝀𝟑 Freezing point depth slope 0.000761

𝑻𝒊 Ice temperature -10 ℃

𝒖𝒃𝒌𝒈 Minimum background velocity 0.3 m ∙ s−1

Table C.1. Constants and parameter values of the ICEPLUME model.

The background melt rate is calculated following the same Equations (C.2) by substituting

the plume temperature, salinity and velocity 𝑇𝑝, 𝑆𝑝 with the ambient temperature and salinity 𝑇𝑎

and 𝑆𝑎. The melt rate is then corrected for the surface area since portion of the grid cell is covered

by the buoyant plume where the melting rate is calculated already.

This background melt parameterization is known to underestimate the melt rate by at least

one order of magnitude (Sutherland et al., 2019). The parameterization of Holland and Jenkins

(1999) is first developed to estimate melt rate below ice shelves, which are horizontally aligned

on top of seawater. Since the meltwater is fresher than seawater below, it forms a hydrostatically

stable layer, which tends to prevent further melting. Near a vertically aligned glacier/ocean

boundary, melt water is hydrostatically unstable compared to ambient water, and convection cells

tend to form which accelerates melting. This may partly explain the underestimated melt rate. A

temporary solution is to increase the background velocity 𝑢𝑏𝑘𝑔, which guarantees a minimum

102

amount of melting; however, the melt rate produced is still significantly lower than field

observations.

The detrainment model aims to spread the buoyant plume vertically in several layers. It uses

the parameterization of Ching et al. (1993) to determine the ‘thickness’ of buoyant plume

detrainment. The coupler integrates the ICEPLUME model and ROMS ocean model. The two

method of coupling are through horizontal advection or vertical convergence/divergence. Details

of the detrainment parameterization and coupler settings are discussed in Chapter 2.

C.3 Code Structure

ROMS-ICEPLUME is forked from the Rutgers ROMS. It contains several additional

FORTRAN source code files as well as modifications to the original code. New source code files

are located in ROMS/IcePlume. The module is written in six source code files:

• mod_iceplume.F: This file contains model constants, parameters, shared variables

and arrays for ICEPLUME. All profiles of plume status are stored in TYPE PLUME,

which is also used to communicate with the ocean model.

• iceplume_opkd.F: This file contains an ODE solver, ODEPACK, developed by

Hindmarsh (1982). It is originally written in FORTRAN 77, here it is revamped to be

compatible with the ROMS C preprocessor.

• iceplume_entrain.F: This file contains the entrainment model based on the BPT. The

main subroutine is ICEPLUME_ENTRAIN, which calculates entrainment and melt

rates; the subroutine GENERAL_ENTRAIN_MODEL contains the BPT equation

code, which is called repeatedly by the ODE solver; the subroutine

PLUME_METRICS is used to calculate 𝐿𝑚 and 𝐿𝑐 based on the model type.

103

• iceplume_detrain.F: This file contains the detrainment model. The main subroutine is

ICEPLUME_DETRAIN, which calculates velocity and tracer profiles of the

outflow.

• iceplume_calc.F: This file contains the wrapper function that calls the plume

entrainment and detrainment melting models. It also calculates volume and tracer

fluxes and background melt rates.

• iceplume.F: This file contains the main function ICEPLUME, which communicates

with the ocean model.

To couple ICEPLUME to ROMS, a few pieces of code are added in the following source

codes:

• ROMS/Utility/checkdefs.F: Add ICEPLUME as a C-preprocessor definition.

• ROMS/Modules/mod_arrays.F: Allocate types and arrays to initiate ICEPLUME

model.

• ROMS/Modules/mod_scalars.F: Relax maximum speed threshold to 100 m/s when

ICEPLUME is activated.

• ROMS/Modules/mod_sources.F: Define ICEPLUME input variables in TYPE

SOURCES.

• ROMS/Modules/mod_ncparam.F: Define variables to read plume model

parameters/forcing from the river netCDF file.

• ROMS/Nonlinear/set_idata.F: Read plume model parameters (depth/length/type, etc.)

from river netCDF file.

• ROMS/Nonlinear/get_data.F: Read raw plume model forcing (subglacial discharge,

tracer concentration, etc.) data from river netCDF file.

104

• ROMS/Nonlinear/set_data.F: Interpolate plume forcing data to time of calculation

from raw forcing data.

• ROMS/Nonlinear/initial.F: Initiate variables for plume model.

• ROMS/Nonlinear/main3d.F: Call the ICEPLUME model.

• ROMS/Nonlinear/step2d_LF_AM3.F: If use Luv coupler, set the 2-D horizontal

advection forced by plume; If use Lw coupler, set the increment in Zeta forced by

plume.

• ROMS/Nonlinear/step3d_uv.F: If use Luv coupler, set the 3-D horizontal advection

forced by plume.

• ROMS/Nonlinear/omega.F: If use Lw coupler, set the 3-D vertical velocity forced by

plume.

• ROMS/Nonlinear/step3d_t.F: Set 3-D tracer fluxes forced by plume.

• ROMS/Nonlinear/pre_step3d.F: Pre-step for 3-D velocity and tracer fluxes.

• ROMS/External/varinfo.dat: Add variable ids for plume model.

C.4 User Instruction

The current version of ICEPLUME is based on Rutgers ROMS version 3.8. To activate the

basic functionality of ICEPLUME, add #define ICEPLUME in your application header file

before compiling. Once ICEPLUME itself is defined, the following options can be used to add

additional configurations:

• ICEPLUME_MELT: By default, background melt is not calculated in ICEPLUME.

Use this option to activate calculation of background melt rate.

105

• ICEPLUME_TRACER: Use this option to activate tracer computation of subglacial

discharge. Requires T_PASSIVE.

• ICEPLUME_MELT_TRACER: Use this activate tracer computation of plume and

ambient melt water. When this method is used, the last two dye types (NT(ng)-1,

NT(ng)) will be used for the plume melt and background melt water, respectively; do

not use them for other purpose, and make sure NPT>2 in your ocean.in file.

• ICEPLUME_DET_NO: Use this option to deactivate the detrainment model. Not

recommended for general use.

• ICEPLUME_DET_AVERAGE: When determining the neutral buoyancy depth of

buoyant plume, profiles of tracers from the ocean model are used to determine the

ambient density. If this option is activated, the ambient density profile will be

determined by averaging over a domain near the point source.

• ICEPLUME_DET_NEUTRAL: Use this option to make tracer concentrations of

buoyant plume at different levels match the ambient water.

• ICEPLUME_MIX: Use this option to activate the Mix coupler and deactivate Luv

coupler.

• ICEPLUME_SLOPE: Use this option to activate the Slope coupler. Requires

ICEPLUME_MIX.

When ICEPLUME is activated, ROMS needs to read in additional variables from the river

netCDF file, including:

• subglacial_depth: Dimension (‘river’), grounding line depth. This value is negative

and is in accordance with the ROMS depth metrics. If the value is greater than 0, the

model assumes grounding line is at the bottom, and the grounding line depth is equal

106

to water depth. If it is a negative number, the grounding line depth will be the first ω

layer above this number.

• subglacial_type: Dimension (‘river’), a numerical value determines the type of BPT

model. Acceptable values are

- 0, no subglacial runoff or melting;

- 1, no subglacial discharge, only background melting;

- 2, point source (half-cone, Cowton et al., 2015) model;

- 3, finite-line source model;

- 4, infinite-line source (sheet, Jenkins, 2011) model;

- 5, detached full cone model;

- 6, half ellipse model.

• subglacial_length: Dimension (‘river’), the length of ‘crack’ on glacier, which is also

the width of subglacial river. It is used by subglacial_type 3 and 4.

• subglacial_transport: Dimension (‘river_time’, ‘river’), subglacial discharge rate.

• subglacial_temp: Dimension (‘river_time’, ‘river’), subglacial discharge initial

temperature.

• subglacial_salt: Dimension (‘river_time’, ‘river’), subglacial discharge initial salinity.

• subglacial_dye_: Dimension (‘river_time’, ‘river’), subglacial discharge passive trace

concentration.

• subglacial_Erange: Dimension (‘loc’, ‘river’), average span in Eta direction for the

detrainment model. This is only required by ICEPLUME_DET_AVERAGE. The

dimension ‘loc’ is constantly 2.

107

• subglacial_Xrange: Dimension (‘loc’, ‘river’), average span in Xi direction for the

detrainment model.

• subglacial_angle: Dimension (‘river’), angle of glacier/ocean boundary. This is only

used by ICEPLUME_SLOPE.

For more information, contact the developer for a sample river netCDF file or a python script.

The ICEPLUME model has a standalone version that is able to calculate the plume status

offline using ROMS forcing and history files. It has a series of python-based scripts that helps to

automate the calculation. Contact the developer if you need a tryout.

108

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