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A Heuristic Model of Dansgaard-Oeschger Cycles: Description,1

Results, and Sensitivity Studies: Part I2

Hansi A. Singh, ∗

Department of Atmospheric Sciences, University of Washington, Seattle, Washington

3

David S. Battisti

Department of Atmospheric Sciences, University of Washington, Seattle, Washington

The Geophysical Institute, University of Bergen, 5007 Bergen, Norway

4

Cecilia M. Bitz

Department of Atmospheric Sciences, University of Washington, Seattle, Washington

5

∗Corresponding author address: Hansi Singh, University of Washington Department of Atmospheric

Sciences, 408 Atmospheric Sciences-Geophysics (ATG) Building Box 351640, Seattle, WA 98195-1640.

hansi@atmos.washington.edu

1

ABSTRACT6

We present a simple model for studying the D-O cycles of the last glacial period, based on7

the Dokken Et Al Hypothesis for D-O Cycles. The model is a column model representing the8

Nordic Seas, and is composed of ocean boxes stacked below a one-layer sea ice model with an9

energy-balance atmosphere; no changes in the large-scale ocean overturning circulation are10

invoked. Parameterizations are included for latent heat polynyas and sea ice export from the11

column. The resulting heuristic model was found to cycle between stadial and interstadial12

states at times scales similar to those seen in the proxy observational data, with the presence13

or absence of perennial sea ice in the Nordic Seas being the defining characteristic for each14

of these states. The major discrepancy between the modelled oscillations and the proxy15

record is in the length of the interstadial phase, which is shorter than that observed. The16

modelled oscillations were found to be robust to parameter changes, including those related17

to the ocean heat flux convergence (OHFC) into the column. Production of polynya ice18

was found to be an essential ingredient for such sustained oscillatory behavior. A simple19

parameterization of natural variability in the OHFC enhances the robustness of the modelled20

oscillations. We conclude by discussing the implications of such a hypothesis for the state21

of the Nordic Seas today and its state during the Last Glacial Maximum, and contrasting22

our model to other hypotheses that invoke large-scale changes in the Atlantic meridional23

overturning circulation for explaining millennial scale variability in the climate system. An24

extensive time scale analysis will be presented in Part II.25

1

1. Introduction and Motivation26

a. Evidence for Dansgaard-Oeschger (D-O) Cycles27

Ice cores from Greenland (GRIP, NGRIP, and GISP) show that the high latitude cli-28

mate system experienced unusually large, abrupt warmings during the last ice age. These29

abrupt warmings are now known as Dansgaard-Oeschger (D-O) events, in honor of scientists30

W. Dansgaard and H. Oeschger (see, e.g. Dansgaard et al. 1993, 1984). The Greenland ice31

cores suggest that a typical D-O event is characterized by an abrupt warming of surface32

temperatures from glacial conditions (approximately -30°C) to near-interglacial conditions33

(approximately -20°C) over the Greenland Ice Sheet. This transition from glacial to inter-34

glacial conditions occurs rapidly, on a timescale of 20 years or less. The ensuing period of35

interglacial conditions persists anywhere from 100 to 600 years, over which time gradual36

surface cooling occurs. At the end of this interstadial phase, a more rapid decline in temper-37

ature harkens a return to glacial conditions. The sum of the stadial and interstadial phases,38

corresponding to a full cycle period, appears to be on the order of 1500 years, which we will39

henceforth call a D-O cycle.40

D-O cycles are evident in the ice cores and other records between 30,000 and 70,00041

years ago, and they are notably absent during the Last Glacial Maximum itself. Authors42

have, variously, reported anywhere between 20 and 27 D-O cycles, depending on the criteria43

for defining a D-O cycle (see Dansgaard et al. 1993; Rahmstorf 2002). From our current44

collection of proxy records of the geologic past, the D-O cycles constitute the most robust45

form of millennial-scale variability in the climate system. We point the reader to the recent46

review papers by Rahmstorf (2002), Alley (2007) and Clement and Peterson (2008) for47

evidence of the global signature of D-O cycles during the last glacial period.48

Evidence for large-scale ocean circulation changes occurring in concert with D-O events49

is mixed. There is ample evidence for abrupt changes in the overturning circulation in50

the North Atlantic associated with the abrupt cooling events that occurred during the last51

2

deglaciation known as Heinrich 1 and the Younger Dryas (see the review by Alley 2007).52

Arguably, only one published study provides convincing evidence that D-O events themselves53

are coordinated with changes in the ocean circulation. Keigwin and Boyle (1999) examined54

planktonic and benthic organisms at Bermuda Rise in the Atlantic and found that marked55

increases in sea surface temperature (SST) in the North Atlantic during Marine Isotope56

Stage 3 (about 32-58 kyr BP) were concommitent with increases in deep water formation and57

ventilation of the North Atlantic, and that these changes plausibly align with the stadial-to-58

interstadial transitions in Greenland. In contrast, a review by Elliot et al. (2002) concluded59

there is little relationship between deep water formation and D-O cycles in the northeast60

Atlantic, suggesting that large-scale ocean circulation changes are not coincident with the61

D-O events themselves.62

In contrast to the mixed evidence for D-O related changes in the North Atlantic Ocean,63

there is ample evidence for large changes in the hydrographic structure in the Nordic Seas.64

For example, Rasmussen and Thomsen (2004) studied planktonic and benthic foraminifera65

from several sedimentary cores from the Nordic Seas and found that significant differences66

in the local temperature and salinity profile accompanied the stadial and interstadial phases67

of the D-O cycles: high SSTs (5 to 7 ℃) and cool deep waters (−0.5 ℃) at the start of68

the interstadial phase, decreasing SSTs and slightly warmer deep waters at the end of the69

interstadial, and ice-covered surface waters and warming deep waters (2 to 8 ℃) during the70

stadial. Recent isotopic planktonic and benthic foraminiferal data published in Dokken et al.71

(2013) also link the stadial and interstadial phases of the D-O cycles to strong and weak brine72

production, respectively, suggesting differences in sea ice growth, both locally and offshore,73

during the stadial and interstadial phases. In both of these studies, the interstadial phase74

coincides with a hydrographic structure in the Nordic Seas similar to that seen today; in75

contrast, the stadial phase coincides with one similar to that seen in the present-day Arctic76

Ocean.77

3

b. Proposed Mechanisms78

One major hypothesis for D-O cycles invokes an instability in the Atlantic Meridional79

Overturning Circulation (AMOC) in glacial times (despite the mixed proxy evidence for80

changes in the North Atlantic circulation, as described in §1a). The AMOC hypothesis81

explains D-O cycles by a hysteresis in the state of the AMOC, wherein the meridional82

circulation abruptly switches between a warm state, where overturning is robust and sinking83

occurs at high latitudes, to a cold state, where overturning is weak and sinking occurs at84

mid-latitudes. Weaver et al. (1991) and Winton and Sarachik (1993) found that it is possible85

to make the AMOC oscillate in an ocean-only model with steady, mixed surface boundary86

conditions.87

Later work by de Verdiere and Raa (2010) found that such oscillations were still a robust88

feature of an Earth system Model of Intermediate Complexity (EMIC); they also found that89

the magnitude of the oscillations was damped by the sea ice response. A large increase90

in ocean heat transport to the high-latitudes and a salinity-driven jump in the equator-91

to-pole density gradient are both features of the AMOC switch mechanism in this model.92

Furthermore, oscillations were found to exist only when a sufficiently large freshwater forcing93

was imposed; forcings lower than this threshold did not produce oscillations.94

Along a similar vein, Loving and Vallis (2005) conducted a series of experiments with95

another EMIC, and found that when the atmospheric boundary was sufficiently cold, the96

AMOC spontaneously underwent millennial time-scale oscillations. Large shifts in the sea97

ice edge enhanced the magnitude of the oscillations and widened the regime over which they98

occurred. While the system was found to oscillate in the cold state (i.e., when the atmo-99

spheric emissivity was set appropriately low), surface freshening did not support oscillations:100

the key determinant of whether the meridional overturning would oscillate between weak and101

strong states was the temperature state of the atmosphere, not the surface salinity forcing.102

The reason for the different results obtained by Loving and Vallis (2005) and de Verdiere103

and Raa (2010) regarding the importance of sea ice for the millennial oscillations, and the104

4

necessity for a strong salinity forcing, are unclear.105

Another class of hypotheses invokes some form of time-varying external forcing to cause106

the AMOC to switch between its warm and cold states, usually a periodic freshwater release107

from Northern Hemispheric ice sheets into the Atlantic Ocean (see Rahmstorf 2002, and108

references therein). While the behavior of ice sheets on millennial time scales is not well109

understood, modeling work on ice sheet binge-purge cycles (see MacAyeal 1993, for example)110

suggests one possible mechanism for such a periodic freshwater forcing. Although periodic111

freshwater forcing of the AMOC is the most widely-accepted modus operandi of the D-112

O cycles, observational evidence for ice-rafted debris (IRD) accompanying each D-O cycle113

remains mixed. Bond and Lotti (1995) examined sediment cores in the North Atlantic114

and found that enhanced IRD is associated with the cold phase of the D-O cycles, albeit115

using only three tie-points between the sediment core chronologies and the chronology of116

the Greenland record of the D-O cycles. Elliot et al. (2001) found that millennial-scale117

excursions in IRD were coordinated with magnetic susceptibility in sediment cores in the118

North Atlantic, with high IRD associated with cold conditions (low magnetic susceptibility)119

in the North Atlantic. On the other hand, Dokken and Jansen (1999) and Dokken et al.120

(2013) examined a high-sedimentation-rate core taken from the Nordic Seas and concluded121

enhanced IRD was associated with the warm phase of the D-O cycle.122

The modeling evidence for episodic freshwater forcing of the AMOC is also mixed. Most123

EMICs and Global Circulation Models (GCMs) subject to freshwater forcing in the North124

Atlantic exhibit air temperature changes over Greenland that are less than half the mag-125

nitude of that seen in the proxy record (see Okumura et al. 2009; Schmittner and Stocker126

2001; Rahmstorf 2002; Vellinga and Wood 2002, for examples), with EMICs displaying sig-127

nificantly lower sensitivity to the hosing than GCMs (Stouffer et al. 2006). The only GCM128

simulations that record a strong, persistent response to freshwater forcing in the North At-129

lantic are those performed in a Last Glacial Maximum (LGM) configuration (Bitz et al. 2007;130

Otto-Bliesner and Brady 2010), suggesting that the sensitivity of the AMOC to freshwater131

5

input depends on the initial state of the climate system (Swingedouw et al. 2009). Most132

notably, Ganopolski and Rahmstorf (2001) find that a smooth, periodic forcing can give rise133

to D-O-like cycles in an EMIC, with abrupt warming similar in magnitude to that seen in134

the proxy record. In sum, observational and modelling evidence supporting the hypothesis135

that a periodic freshwater forcing of the AMOC could be responsible for the D-O cycles is136

promising, but remains inconclusive.137

Another hypothesis invoked for explaining the D-O cycles invokes excursions in the extent138

and thickness of sea ice in the high latitudes, and the resulting shifts in position of the sea139

ice edge in the North Atlantic, as the driving force for the D-O oscillations (see Clement140

and Peterson 2008, for a review of such hypotheses). Such hypotheses sometimes include141

secondary changes in the AMOC. Modeling studies show that changes in sea ice appear to be142

integral to D-O cycles. Li et al. (2005) showed that changes in wintertime sea ice extent in143

the Nordic Seas can account for the changes in temperature and accumulation at Greenland144

like those seen in the proxy data. Chiang et al. (2003) and Chiang and Bitz (2005) showed145

that changes in sea ice extent also affect SSTs throughout the Northern Hemisphere and146

cause meridional displacement in the position of the intertropical convergence zone that147

could explain precipitation changes seen during a D-O cycle. Changes in sea ice extent in148

the Northern Hemisphere have also been linked to the position of the midlatitude jet, the149

strength of the Hadley cell, the position and intensity of the midlatitude storm tracks (Li150

and Battisti 2008), and changes in monsoonal precipitation patterns (Pausata et al. 2011).151

The sea ice hypothesis, however, is not without drawbacks. The major issues are two-152

fold: (1) as of yet, there are no proxy records that can be used to infer directly the thickness153

and extent of sea ice during the D-O cycles, and (2) the exact mechanism by which sea ice154

changes on millennial time scales remains unknown.155

6

c. The Dokken Et Al Hypothesis for D-O Cycles156

Motivated by a lack of clear proxy evidence for changes in the AMOC associated with157

D-O events, in this study we examine whether the D-O cycles can be understood as an158

intrinsic oscillation of the atmosphere-sea ice-ocean system that is local to the Nordic Seas,159

as hypothesized by Dokken et al. (2013), and independent of large-scale changes in the160

overturning circulation. This idea stems from earlier laboratory and theoretical work by161

Welander and colleagues (see Welander 1977; Welander and Bauer 1977; Welander 1982),162

who showed that a tank of freshwater heated from below and cooled from above could exhibit163

self-sustaining overturning oscillations indefinitely. They theorized that the driving force for164

such oscillations is the competing effects related to how heat is transferred from the base165

of the fluid to the surface: heat transfer is efficient when the base is warm and the surface166

is cold until ice forms, and then surface heat loss is inhibited. Eventually, however, enough167

heat builds below to initiate convection, thereby melting the ice. Later experimental work168

showed that such self-sustained relaxation oscillations are a robust feature of seawater that169

is heated from below and cooled from above. A tank of seawater will oscillate indefinitely170

with a periodicity that depends on the imposed heat fluxes, the vertical length scale of the171

tank, and the salinity of the seawater. In their experiments, a fresher, well-mixed layer in172

contact with ice lies above a saltier reservoir below. Heating of the bottom reservoirs results173

in eventual instability, and the entire water column overturns rapidly, resulting in abrupt174

loss of the sea ice. Eventually, however, persistent surface cooling allows ice to re-grow, and175

the cycle continues ad infinitum. While such oscillations may be found to persist indefinitely176

in theoretical models and in laboratory studies, it has remained unclear what relevance such177

experiments have for the Earth’s atmosphere-ocean-ice system.178

We introduce a new hypothesis to describe the D-O cycles invoking the characteris-179

tics of atmosphere-sea ice-ocean oscillations as described by Welander and colleagues, and180

the specific state of the regional climate (perennially ice-covered, seasonally ice-covered, or181

perennially ice-free) within the Nordic Seas during glacial times. In the ocean, our hypothesis182

7

invokes local hydrographic changes in the Nordic Seas, as evidenced by regional paleo-proxy183

sedimentary data from Rasmussen and Thomsen (2004) and Dokken et al. (2013), changes184

that coincide with thermohaline oscillations within the region; however, no large-scale ocean185

circulation changes in the Atlantic are invoked.186

Following Dokken et al. (2013), we propose that the precise etiology of the D-O cycles is187

as follows. During the stadial phase of the D-O cycles, the Nordic Seas are covered in sea188

ice, and conditions are relatively quiescent in the North Atlantic. The seasonal cycle of sea189

ice growth and melt is enough to enhance stratification in the upper ocean: brine rejection190

during sea ice growth enhances the cold halocline while sea ice melt serves to lighten the191

surface mixed layer. Vertical mixing below the halocline is merely diffusive. On the other192

hand, a small amount of ocean heat convergence below the pycnocline slowly raises the193

temperature of this deep region.194

For a millennium or so, the water column remains stable. When the deep ocean warms195

enough to render the water column unstable, the entire water column overturns, bringing196

warm water into contact with the sea ice. As a consequence, the climate system transitions197

into the interstadial phase of the cycle.198

During the interstadial phase, the Nordic seas are seasonally ice free. The stirring of the199

upper ocean by surface-level winds is vigorous. The water column is poorly stratified, and200

the cold halocline is nonexistent. The mixed layer is deep and warm, heated effectively by201

incoming solar radiation and by a persistent but small amount of ocean heat transport, which202

is now focused near the surface. This ice-free state, however, does not last: as turbulent203

surface fluxes cool the surface, sea ice re-forms, the upper ocean begins to re-stratify, and the204

cold halocline begins to re-emerge. On the other hand, the growth of sea ice also prevents205

the upper ocean from releasing stored heat, thereby slowing the stratification process and206

prolonging the onset of the stadial phase of the cycle. Eventually, the surface of the water207

column is cool enough for perennial sea ice cover to return to the Nordic Seas, ending the208

interstadial phase.209

8

Throughout the D-O cycle, we assume that the sinking branch of the AMOC termi-210

nates south of the Nordic Seas, in agreement with long-equilibration GCM LGM simulations211

(Brandefelt and Otto-Bliesner 2009). As a consequence, only a small amount of heat is212

transported into the Nordic Seas compared to the transport in the modern climate (less213

than 10%), and the heat transport is only weakly dependent on the phase of the D-O cycle.214

This paper and one to follow will develop a numerical model of the D-O cycles based215

on the conceptual picture presented so far. The D-O cycles occurred over a large span of216

the last glacial period, which suggests that a D-O model must exhibit such cycling behavior217

over a substantial, physically-relevant swath of the model’s parameter space. An advantage218

of developing a simple, relatively tractable model is that the timescales involved in the D-O219

cycles can be analyzed and explained: why does each phase of the cycle have its own distinct220

timescale, and why are the transitions between the phases asymmetric? Such conundrums221

could be used to test the validity of the conceptual hypothesis and could provide a physical222

understanding of the D-O events themselves.223

This paper will introduce the numerical model in detail, present the basic behavior of224

the model in a control parameter regime, and will go on to describe the sensitivity of the225

model to key parameter values and to validate its robustness. An ensuing paper will focus226

on providing both mathematical expressions and conceptual explanations for the time scales227

seen in the models, including the length of the stadial and interstadial phases and the228

timescale over which transitions between the phases occur. Taken together, these studies229

will serve to introduce a new quantitative framework for understanding the D-O events.230

2. Model Description231

a. Model Overview, Geometry, and State Variables232

Figure 1 provides an overview of the model geometry and the state variables associated233

with each component of the system. The model is columnar, representing the area-averaged234

9

Nordic Seas, and consists of a simple seasonally-varying energy-balance atmosphere overlying235

evolving sea ice and four stacked oceanic boxes. The surface atmospheric temperature, Tatm,236

is the principal state variable for the atmosphere and consists of two key components: a237

surface air temperature over sea ice, T seaIceatm , and a surface air temperature over open water,238

T leadatm . The sea ice model has associated state variables for the thickness of the sea ice, hice,239

and the fraction of the column covered by sea ice, χice. The ocean consists of four boxes. The240

state variables associated with the i-th ocean box are its temperature, Ti, and salt content,241

Φi. All boxes have constant depths with default values set similarly to the present-day242

vertical distribution of temperature and salinity in the Arctic (as described by Aagard et al.243

1981): a mixed layer (ML) of depth 50 m, a pycnocline layer (PC) of depth 300 m, a deep244

layer (DP) of depth 800 m, and an abyssal layer (AB) of depth 3000 m.245

When developing this heuristic column model, we have made the following major as-246

sumptions:247

• Positive or negative energy imbalances at the top and bottom faces of the sea ice drive248

ice melt or growth, respectively. In other words, we assume that the sea ice has zero249

heat capacity (see Semtner 1976).250

• We assume that the brine rejected from growing sea ice interleaves within the PC box251

rather than the ML box (as in Duffy et al. 1999; Nguyen et al. 2009). On the other252

hand, we assume that sea ice melt freshens the ML box. In this way, sea ice growth253

and melt acts as a salt pump and drives ocean stratification.254

• The amount of sea ice in the column plays a role in determining the amount and255

distribution of OHFC (Aagard et al. 1981). An ice-free column will converge heat256

within surface layers, while an ice-covered column will converge smaller amounts of257

heat at depth.258

• The amount of sea ice in the column governs the strength of wind-driven mixing be-259

tween the surface layers (Kraus and Turner 1967).260

10

• We parameterize sea ice production in polynyas and sea ice export processes. Ad-261

ditional sea ice (and its attendant brine) is imported into the column from offshore262

polynyas, while winds and currents export ice from the column into the North Atlantic.263

• We assume that during the last glacial period, precipitation and run-off into the Nordic264

Seas are negligible. The former point is consistent with GCM simulations, which dis-265

play topographical steering of the jet and the midlatitude stormtrack by the Laurentide266

Ice Sheet such that they are positioned well to the south of the Nordic Seas (as in Li267

and Battisti 2008; Braconnot et al. 2007; Laıne et al. 2009). As a consequence, advec-268

tive salt fluxes into the upper ocean layers (which would be expected to balance such269

surface freshwater fluxes) are also assumed to be negligible.270

• As seen in GCM simulations of the last glacial period, we assume that the AMOC is271

shifted south of the Nordic Seas (see Li and Battisti 2008; Brandefelt and Otto-Bliesner272

2009). Thus, the AMOC has little influence on conditions in the Nordic Seas during273

the both the stadial and interstadial phases of the D-O cycles.274

• Over millennial time scales, the column is salt-conserving. This allows the model to275

be integrated for thousands of years.276

Figure 2 provides an overview of the different salt and heat fluxes or flux convergences277

(designated with Q and W , respectively) that determine the temperature and salinity evo-278

lution of the model components.279

b. Ocean Box Temperature Evolution280

In general, the temperature evolution of each ocean box may be described by the Ordinary281

Differential Equation (ODE)282

micwdTidt

= Qtransi +Qintface

i,i−1 −Qintfacei+1,i , (1)

11

where Ti is the temperature of the layer, mi is the mass of the layer, cw is the specific283

heat capacity of water, Qtransi is the net ocean heat flux convergence (hereafter OHFC) into284

the box, and Qintfacei,i−1 and Qintface

i+1,i are the interfacial eddy diffusive heat flux from the box285

below and above. We describe the OHFC parameterization in §b1, and the interfacial flux286

formulation in §e.287

We apply (1) to each ocean box as follows. The AB layer is relatively isolated, lying below288

the level of the Greenland-Scotland ridge. Therefore, we posit that there is negligible OHFC289

directly into the AB layer, and only one interfacial flux, that from the DP box that lies290

directly above it: QintfaceAB−DP . In contrast, the temperature of the DP box evolves as a result291

of both interfacial eddy diffusive heat fluxes from the boxes directly above (QintfaceDP−PC) and292

below (QintfaceAB−DP ), and from OHFC into the layer, Qtrans

DP . The temperature of the pycnocline293

evolves similarly to the DP box, driven by an OHFC, QtransPC , and interfacial fluxes from294

neighboring boxes above and below, QintfacePC−ML and Qintface

DP−PC .295

The mixed layer interacts with the atmosphere, sea ice, and pycnocline box, exchanging296

heat and salt. Consequently, the temperature of the ML evolves in a more complex manner297

than that of the other ocean boxes. The following equation for evolution of the ML follows298

that outlined by Thorndike (1992), but allows for fractional (incomplete) sea ice cover. In299

general, we posit that the temperature of the ML changes according to300

mMLcwdTML

dt= (1− χice)

(QnetSW −Qnet

LW −Q↑turb

)− χiceQML−ice +Qtrans

ML +QintfacePC−ML. (2)

In (2), QtransML is the OHFC into the ML, Qnet

SW is the net incoming shortwave solar radiation301

absorbed by the ML,QnetLW is the net outgoing longwave radiation emitted by the ML,QML−ice302

is the heat flux between the ML and sea ice, and Q↑turb is the turbulent heat flux from the303

ML into the atmosphere. The surface fluxes (shortwave, longwave, and turbulent fluxes) are304

all described in section §2f. The heat flux between the ML and the overlying sea ice (when305

sea ice is present) is formulated using306

QML−ice = C0(TML − T bottomice ), (3)

12

where C0 = 20 W m−2 ℃−1, and T bottomice = −1.8 ℃.307

1) OHFC Formulation308

We formulate the OHFC in a manner consistent with our hypothesis, namely that the309

sinking branch of the AMOC is shifted south of the Nordic Seas during the glacial period,310

and is relatively impervious to changes in sea ice and hydrography therein. As a result, the311

OHFC into the Nordic Seas is reckoned to be a fraction of that seen in the modern climate:312

on the order of 1 W/m2, rather than the 35 W/m2 converging into the region today (Hansen313

and Osterhus 2000). In fact, the OHFC values we use in our model are comparable to that314

seen in the present-day Arctic (Aagard et al. 1981). Furthermore, we distribute this heat315

between the ocean layers in a manner that is dependent on the sea ice cover: into the top316

layers of the ocean (the ML and PC boxes) when the sea ice cover is thin and seasonal, and317

into the deep ocean (the DP box) when the sea ice cover is substantial and the surface ocean318

is well-stratified. The latter point is consistent with the OHFC into the Arctic as described319

by Aagard et al. (1981), who showed that warm, salty water transported from the Atlantic320

into the Arctic will subduct beneath a cold halocline (as posited to exist in the Nordic Seas321

during stadial conditions, when sea ice cover is perpetual), and enter the water column at322

great depth (> 500 m).323

Table 1 describes the notation for, and typical values of, the OHFC into the model ocean324

boxes. The vertical distribution of OHFC depends strongly on χice, the sea ice fraction in325

the column, while the magnitude of the OHFC depends only mildly on this quantity. In326

particular, we parameterize the OHFC into the DP layer as QtransDP = χiceQ

Icetot , where QIce

tot is327

0.5 W m−2 and χice is the sea ice fraction. For the PC layer,328

QtransPC = (1− χice) ·

1

3QNoIcetot , (4)

where QNoIcetot is the total OHFC when the system is in an ice-free state, reckoned to be 3 W329

m−2 in the control simulation. Analogous to the PC, the OHFC into the ML is nil when the330

13

column is ice-covered, but is non-zero when the column is ice-free:331

QtransML = (1− χice) ·

2

3QNoIcetot . (5)

In equations (4) and (5), the factors of 1/3 and 2/3 permit us to partition the OHFC between332

the PC and ML boxes, respectively, allowing for the heating to be surface-weighted such that333

a greater fraction converges in the ML box than in the PC box.334

c. Sea Ice Evolution335

We model the seasonal growth and melt of the sea ice after Thorndike (1992). When sea336

ice cover is complete (there is no open water), the sea ice evolves according to337

ρiceLfdVicedt

= −QML−ice +QnetLW −Qnet

SW + ρiceLfdpicedt− Iexport, (6)

where ρice is the density of sea ice, Lf is the heat of fusion for seawater, Vice is the volume338

of the ice, QML−ice is the heat flux into the ice from the ML (formulated according to (3)),339

QnetSW is the net shortwave radiation incident on the ice, Qnet

LW is the net longwave radiation340

at the top surface of the ice, dpice/dt is a sea ice source term to model distally-formed sea341

ice advected into the column from polynyas, and Iexport is a sea ice sink term to model ice342

export from the Nordic Seas. The formulation of the shortwave, longwave, and atmospheric343

heat transport fluxes in (6) will be described in §2f. We describe the polynya source and ice344

export sink terms in the following two paragraphs.345

As described in Martin et al. (1998) and Cavalieri and Martin (1994), we posit that346

latent heat polynyas driven by katabatic winds blowing off the Fennoscandian Ice Sheet347

(FSIS) were responsible for the production of sea ice along the FSIS during the last glacial.348

This ice is then swept into the wider Nordic Seas by winds and currents, accelerating ice349

growth beyond that expected from local conditions. In our model, we set∫

(dpice/dt)dt = 1.0350

m per cold season, provided that the sea ice thickness within the column is greater than or351

equal to 1.0 meter. When hice is less than 1.0 m, there is no extra polynya ice added to the352

14

column. Note that brine rejected from polynya ice is also assumed to interleave into the PC353

layer, just like brine rejected from sea ice formed in situ.354

We hypothesize that some of the sea ice formed within the Nordic seas during the last355

glacial cycle was exported from the Nordic Seas into the North Atlantic. In order to account356

for the export of sea ice from the column, we formulate a yearly cumulative export term,357

Iexport as the integrated sum of the ice grown within the column per year, multiplied by an358

ice export factor Cr:359

Iexport = Cr

∫1year

(dV growth

ice

dt

)dt. (7)

Each year, we remove a quantity of ice from the column, Iexport, which corresponds to the360

yearly export of (100 × Cr)% of the ice formed within the column per annum. We use361

Cr = 0.05, corresponding to the export of 5.0% of the ice formed within the Nordic Seas per362

year.363

When sea ice cover is incomplete and open water is present, energy conservation requires364

that the sea ice evolve according to365

ρiceLfdVicedt

= χice(QnetLW −Qnet

SW −QML−ice) + ρiceLfdpicedt− Iexport, (8)

where χice is the fraction of the column covered by sea ice. In general, the volume of sea ice366

present is related to the ice thickness (hice) and ice area by Vice = hiceχice. The ice fraction,367

χice may be diagnosed for a given hice using the following formulation:368

χice =

0 for hice = 0m

2hice for hice ∈ (0, 0.5]m

1 for hice > 0.5m .

(9)

Equation (8) reduces to equation (6) when hice > 0.5 m, meaning that χice = 1.369

15

d. Ocean Salinity370

We formulate the salinity evolution of the different components of the system using an371

equation for salt content (in units of grams salt) within the i-th layer:372

dΦi

dt= W intface

i,i−1 −W intfacei+1,i +W x

i , (10)

where W intfacei,i−1 and W intface

i+1,i refer to the interfacial salt fluxes with layers below and above,373

and W xi is a term representing horizontal or vertical advective fluxes of salt due to sea ice374

growth and melt, as well as transport processes (which we will detail further below). The375

salinity may be derived from the salt content using Si = Φi

ρihi, where hi is the depth of the376

i-th layer, and ρi is its density. We compute the density of the box in this and all subsequent377

calculations using the full equation of state for seawater (from Gill 1982).378

The salt content of the ML evolves due an interfacial salt flux from the PC, W intfaceML−PC ,379

and a negative salt flux from sea ice melt, WmeltML . Given a particular rate of sea ice melt,380

dVice/dt, and assuming (for simplicity) that sea ice has no salt content, the meltwater flux381

may be formulated as382

WmeltML = ρice

S0ML(

1− S0ML

1000

) dV meltice

dt, (11)

where S0ML is a reference salinity (taken to be the initial salinity of the mixed layer). We383

derive equation (11) from the assumption that the mass of ice melted per unit time results384

in a mass flux of salt-free water (equal to the mass flux of the melted ice) added to the mixed385

layer, thereby lowering its total salt content. Note that we use the reference salinity, S0ML386

to estimate the negative mass flux of salt associated with this sea ice melt rate. This allows387

the column to be salt-conserving over time scales exceeding one year, since the negative salt388

flux associated with sea ice melt is balanced by a positive salt flux into the pycnocline due389

to brine rejected when the sea ice was formed.390

The salt content of the PC evolves due to both brine rejected as a result of sea ice391

formation, W brinePC , a freshwater recirculation flux resulting from ice export, W export

PC,DP (see the392

following paragraph for a description of this term), and interfacial fluxes from the neighboring393

16

ML and DP boxes, W intfaceML−PC and W intface

PC−DP , respectively. As described in coupled modeling394

studies by Duffy et al. (1999); Nguyen et al. (2009), we formulate our model so that all the395

brine rejected from sea ice formation (both within the column and in polynyas) is assumed396

to sink to the depth of the pycnocline and interleave therein, thereby enhancing the cold397

halocline. We formulate this salt flux into the pycnocline analogously to the negative salt398

flux into the ML due to sea ice melt, equation (11):399

W brinePC = ρice

S0ML(

1− S0ML

1000

) dV growice

dt(12)

The salt content of the DP box evolves via interfacial fluxes from the neighboring boxes400

PC and AB boxes (W intfacePC−DP and W intface

DP−AB, respectively), as well as a negative salt flux401

arising from ice export from the column, W exportPC,DP . We model this ice export term as follows:402 ∫

1 year

W exportPC,DPdt = ρice

S0ML(

1− S0ML

1000

)Iexport. (13)

The rationale behind this term is that ice exported from the Nordic Seas into the North403

Atlantic where it melts and is entrained into the sinking branch of the AMOC south of404

the Nordic Seas, creating fresher water below that subsequently recirculates back into the405

Nordic Seas at the level of the PC and DP layers. The ice export sink term causes the406

column to salinize over long time scales; including this freshening term allows the column to407

be salt-conserving. The fraction of the negative salt flux added to the PC and DP is FPC408

and (1− FPC), respectively, where the default value of FPC is taken to be 0.8. Finally, the409

AB box evolves solely through an interfacial diffusive salt flux with the DP box, W intfaceDP−AB.410

e. Formulation of Interfacial Fluxes and Convective Mixing411

We parameterize vertical mixing processes between ocean boxes as follows. When the412

system is stably stratified, the density of the underlying box j is greater than that of the413

overlying box i: ρj > ρi. In this case, mixing between boxes will be either eddy-driven414

or diffusive, depending on the depth of the boxes involved. In general, interfacial mixing415

17

between the i-th and j-th box results in the following heat and salt fluxes:416

Qintfaciali,j =

2KTi,jcw(ρiTi − ρjTj)

hi + hj(14)

W intfaciali,j =

2KSi,j(ρiSi − ρjSj)hi + hj

, (15)

where hi is the depth of the i-th box, and KXi,j is the mixing coefficient for property X across417

the interface between boxes i and j. The assumption behind this formulation is that mixing418

across interfaces results in the exchange of equal volumes of seawater between boxes, water419

that carries the temperature and salinity properties of its parent box into its new residence,420

thereby changing the state variables of both boxes.421

We have assigned the coefficients of vertical mixing, KTi,j and KS

i,j, between each set422

of boxes based on the depth of the layers in question. As described in Wunsch and Ferrari423

(2004) and Gargett (1984), turbulent eddies are the dominant mode of vertical mixing within424

the the top 200 m of the ocean. Such eddies mix heat and salt in the upper ocean, with425

the magnitude of the mixing coefficient on the order of 1× 10−4 m2s−1 (as in Gargett 1984).426

In such a case, we expect that KT = KS, since both properties will be transported equally427

by eddy-driven mixing processes. We posit mixing at the interface between the ML and PC428

boxes is eddy-driven, so KTML−PC = KS

ML−PC = O(10−4) m2 s−1.429

As described in Kraus and Turner (1967), the extent of mixing in the upper layer of the430

ocean is wind-driven, and therefore moderated by the fraction of ocean surface covered with431

sea ice. Therefore, we assign specific values to KT,SML−PC based on the extent of sea ice cover,432

as follows:433

KT,SML−PC = χiceK

T,S−IceML−PC + (1− χice)KT,S−NoIce

ML−PC (16)

In equation (16), we have assigned KT,S−IceML−PC = 1 × 10−4 m2 s−1 and KT,S−NoIce

ML−PC = 6 ×434

10−4 m2 s−1. Equation (16) posits a larger amount of upper ocean mixing when χice < 1435

and a smaller amount of upper ocean mixing when χice ≈ 1.436

On the other hand, vertical mixing at depths greater than 400 m is dominated by diffusive437

processes, as described by Huppert and Turner (1981). In this realm, heat and salt diffuse438

18

at separate rates, based on their molecular diffusivity. At a given depth, the molecular439

diffusivity of heat exceeds the molecular diffusivity of salt by a factor of at least 10, giving440

KT = 10KS. Furthermore, we expect that the magnitude of mixing between the PC and441

DP boxes is larger than that between the DP and AB boxes, due to the greater depth of442

the latter. Therefore, we have set the remainder of the vertical eddy diffusive coefficients443

as follows: KTPC−DP = 10−5 m2 s−1, KS

PC−DP = 10−6 m2 s−1, KTDP−AB = 10−7 m2 s−1, and444

KSDP−AB = 10−8 m2 s−1.445

When the density of an overlying box exceeds that of an underlying box, we assume446

mixing is done by convection. In this case, we completely mix the temperature and salinity447

in the two layers.448

f. Formulation of Surface Conditions, Processes, and Fluxes449

We diagnose the surface air temperature as an average of the surface temperature over450

open water, TML, and the surface air temperature over sea ice, T topice , weighted by the fraction451

of sea ice cover in the column:452

Tatm = χiceTtopice + (1− χice)TML (17)

We describe the calculation of T topice , the temperature at the top of the ice, in equation (22).453

We formulate the surface processes included in equation (2), the equation governing the454

evolution of the ML box, and (6), the equation governing the evolution of the sea ice, as455

follows. First, following Thorndike (1992) we assign the mean shortwave flux reaching the456

surface during the warm half of the year as FSW ≈ 200 W m−2 and as 0 W m−2 during457

the cold half. Taking into account the albedo, αi, of the surface receiving the radiation, we458

19

obtain the following formulation for the net shortwave fluxes:459

QnetSW =

(1− αice)FSW over sea ice in summer

(1− αocn)FSW over open ocean in summer

0 over sea ice or open ocean, in winter

(18)

Since αice ≈ 0.6 and αocn ≈ 0.1, the net shortwave flux absorbed during the warm half of460

the year by the ML and sea ice, respectively, are 180 W m−2 and 80 W m−2.461

Next, the net longwave flux in (2) and (6) can be approximated as per Thorndike (1992)462

as follows:463

QnetLW ≈

A+BTsurfn

−D/2, (19)

assuming the surface temperature (either ocean mixed layer, sea ice, or both) is in equilibrium464

with the overlying atmosphere, where Tsurf is the temperature of the surface, the constants465

A and B are equal to 320 W m−2 and 4.6 W m−2 ℃−1, respectively, and D, the atmospheric466

heat transport flux convergence is taken to be 90 W m−2. The quantity n refers to the optical467

depth of the atmosphere, and may be used to approximate the effects of re-absorption and468

re-emission between the surface and atmosphere. As in Thorndike (1992), n depends on469

season and sea ice cover. Thus, equation (19) becomes470

QnetLW =

(A+BTML)/ns −D/2 over open ocean, summer

(A+BTML)/nw −D/2 over open ocean, winter

A/ns −D/2 over sea ice, summer

(A+BT topice )/nw −D/2 over sea ice, winter,

(20)

where nw = 2.5 during the cold half of the year for both ice-covered and ice-free periods,471

and ns = 2.8 during the warm half of the year during ice-covered periods, and ns = 2.9472

during the warm half of the year during ice-free periods. During the warm half of the year,473

we assume T topice = 0 ℃.474

20

The temperature at the top of the ice during the cold half of the year is found from475

equating the energy fluxes at the top surface of the ice QnetLW = Q↑cond, where Q↑cond is the476

conductive heat flux through the ice. From Thorndike (1992),477

Q↑cond =k(T topice − T botice )

hice. (21)

Substituting (21) and (20) into QnetLW = Q↑cond, and solving for T topice during the cold half of478

the year gives479

T topice =1

B/nwinter + k/hice

(kT botice

hice− A

nwinter+D

2

). (22)

Finally, we approximate the atmosphere-to-ocean turbulent heat flux, Q↑turb, using a bulk480

formulation (Peixoto and Oort 1992, as in): Q↑turb = BT (TML − Tref ), where BT ≈ 5.0 W481

m−2 ℃−1, and Tref is the minimum temperature the ML box may attain (-1.8 ℃).482

g. Numerical Methods483

We used the forward Euler method to discretize the time derivatives found in the model484

equations. We integrated the model asynchronously in order to accommodate the very485

different time scales involved in the components. Surface processes, including those involving486

the atmosphere, mixed layer, and sea ice, evolve rapidly on scales shorter than a season (as487

described in Thorndike 1992). For these components, we used a time step of ∆t = 1/200488

yr. Middle and deep oceanic processes, however, have much slower time scales (Wunsch and489

Ferrari 2004), allowing integration of these components with a time step of ∆t = 1/20 yr.490

3. Model Results491

The model, as detailed in previous sections, replicates many of the essential features of492

the D-O cycles (see Figure 3 for a 6,000 year integration of the standard model).493

We begin by considering the stadial phase in the standard model output, which is ap-494

proximately 1100 years long with perennial sea ice cover in the Nordic seas. Seasonally, the495

21

sea ice has a maximal thickness of approximately 3.0 m at the end of winter and a minimum496

thickness of 2.3 m at the end of summer (Figure 4). With this perennial sea ice, surface air497

temperature is approximately -22 ℃ during the cold season, for an average annual temper-498

ature of -11 ℃ (Figure 5). The temperatures in the upper ocean boxes are also close to the499

freezing point: TML ≈ −2 ℃ and TPC ≈ −1 ℃. The temperature in the PC box, however,500

increases slightly over the course of the stadial period due to diffusive heat fluxes between501

the PC and DP boxes (not shown). The temperature in the DP box rises substantially,502

increasing from approximately 4.4 to 4.9 ℃ over the 1100 years of the stadial period (Figure503

6). Though the temperatures of the ML and PC are close throughout the stadial phase,504

their salinities are very different (Figure 7). During the stadial, SPC increases from 34.2505

to 34.9 psu, while SML increases from 33.5 to 34.0 psu (Figure 7), resulting in extremely506

strong density stratification in the upper ocean throughout this period (Figure 8). Such a507

stable cold halocline is similar to that seen today in the Arctic Ocean under its perennial508

sea ice cover (Aagard et al. 1981). Finally, the salinity of the DP decreases slowly, from509

32.8 psu to 32.2 psu, over the course of the stadial period (Figure 7) due to sea ice exported510

into the Atlantic: sea ice export causes freshening of the Atlantic water that circulates in511

this layer (as diffusive exchange with the PC box would increase salinity, and exchange with512

the AB box is negligible). Indeed, making the export coefficient Cr zero results in little net513

freshening of the DP layer (not shown).514

The transition from the stadial to interstadial period is extremely rapid (as seen in Figures515

6 and 7), occurring over a brief three year span (not shown). Convective overturning of the516

PC and DP boxes initiates the transition to the interstadial period. This overturning event517

is caused by a combination of factors, all of which conspire to decrease ρDP and increase ρPC ,518

thereby equalizing their densities and initiating convection (Figures 8 and 9): the increase519

in the temperature of the DP layer (due to weak OHFC into the DP box), the decrease520

in the salinity of the DP layer, and the increase in the salinity of the PC layer (due to521

brine rejection from sea ice formation and subsequent export). The overturning of the PC522

22

and DP boxes quickly raises the temperature of the PC box to about 3 ℃ at the start of523

the interstadial (Figure 5). Next, the large temperature gradient between the ML and PC524

drives an upward flux of heat into the ML box through eddy-driven vertical mixing, thereby525

increasing the temperature of the mixed layer, increasing the ice-ocean conductive heat flux526

by several orders of magnitude, and allowing the perennial sea ice cover to melt rapidly,527

giving way to seasonal sea ice (Figures 4 and 5). The overturning event also causes the528

salinities of the PC and DP boxes to equalize. The salinity of the mixed layer decreases529

rapidly due to abrupt sea ice melt during the stadial-to-interstadial transition, hitting a low530

point of SML ≈ 31.8 psu (Figure 4).531

The interstadial phase is approximately 150 years in length. The model output, par-532

ticularly from Figures 4 and 5, suggests that the interstadial period has several different533

phases. The first phase corresponds to the rapid decay of the PC box temperature from its534

maximum value of 3 ℃ to about 2 ℃, a process that takes approximately 20 years. After535

an initial adjustment process (< 5 years) in which the temperature of the ML box increases536

from -1.8 ℃ to 3 ℃ (annually averaged), the ML box evolves similarly to the PC box, though537

it experiences substantial seasonal cycling due to its interaction with the atmosphere and538

sea ice (see Figures 5 and 6). Following this initial rapid decay, we also discern that there539

is a phase of slow decay (with a long time scale of several hundred years), in which the540

maximum thickness of the sea ice increases gradually (from 2.2 m to 2.5 m; Figure 4), and541

the temperatures of the ML and PC boxes decay slowly (from 1.5 ℃ to 1.2 ℃ in both the542

ML and PC boxes; Figure 6). The salinities of the PC and ML boxes, which briefly equalized543

at the beginning of the interstadial period due to rapid sea ice melt and eddy-driven vertical544

exchange, diverge during the interstadial, and the ML and PC layers begin to re-stratify: as545

the maximum winter thickness of the sea ice increases throughout this period, this results546

in a net flux of brine into the PC box, thereby increasing its salinity (Figures 7, 8, and 9).547

The transition back to the stadial phase, in which the system has year-round sea ice548

cover, is slower than the abrupt transition to the interstadial phase. There appears to be549

23

two time scales associated with this transition. There is short time scale of approximately 10550

to 20 years associated with the summer temperature of the ML box returning to below -1.0551

℃, the sea ice thickness returning to its stadial range, and the end-of-winter atmospheric552

temperature returning to -22 ℃ (Figures 4 and 6). There is also a longer time scale of553

approximately 50 years associated with the time required for the pycnocline temperature554

and the end-of-summer ice thickness to relax to their averaged stadial values, -1.0 ℃ and555

2.3 m, respectively.556

Figure 10 provides a pictorial summary of the major processes that characterize the557

stadial and interstadial phases in the model. The difference between the fluxes in the two558

phases results from major differences in the sea ice cover: a substantial, perennial sea ice559

cover during the stadial period, and a thinner, seasonal sea ice cover during the intersta-560

dial. During the stadial phase, a robust perennial ice cover allows for vigorous ice export561

and accompanying freshwater recirculation into the DP layer; during the interstadial, both562

export and recirculation are comparatively weaker as there is less ice to export. During the563

interstadial phase, the column spends a significant amount of time in an ice-free state, and564

the resulting wind-driven mixing between the ML and PC boxes is strong compared to that565

during the stadial phase. The perennial sea ice cover during the stadial is augmented by566

polynya ice, which is imported into the column (with its accompanying brine), while the567

interstadial column, which only possesses a seasonal sea ice cover, does not receive extra ice568

and brine input from polynyas. During the interstadial phase, large amounts of sea ice grow569

and melt annually with the seasonal cycle, while during the stadial phase, this seasonal cycle570

in sea ice cover is much smaller. As a consequence, the input of freshwater into the ML box571

from melt and the PC box from brine rejection is much larger during the interstadial than572

the stadial. Finally, as we have summarized in Table 1, the OHFC during the stadial period573

is into the DP box, while the OHFC during the interstadial is shared by the ML and PC574

boxes.575

24

4. Sensitivity Studies576

a. OHFC Parameters577

Figure 11 provides an overview of the sensitivity of the stadial and interstadial time scales578

to changes in the OHFC parameters, QNoIcetot and QIce

tot . First and foremost, we see that for579

the model to support D-O-like oscillations, the net heat transport into the Nordic Seas must580

be small throughout the cycle - comparable to that seen today in the Arctic, and a small581

fraction of that into the Nordic Seas today. This finding is consistent with our supposition582

that the sinking branch of the AMOC terminates south of the Nordic Seas throughout the583

glacial period, and that the position of this sinking branch is relatively insensitive to the584

state of the Nordic Seas.585

In general, we see that the model produces robust oscillations over a reasonable range586

of values of these OHFC parameters, and that this range is between one to two orders587

of magnitude less than the OHFC into the Nordic Seas today. In particular, we see that588

oscillations reminiscent of D-O events (with interstadial phases greater than 75 years in589

length and with total cycles between 1000 and 1500 years in length) occur over a 50%590

change in QNoIcetot and a 40% change in QIce

tot .591

The ranges of QNoIcetot and QIce

tot in Figure 11 were chosen to highlight the central region592

about which physically relevant oscillations exist, given the default parameter values listed593

in Table 2. We note that oscillations still occur for QIcetot < 0.4 W/m2 and QNoIce

tot < 0.75594

W/m2 (even though the length of the interstadial phase is less than 75 years), suggesting595

that the oscillations themselves are robust to changes in OHFC values; the length of the596

interstadial phase, however, is more sensitive to such changes.597

In order to produce Figure 11, we have defined the onset of the interstadial phase when598

the PC and DP boxes become buoyantly unstable and overturn (as described in §3); the599

end of the interstadial phase (and beginning of the stadial phase) is defined to be when the600

temperature of the PC box has relaxed to 0.1 ℃ of its stadial value; and the time required for601

25

a single cycle (stadial plus interstadial phases) is taken to be the time between overturning602

events. It is possible for the interstadial phase to go on forever (hatched regions of Figure 11)603

when the ML and PC boxes cannot rid themselves of excess heat, a requirement if perennial604

sea ice is to re-form. Whether or not the model re-forms perennial sea ice depends on several605

bifurcation parameters. We will carefully consider these parameters and their implications606

in a forthcoming paper.607

From examining Figure 11, we see that the total OHFC when ice is present (QIcetot ) affects608

the length of the stadial, and hence how long it takes for the ocean column to become609

convectively unstable and overturn. Such a process brings deep ocean heat up to the surface610

and melts the ice, initiating the interstadial phase. On the other hand, we see that the total611

OHFC when ice is absent (QNoIcetot ) affects the interstadial time scale, the time required for612

the column to cool and support the re-growth of perennial sea ice. We will further describe613

the relationship of these and other parameters to the lengths of the interstadial and stadial614

phases in §4e.615

b. Vertical Distribution of OHFC during Ice-Free Conditions616

In our control simulation, we have apportioned QIceFreetot , the total (vertically-integrated)617

OHFC during ice-free conditions, as follows: two-thirds of the total heat flux enters the ML618

box, and one-third of the total heat flux enters the PC box. The sensitivity to the choice of619

how we distribute the OHFC vertically is tested in two alternate model configurations: one620

with a larger proportion of the ice-free OHFC concentrated in the ML box and less in the621

PC box, and one with a smaller proportion of ice-free OHFC concentrated in the ML box622

and more in the PC.623

Table 3 shows the sensitivity of the model to such changes in the distribution of QIceFreetot .624

Here, we present the timescales produced by the model in the case where we have modified625

the value of BT to 5.5 W m−2 ℃−1 (from 5.0 W m−2 ℃−1) and have left the values of other626

control parameters unchanged. We record the length of the interstadial phase and cycle627

26

duration when the ratio of OHFC into the ML and PC boxes are shifted from 2:1 (as in628

the control) to 1:1 (more of total OHFC into the PC box) and 3:1 (more of total OHFC629

into the ML box). We find that while the lengths of the stadial and interstadial phases are630

modified by this choice, the fundamental behavior of the model (oscillations between stadial631

and interstadial phases on a millennial time scale) does not change appreciably (the changes632

in the length of the stadial and interstadial phases in the 3:1 and 1:1 cases are within 5%633

and 60%, respectively, of those in the 2:1 control case). Therefore, we conclude that the634

model is robust to changes in the vertical distribution of OHFC during ice-free conditions.635

c. Temporal Variability in Ocean & Atmospheric Heat Flux Convergence636

1) OHFC637

Observational studies show that the OHFC into the Nordic Seas today is variable on638

interannual time scales on the order of 5 to 10 years (see Hatun et al. 2005). Although639

the North Atlantic gyre is displaced southward in the glacial climate due to changes in the640

windstress curl (see, e.g. Li and Battisti 2008; Braconnot et al. 2007; Laıne et al. 2009), we641

explore the impact of general stochastic variability in the OHFC on the modelled D-O cycles642

by adding a red noise component to QNoIcetot :643

(QNoIcetot )i = λ(QNoIce

tot )0 + (1− λ)(QNoIcetot )i−1 + ε, (23)

where λ is ∆t over the time scale over which OHFC is said to be variable (for ∆t = 1 year,644

realistic modern variability in the OHFC is simulated when λ ≈ 1/5 to 1/10), and ε is the645

appropriately-scaled white noise process. Substituting such a process for our static QNoIcetot646

with other parameters held at their control values results in an increase in the length of647

the interstadial phase, from 150 years to nearly 180 years. Furthermore, we find the phase648

space that supports physically relevant oscillations (as defined in §4a above) is widened in649

the QNoIcetot dimension by approximately 10% with such a modification (not shown). Thus650

simulating the interstadial heat flux convergence as a red noise process tends to increase the651

27

robustness of the model.652

2) Atmospheric Heat Flux Convergence653

It is known that the atmospheric heat transport into the polar regions is also variable654

(see Overland and Turet 1994). This variability is well-approximated as white noise and may655

be added to D as a good representation of this real physical phenomenon. When white noise656

with a standard deviation of 5 W m−2 is added to D with all model parameters maintained as657

in Table 2 (the ‘default’ parameter regime), the average length of the interstadials increases658

from 180 years to 210 years (with a standard deviation of 100 years), while the average659

length of a cycle remains approximately unchanged.660

d. Polynya Activity661

Removing the polynya parameterization in the model (setting∫

1 year(dpice/dt)dt = 0),662

while leaving all other parameters at their default values prevents cycling between stadial663

and interstadial states. Rather than oscillating, the model remains in an interstadial state664

perpetually. Re-tuning other model parameters does not prevent this. Thus, one robust665

result from our model is that ice production in latent heat polynyas are a prerequisite for666

D-O cycles: removing this term precludes the system from oscillating between stadial and667

interstadial states.668

e. General Sensitivity to Parameter Choices669

Table 4 displays the general sensitivity of the model to key parameter values. From670

examining the table, we see that the parameters broadly fall into two categories: those that671

tend to change the length of the stadial phase, and those that tend to change the length of672

the interstadial phase. In a subsequent paper, we will will provide a detailed analysis of how673

changes in these parameters affect the lengths of the stadial and interstadial phases, and the674

28

time scales therein. Here, we provide a brief overview of these parameter dependencies, and675

attempt to obtain an intuitive sense of how the behavior of the model is affected by these676

different classes of parameters.677

The parameters that change the length of the stadial phase are those that affect the678

rate the column becomes buoyantly unstable and overturns. Overturning is initiated by679

buoyant instability between the PC and DP boxes (see §3). Thus, parameters that affect680

ρPC and ρDP , the densities of the pycnocline and deep layers, will tend to affect the length681

of the stadial phase of the oscillation; these include Cr, the ice export factor, and QIcetot , the682

OHFC. If Cr is increased, the salinity of the PC box increases and the salinity of the DP box683

decreases at a faster pace, shortening the time it takes for ρPC to equal ρDP and shortening684

the stadial phase. Similarly, if QIcetot is increased, the temperature of the DP box increases at685

a faster pace, again shortening the time required for ρPC to equal ρDP and, thus, shortening686

the length of the stadial phase.687

The length of the interstadial phase is determined by the longest interstadial time scale,688

which depends on a combination of parameters related to the delivery of heat into the mixed689

layer. During the interstadial phase, there is a delicate balance between the rate at which690

heat is delivered into the mixed layer and the rate at which it is fluxed from the mixed layer691

to the atmosphere; shifting this balance towards the former lengthens the interstadial, while692

shifting it towards the latter tends to shorten it. In general, QNoIcetot (the OHFC during ice-693

free conditions) and KNoIceML−PC (the mixing coefficient between the ML and PC boxes during694

ice-free conditions) are both related to the rate at which heat is delivered into the ML,695

while BT is related to the rate at which heat is fluxed out of the ML. Increasing QNoIcetot and696

KNoIceML−PC tends to lengthen the interstadial time scale up to a certain point beyond which697

further increases in these parameters prevent the system from returning to a stadial phase.698

Similarly, decreasing BT tends to lengthen the interstadial phase up to a certain point, when699

any further decreases tend to prevent the system from returning to the stadial.700

29

5. Discussion and Implications701

a. Discussion702

As with any idealized model, the purpose of our model is to illuminate the essential703

ingredients responsible for the gross characteristics D-O cycles, in particular the time scales704

of the stadial and interstadial phases, the abrupt transition from stadial to interstadial phase,705

and the gentler transition back from the interstadial to the stadial phase. Our results show706

that a model of the area-averaged conditions in the Nordic Seas that combines an energy707

balance atmosphere, a one-layer sea ice model (which includes parameterizations of polynyas,708

ice export, and fractional ice cover), and four stacked ocean boxes (representing the gross709

hydrographic structure of the Nordic Seas) can exhibit the gross characteristics of the D-710

O cycles, including the observed hydrographic structure (Rasmussen and Thomsen 2004;711

Dokken et al. 2013), the time scales associated with the stadial and interstadial phases, and712

the time scales associated with the transitions between phases.713

Proxy data from the GRIP and GISP ice cores show that the D-O oscillations occur with714

a periodicity of approximately 1500 years, with the length of the interstadial phase persisting715

anywhere between 100 and 1000 years and the stadial phase occupying the remainder of the716

period (Rahmstorf 2002). Our model compares reasonably with these observations: within717

the explored parameter space where physically relevant oscillations exist, the total length of718

a modelled D-O cycle is anywhere from 1000 to 1500 years, and the length of the interstadial719

phase is anywhere from 75 years to 500 years. While the total length of the modelled cycle720

and the length of the stadial phase agree well with observational data, the length of the721

modelled interstadial phase is significantly shorter than that seen in the proxy record.722

According to the ice core record, the transitions from the stadial to interstadial phase723

appear to be nearly instantaneous in the Greenland record, suggesting a timescale of less724

than 20 years for this transition (i.e. the temporal resolution of the Greenland ice cores725

record). Our model results show transitions from the stadial to the interstadial phase occur726

30

in less than 5 years, consistent with the observations. In contrast, the transition from the727

interstadial to the stadial phase in the ice core records appears to take over 100 years, and728

possibly longer. Our model results, however, show this transition occurs within 20 to 40729

years, which represents a significant disagreement with observations.730

In summary, we find that there are two major discrepancies between our modelled oscil-731

lations and those seen in the proxy record: in our model the length of interstadial phase is732

too short compared to observations, and the length of the transition from the interstadial733

to stadial phase is too abrupt. These differences may be a result of missing physics in our734

model, or they may be due to the coarse layering of the ocean column, the lack of hori-735

zontal resolution within the Nordic Seas, or the absence of stochastic processes. We have736

gauged the sensitivity of our model to stochastic variations in the OHFC and atmospheric737

heat transport (see §4a) and found that adding this variability increased the length of the738

interstadial phase, though not enough to account for the discrepancy from observations. We739

believe that future work on this hypothesis could focus on accurate modeling of horizontal740

transport and mixing processes within the Nordic Seas as well as on increasing the vertical741

resolution of the ocean column.742

We also note that our model results are consistent with observational inferences (see §1a)743

that indicate a stadial increase in deep temperature of 2 ℃ prior to overturning (Rasmussen744

and Thomsen 2004; Dokken et al. 2013). Additionally, our results corroborate other obser-745

vational inferences suggesting rigorous sea ice production and accompanying brine rejection746

along the FSIS is essential for D-O cycles (Dokken et al. 2013).747

The model results of Winton and Sarachik (1993), Loving and Vallis (2005), and de Verdiere748

and Raa (2010) feature oscillations in the AMOC that are reminiscent of the observed D-O749

cycles. By contrast, our model features D-O cycles due to interactions within an atmosphere-750

sea ice-ocean column, rather than within an Atlantic Ocean scale model: changes in the ocean751

stratification and sea ice cover within the Nordic Seas that are the hallmark of our model752

D-O cycles are independent of any changes in the heat and salt transport associated with753

31

the AMOC. In fact, our model suggests that D-O cycles will only exist if the sinking branch754

of the AMOC does not extend into the Nordic Seas (as it does in the present climate): the755

system will only support a stadial state when ocean heat transport into the Nordic Seas is756

less than 0.005 PW (3 W m−2 averaged over the area of the Nordic Seas) – which is a small757

fraction of what is delivered into the Nordic Seas today.758

In general, we hypothesize that ocean heat transport incident on an ice-covered region759

will tend to subduct and circulate under a cold halocline, while ocean heat transport incident760

on a region that is ice-free will tend to concentrate in the upper ocean and converge there761

while it fluxes into the atmosphere. In the former case, very little heat will converge into762

the region, while in the latter case, a slightly larger amount of heat will converge. Hence,763

the sea ice cover causes changes in the distribution of OHFC that feed back on the sea764

ice cover itself. Work by Bitz et al. (2006) used GCM results to argue that transitioning765

from perennial to seasonal sea ice in the Arctic enhanced brine rejection along the Siberian766

shelf, resulting in an increase in the OHFC into the region. Such changes occurred despite a767

weakening in the strength of the AMOC south of the subpolar seas (Bitz et al. 2006). Thus,768

the OHFC changes we prescribe account for differences in the vertical distribution of how769

heat converges locally in the presence or absence of sea ice and a cold halocline, and are770

independent of the AMOC.771

Consistent with the small value of OHFC, the only advective salt flux convergence (ASFC)772

in our model is into the DP box, and is used to balance the freshwater flux out of the Nordic773

Seas associated with sea ice export. Physically, we envision that exported sea ice melts in774

the North Atlantic, entrains into the sinking branch of the AMOC, and returns at depth to775

the Nordic Seas. The lack of ASFC into the top two boxes is supported by GCM simulations776

run with Last Glacial Maximum (LGM) boundary conditions, which show that the ASFC777

into the Nordic Seas is very small (see Braconnot et al. 2007, and the PMIP2 archive).778

In our model, the slight difference in the magnitudes of the ice-covered and ice-free779

OHFC only affects the details of the modelled oscillations, not the oscillations themselves or780

32

the hydrographic structures of the stadial and interstadial phases. Imposing an ice-covered781

and ice-free OHFC that are both equal to 1 W m−2, for example, results in modeled D-O782

oscillations with an interstadial phase of 150 years and a full cycle period of 850 years.783

Modelling results (Brandefelt and Otto-Bliesner 2009) support our assumption that the784

Nordic Seas were relatively disconnected from the AMOC throughout the last glacial period.785

In the modern climate, the wind-driven subpolar gyre efficiently delivers heat and salt into the786

Nordic Seas; as a result, the sinking branch of the AMOC resides in the Nordic Seas. In the787

glacial climate however, GCM simulations suggest that the Laurentide ice sheet deflected the788

atmospheric jet southward and reduced the storminess over the North Atlantic (Braconnot789

et al. 2007; Laıne et al. 2009). Consistent with the southward deflection of the jet, long-790

equilibration coupled climate model simulations (Brandefelt and Otto-Bliesner 2009) suggest791

the AMOC was weaker and shallower during the glacial period and the sinking branch was792

shifted southward from today’s position in the Nordic Seas.793

b. Implications794

Our model helps explain why D-O cycles are not present in the Holocene. As discussed795

in Section 4e, D-O cycles are only possible when the ocean heat imported to the Nordic Seas796

is small, less than 0.005 PW (3 W m−2 multiplied by the area of the Nordic Seas). The797

OHFC into the Nordic Seas today is about 0.05 PW (Hansen and Osterhus 2000), which is798

an order of magnitude higher.799

Further reasons why the Nordic Seas do not experience stadial conditiona and D-O type800

oscillations during the Holocene may also include the current position of the midlatitude801

storm track and the lack of substantial polynya activity, both of which make it difficult to802

form a strong halocline in the region today. Today, the midlatitude storm track traverses the803

Nordic Seas; during glacial times, on the other hand, proxy evidence (see Pailler and Bard804

2002) and modeling results (see, e.g. Li and Battisti 2008; Braconnot et al. 2007; Laıne et al.805

2009) suggests that the midlatitude storm track was shifted equatorwards from its current806

33

position. While conditions within the Nordic Seas during the last glacial period may have807

been quiescent enough to support the formation of a strong halocline, the current position808

of the storm track may preclude the existence of a halocline due to significant wind-driven809

mixing of the upper ocean in the region. Without a halocline, sea ice formed in the region (if810

any) would be unable to persist due to heat fluxes from warmer waters below. In the Arctic811

basin today, the presence of a strong halocline is crucial for preventing warmer circulating812

Atlantic waters below from melting the sea ice (Aagard et al. 1981). In addition to the813

current position of the storm track, the lack of polynya activity in the Holocene would also814

make it more difficult to form a strong halocline.815

The proxy record of the last glacial period presents another interesting conundrum re-816

garding millennial scale variability: D-O oscillations, which were robust and plentiful during817

the glacial period prior to 24 kyr before present, were absent during the Last Glacial Max-818

imum (LGM) itself. The results of our model suggest that latent heat polynyas, which819

formed off the edge of the Fennoscandian Ice Sheet (FSIS) during the last glacial period,820

were essential for the D-O oscillations: polynyas are required for returning the Nordic Seas821

to the stadial phase of the D-O cycle, and for maintaining the stratification between the ML822

and PC throughout the stadial phase. During the LGM, the FSIS was believed to extend to823

the continental shelf. While katabatic winds would have still emanated from the FSIS in this824

case, and latent heat polynyas would still result, conditions would have been much colder825

with more brine rejected. The continental slope bathymetry over which the brine rejected826

from such polynyas would traverse would be much different from the extensive continental827

shelf over which it flowed and entrained during the glacial. In the former case, it is possible828

that the resulting large volume of cold brine flowing down the continental slope could not829

entrain enough ambient water to interleave into the halocline. Consequently, the brine would830

sink to the bottom of the water column, in a manner analagous to deep water formation831

in coastal polynyas in the present-day Weddell Sea. Thus, it is possible that the region832

only supported a weak halocline, and incoming heat from the North Atlantic converged into833

34

the deep would not be trapped and decoupled from the surface (as we hypothesize is char-834

acteristic of the stadial phases of the D-O cycles), but rather, gradually vented, reducing835

the thickness of the sea ice in the Nordic Seas. In such a case, there would be insufficient836

build-up of deep ocean heat and pycnocline salt to initiate a convective overturning event,837

and the Nordic Seas would remain in a perpetual stadial phase.838

Acknowledgments.839

We wish to acknowledge grants from the Comer Foundation and the U.S. Department of840

Energy Computational Science Graduate Fellowship (DOE CSGF); Aaron Barker for com-841

posing Figure 10; and the helpful comments, suggestions, and critiques of three anonymous842

reviewers, whose thoughtful input improved the quality of the work substantially.843

35

844

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41

List of Tables974

1 Values of QtransX , the OHFC, into the model ocean boxes. Both the general975

formulation and the specific formulation (during ice-covered and ice-free con-976

ditions) are shown. 43977

2 Values of constants and default parameter values used to generate model results. 44978

3 Sensitivity of the model to changes in the relative distribution of the ocean979

heat flux distribution in the mixed layer and pycnocline boxes during ice-free980

conditions. All parameters are as in the control case, but with BT = 5.5 W981

m−2 ℃−1. 45982

4 General sensitivity of the model time scales (length of stadial phase and length983

of interstadial phase) to changes in model parameters. 46984

42

Table 1. Values of QtransX , the OHFC, into the model ocean boxes. Both the general

formulation and the specific formulation (during ice-covered and ice-free conditions) areshown.

Ocean Box General χice = 1 (Ice Present) χice = 0 (Ice Free)

ML, QtransML

23QNoIcetot (1− χice) 0 2.0 W m−2

PC, QtransPC

13QNoIcetot (1− χice) 0 1.0 W m−2

DP, QtransDP QIce

totχice 0.5 W m−2 0

AB 0 0 0

Total sum of above QIcetot = 0.5 W m−2 QNoIce

tot = 3.0 W m−2

43

Table 2. Values of constants and default parameter values used to generate model results.

Depth of Ocean BoxesDepth of ML, hML 50 mDepth of PC, hPC 300 mDepth of DP, hDP 800 mDepth of AB, hAB 3000 m

Mixing CoefficientsKT

(ML−PC),Ice 1.0× 10−4 m2 s−1

KS(ML−PC),Ice 1.0× 10−4 m2 s−1

KT(ML−PC),NoIce 6.0× 10−4 m2 s−1

KS(ML−PC),NoIce 6.0× 10−4 m2 s−1

KTPC−DP 1.0× 10−5 m2 s−1

KSPC−DP 1.0× 10−6 m2 s−1

KTDP−AB 1.0× 10−7 m2 s−1

KSDP−AB 1.0× 10−8 m2 s−1

Optical Depthsns, ice-covered 2.8ns, ice-free 2.9nw, ice-covered 2.5nw, ice-free 2.5

Other Ice and ML Parametersαice 0.65αocean 0.1Ice-ocean conductive flux constant C0 20 W m−2 ℃−1

Atmospheric heat transport D 90 W m−2

OLR constant A 320 W m−2

Temperature-dependent OLR constantB 4.6 W m−2 ℃−1

Summer SW radiation FSW 200 W m−2

ML turbulent heat flux parameter BT 5.0 W m−2 ℃−1

Thermal conductivity of sea ice, k 2.0 W m−1 ℃−1

Brine and Polynya ParametersRange of hice in which polynyas add ice to the column [1.0,∞) m

Ice added by polynyas each year hpolynasice 1.0 mSea ice export residual brine Cr 0.05Fraction of freshwater recirculation flux into PC, FPC 0.80

OHFC ParametersIce-free OHFC, QNoIce

tot 3.0 W m−2

Ice-covered OHFC, QIcetot 0.5 W m−2

44

Table 3. Sensitivity of the model to changes in the relative distribution of the ocean heatflux distribution in the mixed layer and pycnocline boxes during ice-free conditions. Allparameters are as in the control case, but with BT = 5.5 W m−2 ℃−1.

Heat Flux Convergence Interstadial LengthDistribution, No Ice Cycle Duration

QtransML,NoIce : Qtrans

PC,NoIce 180

1 : 1 1170

QtransML,NoIce : Qtrans

PC,NoIce 110

2 : 1 1210

QtransML,NoIce : Qtrans

PC,NoIce 85

3 : 1 1245

45

Table 4. General sensitivity of the model time scales (length of stadial phase and lengthof interstadial phase) to changes in model parameters.

Parameter Change Effect on Stadial Effect on Interstadial

Increase BT little effect shortenIncrease Cr shorten little effect

Increase QIcetot shorten little effect

Increase QNoIcetot little effect lengthen

Increase dpice/dt more ice shortenIncrease Ice Threshold for Polynyas little effect lengthen

Increase KNoIceML−PC little effect lengthen

46

List of Figures985

1 Basic geometry of the model. The model is columnar, and consists of 4 oceanic986

boxes (with default depths as labelled), sea ice, and a basic energy balance987

atmosphere. 50988

2 In-depth geometry of the model, with surface and interfacial fluxes labelled.989

The state variables for the ocean boxes are Ti and Φi, while the state variables990

for the sea ice are hice and χice. Heat fluxes are represented by Qi, and salt991

fluxes are described by Wi. Ice source and sink terms (from polynyas and ice992

export, respectively) are represented by pice and Iexport, respectively. 51993

3 Results for atmosphere and ocean temperature, sea ice thickness, and ocean994

salinity from 6,000 years of output from the model with default parameter995

values. Except for the atmospheric temperature, the model output, which996

reports all state variables biannually (once at the end of the warm season, and997

again at the end of the cold season), has been smoothed with a 5-year running-998

mean filter; the atmospheric temperature has been smoothed with a 25-year999

running-mean filter to mirror the resolution of the δ18O ice core proxy used1000

to infer Greenland surface temperature. In the top panel, the temperature of1001

the PC box is not shown; at the scale of this figure, the temperatures of the1002

ML and PC are indistinguishable. 521003

4 Un-smoothed model output showing the seasonal evolution of sea ice thickness1004

and salinities of the ML and PC ocean boxes during a typical interstadial1005

event. This model output is biannual, with all state variables reported at the1006

end of the winter (cold) season and at the end of the summer (warm) season. 531007

47

5 Results from one typical model interstadial phase, showing surface atmo-1008

spheric, ML, and PC temperatures. This model output is un-smoothed and1009

biannual, with all state variables reported at the end of the winter (cold) sea-1010

son and at the end of the summer (warm) season. Note the strong seasonal1011

cycle in surface and ML temperatures, and the relative lack of a seasonal cycle1012

in the PC. 541013

6 Results from one model D-O cycle, showing surface temperature and ocean1014

box temperatures. Ocean box temperatures have been smoothed with a two-1015

year running-mean filter, while the surface temperature has been smoothed1016

with a 25-year running-mean filter. Note that the 25-year running-mean filter1017

used on the surface temperature output makes the transition from the sta-1018

dial (cold) phase to interstadial (warm) phase appear earlier on the surface1019

temperature time series than the others. 551020

7 Results from one model D-O cycle, showing sea ice evolution and ocean box1021

salinities. All raw model output, originally reported biannually, has been1022

smoothed with a 5-year running-mean filter. 561023

8 2000 years of un-smoothed model output showing the evolution of the densities1024

of the ML, PC, and DP boxes. 571025

9 2000 years of smoothed model output showing (from top to bottom) surface1026

temperature and the density tendencies of the ML, PC, and DP boxes over1027

time. For a given layer, the density tendency due to temperature is ∆ρT =1028

(∂ρ/∂T )0∆T and the density tendency due to salinity is ∆ρS = (∂ρ/∂S)0∆S,1029

where (∂ρ/∂T )0 and (∂ρ/∂S)0 are computed at reference temperatures and1030

salinities for each layer, and ∆T and ∆S are defined with respect to a reference1031

temperature and salinity for each layer. The residual term is the defined as1032

∆ρres = ρ − ρ0 −∆ρT −∆ρS. Model output, originally reported biannually,1033

has been smoothed with a 25-year running-mean filter. 581034

48

10 Schematic figure depicting the key processes characterizing the stadial and1035

interstadial periods, all of which result from differences in the sea ice cover in1036

the two states. During the interstadial period, the substantial seasonal cycle1037

of sea ice results in large freshwater and brine fluxes into the ML and PC. Fur-1038

thermore, the seasonal ice cover of the interstadial fosters more wind-driven1039

mixing between the ML and PC boxes, less ice export and accompanying1040

freshwater re-circulation, and OHFC into the surface layers. In contrast, the1041

muted seasonal cycle of sea ice during the stadial phase results in smaller1042

melt and brine rejection fluxes. The thicker, year-round sea ice cover of the1043

stadial, which is augmented by polynya ice import, inhibits wind-driven mix-1044

ing between the ML and PC boxes, enhances ice export and accompanying1045

freshwater circulation, and diverts OHFC from the surface layers into the DP1046

box. 591047

11 Model sensitivity to OHFC parameters, QNoIcetot (ice-free) andQIce

tot (ice-covered).1048

The contours represent the average length of the interstadial phase (in years),1049

while the colors represent the average length of the entire cycle (in years).1050

Areas that do not support D-O type oscillations have been hatched in grey. 601051

49

Fig. 1. Basic geometry of the model. The model is columnar, and consists of 4 oceanicboxes (with default depths as labelled), sea ice, and a basic energy balance atmosphere.

50

Fig. 2. In-depth geometry of the model, with surface and interfacial fluxes labelled. Thestate variables for the ocean boxes are Ti and Φi, while the state variables for the sea iceare hice and χice. Heat fluxes are represented by Qi, and salt fluxes are described by Wi. Icesource and sink terms (from polynyas and ice export, respectively) are represented by piceand Iexport, respectively.

51

Fig. 3. Results for atmosphere and ocean temperature, sea ice thickness, and ocean salinityfrom 6,000 years of output from the model with default parameter values. Except for theatmospheric temperature, the model output, which reports all state variables biannually(once at the end of the warm season, and again at the end of the cold season), has beensmoothed with a 5-year running-mean filter; the atmospheric temperature has been smoothedwith a 25-year running-mean filter to mirror the resolution of the δ18O ice core proxy used toinfer Greenland surface temperature. In the top panel, the temperature of the PC box is notshown; at the scale of this figure, the temperatures of the ML and PC are indistinguishable.

11,000 12,000 13,000 14,000 15,000 16,000 17,000

−10

−5

0

5

Tem

pera

ture

(C

)

Atmosphere ML DP AB

11,000 12,000 13,000 14,000 15,000 16,000 17,000

0.5

1

1.5

2

2.5

3

Ice T

hic

kness (

m)

11,000 12,000 13,000 14,000 15,000 16,000 17,000

32

33

34

35

36

Time (yrs)

Salin

ity (

psu)

ML PC DP AB

52

Fig. 4. Un-smoothed model output showing the seasonal evolution of sea ice thickness andsalinities of the ML and PC ocean boxes during a typical interstadial event. This modeloutput is biannual, with all state variables reported at the end of the winter (cold) seasonand at the end of the summer (warm) season.

11,200 11,250 11,300 11,350 11,400 11,450 11,5000

0.5

1

1.5

2

2.5

3

3.5

Ice T

hic

kness (

m)

11,200 11,250 11,300 11,350 11,400 11,450 11,500

32

32.5

33

33.5

34

34.5

35

Time (yrs)

Sa

linity (

psu)

Mixed Layer

Pycnocline

53

Fig. 5. Results from one typical model interstadial phase, showing surface atmospheric,ML, and PC temperatures. This model output is un-smoothed and biannual, with all statevariables reported at the end of the winter (cold) season and at the end of the summer(warm) season. Note the strong seasonal cycle in surface and ML temperatures, and therelative lack of a seasonal cycle in the PC.

11,200 11,250 11,300 11,350 11,400 11,450 11,500

−20

−15

−10

−5

0

5

Tem

pera

ture

(C

)

Atmosphere

11,200 11,250 11,300 11,350 11,400 11,450 11,500

−2

0

2

4

6

Time (yrs)

Tem

pera

ture

(C

)

Mixed Layer

Pycnocline

54

Fig. 6. Results from one model D-O cycle, showing surface temperature and ocean boxtemperatures. Ocean box temperatures have been smoothed with a two-year running-meanfilter, while the surface temperature has been smoothed with a 25-year running-mean filter.Note that the 25-year running-mean filter used on the surface temperature output makesthe transition from the stadial (cold) phase to interstadial (warm) phase appear earlier onthe surface temperature time series than the others.

11,000 11,200 11,400 11,600 11,800 12,000 12,200 12,400 12,600 12,800 13,000−13−12−11−10−9−8−7−6−5−4−3−2−1

01234

Tem

pera

ture

(C

)

Atmosphere Mixed Layer

11,000 11,200 11,400 11,600 11,800 12,000 12,200 12,400 12,600 12,800 13,000

0

2

4

6

Time (yrs)

Tem

pera

ture

(C

)

Pycnocline Deep Atlantic Layer

55

Fig. 7. Results from one model D-O cycle, showing sea ice evolution and ocean box salinities.All raw model output, originally reported biannually, has been smoothed with a 5-yearrunning-mean filter.

11,000 11,200 11,400 11,600 11,800 12,000 12,200 12,400 12,600 12,800 13,000

0.5

1

1.5

2

2.5

3

Ice T

hic

kness (

m)

11,000 11,200 11,400 11,600 11,800 12,000 12,200 12,400 12,600 12,800 13,000

32

33

34

35

Time (yrs)

Salin

ity (

psu)

Mixed Layer Pycnocline Deep Atlantic Layer Abyss

56

Fig. 8. 2000 years of un-smoothed model output showing the evolution of the densities ofthe ML, PC, and DP boxes.

11,000 11,200 11,400 11,600 11,800 12,000 12,200 12,400 12,600 12,800 13,0001025.5

1026

1026.5

1027

1027.5

1028

1028.5

1029

1029.5

1030

Time (yrs)

Density (

kg m

−3)

Mixed Layer Pycnocline Deep Atlantic Layer

57

Fig. 9. 2000 years of smoothed model output showing (from top to bottom) surface tem-perature and the density tendencies of the ML, PC, and DP boxes over time. For a givenlayer, the density tendency due to temperature is ∆ρT = (∂ρ/∂T )0∆T and the density ten-dency due to salinity is ∆ρS = (∂ρ/∂S)0∆S, where (∂ρ/∂T )0 and (∂ρ/∂S)0 are computedat reference temperatures and salinities for each layer, and ∆T and ∆S are defined with re-spect to a reference temperature and salinity for each layer. The residual term is the definedas ∆ρres = ρ − ρ0 − ∆ρT − ∆ρS. Model output, originally reported biannually, has beensmoothed with a 25-year running-mean filter.

11,000 11,200 11,400 11,600 11,800 12,000 12,200 12,400 12,600 12,800 13,000−12

−10

−8

−6

−4

−2

Time (years)

Sfc

Te

mp

(C

)

11,000 11,200 11,400 11,600 11,800 12,000 12,200 12,400 12,600 12,800 13,000−1.5

−1

−0.5

0

0.5

1

∆ ρ

, M

L (

kg

m−

3)

11,000 11,200 11,400 11,600 11,800 12,000 12,200 12,400 12,600 12,800 13,000

−1

−0.5

0

0.5

∆ ρ

, P

C (

kg

m−

3)

11,000 11,200 11,400 11,600 11,800 12,000 12,200 12,400 12,600 12,800 13,000

−0.4

−0.2

0

0.2

Time (years)

∆ ρ

, D

P (

kg

m−

3)

Dens Change due to T Dens Change due to S Residual

58

Fig. 10. Schematic figure depicting the key processes characterizing the stadial and intersta-dial periods, all of which result from differences in the sea ice cover in the two states. Duringthe interstadial period, the substantial seasonal cycle of sea ice results in large freshwaterand brine fluxes into the ML and PC. Furthermore, the seasonal ice cover of the interstadialfosters more wind-driven mixing between the ML and PC boxes, less ice export and accom-panying freshwater re-circulation, and OHFC into the surface layers. In contrast, the mutedseasonal cycle of sea ice during the stadial phase results in smaller melt and brine rejectionfluxes. The thicker, year-round sea ice cover of the stadial, which is augmented by polynyaice import, inhibits wind-driven mixing between the ML and PC boxes, enhances ice exportand accompanying freshwater circulation, and diverts OHFC from the surface layers into theDP box.

59

Fig. 11. Model sensitivity to OHFC parameters, QNoIcetot (ice-free) and QIce

tot (ice-covered).The contours represent the average length of the interstadial phase (in years), while thecolors represent the average length of the entire cycle (in years). Areas that do not supportD-O type oscillations have been hatched in grey.

Q tot

NoIce, W/m

2

Q to

t

Ice, W

/m2

50

50

50

50

75

75

75

75

75

75

75

100

100

100

100

100

100

100

150

150

150

150

150

150

150

150

300 300

30

0

1 1.5 2 2.5 3 3.5

0.7

0.65

0.6

0.55

0.5

0.45

0.4 1100

1150

1200

1250

1300

1350

1400

1450

1500

60