Introduction
It is well known that the albedo difference between seawater and sea ice
leads to a crucial climatic feedback. In a warming climate, melting of high-albedo sea ice exposes low-albedo seawater, thus increasing the fraction of
incoming solar radiation that is absorbed and thereby amplifying warming.
Conversely, in a cooling climate the growth of sea ice increases the
planetary albedo, which amplifies cooling. Classic energy balance models
(EBMs) demonstrate how this well-known ice–albedo feedback can lead to
multiple steady climate states . Given
forcings which resemble present-day conditions, these models are bistable,
with one possible steady state having a partial ice cover and another being
completely ice-free. With a sufficient reduction in the solar input or the
greenhouse effect these energy balance models yield completely ice-covered
steady states, reminiscent of the “Snowball Earth” episodes of the
Neoproterozoic era.
The virtue of simple models, of course, is that they make it possible to
explore ranges of relevant parameters easily. First-order energy balances at
climatic scales are overwhelmingly radiative. The first energy balance models
were not intended to model meridional energy transport, and if included, it
was incorporated as a diffusive process. Diffusion-dominated transport tends
to mitigate the tendency for sea ice to grow in a cooling climate: formation
of extra sea ice would increase the temperature contrast between the
ice-covered high latitudes and the low latitudes, which in turn would
increase the rate of heat transport to the high latitudes. This then slows
the growth of the ice and exports some of the excess cooling due to the
ice–albedo feedback to lower latitudes.
While the small ice cap instability has been a common phenomenon in EBMs
, this bistability has not been seen in more
sophisticated general circulation models (GCMs) . In a
recent study, showed the bistability does not appear when
an EBM was combined with a single column model for ice thermodynamics,
suggesting sufficient complexity, including a seasonal cycle and diffusive
heat transport, eliminates this bifurcation. While the small ice cap
instability is not the primary focus of this study, we will use our model to
investigate this small ice cap bifurcation.
Another class of simple models, box ocean models, have been critical to
understanding meridional ocean circulation, upwelling and mixing, among other
processes . In particular, they have suggested important
climate questions to pursue by means of more sophisticated models, data, or
both. Models and measurements established; however, the importance of the
basic thermohaline circulation and the more complete meridional ocean
circulation. With it, models for energy transport via diffusive processes
were replaced by a more heterogeneous and dynamic mechanism. The Atlantic
Meridional Overturning Circulation (AMOC), in particular, contributes a net
energy transfer into the North Atlantic equivalent to several percent of the
total incoming shortwave solar radiation incident to the region. Models of
the meridional overturning circulation, from the simplest to the most
detailed, agree that under present-day conditions the circulation is bistable
. In addition to the thermally dominated
steady state that is currently observed, a second, salinity-dominated steady
state is also possible, with a much weaker circulation which flows in the
direction opposite to the present flow. It is clearly important to understand
the interaction between this oceanic circulation and the distribution of sea
ice.
Since the present-day AMOC transports heat into the North Atlantic, it tends
to reduce the extent of sea ice. A strengthening of the circulation would
then reduce ice cover, while a weakening would cause it to expand. The
circulation itself is driven by wind stress in and near the Southern Ocean,
as well as in part by mechanical mixing from tidal action
, and it is sustained by variations in the density of
circulating water as it exchanges heat and fresh water with the atmosphere as
it flows along the surface. As water flows northward along the surface
through the tropics it is warmed, and its salinity increases as a result of
excess evaporation. The increase in temperature decreases the water density,
while the increase in salinity increases it. As the water passes to higher
latitudes, it cools and freshens as precipitation exceeds evaporation. Again,
the two effects tend to change the density in opposite directions. In the
present configuration of the AMOC, the thermal effects dominate, so the water
becomes denser as it moves through the subpolar latitudes and ultimately
sinks, returning southward at depth.
The presence of sea ice affects the processes that change the density of
circulating seawater at high latitudes. An ice cover isolates the water from
the atmosphere and thus cuts off the precipitation that otherwise would reduce
the salinity of the water and lower the rate at which its density increases.
It also insulates the water thermally from the atmosphere. This by itself
would not have much effect on the water density, since the water temperature
cannot fall below the freezing temperature anyway. However, it allows the
atmosphere to become colder than it would be if it were in contact with the
seawater. Heat transfer to the cold atmosphere through the ice layer results
in freezing of seawater at the base of the ice. Brine rejection then
increases the salinity and hence the density of the remaining water.
A simple model of ocean circulation with sea ice may be useful in studying
the Snowball Earth episodes in the Neoproterozoic era. Since the discovery of
geologic evidence suggesting that glaciation occurred in the tropics at least
twice in the Neoproterozoic , some 710 and
635 million years ago (Ma), there has been substantial debate about whether
the Earth was ever in a completely ice-covered Snowball Earth state
. These events are thought to have lasted several million
years, raising such questions as how life could have survived a long period
if the Earth were in a completely ice-covered state . Thus,
in trying to explain these signs of apparent tropical glaciation in the
context of global climate dynamics, alternative hypotheses have been proposed
that leave some portion of the ocean either free of ice or covered only in
thin ice .
counts among the first studies to suggest dynamic ocean
heat transport is important to the Snowball Earth hypothesis. Using the fully
coupled Fast Ocean-Atmosphere Model (FOAM), initialized to Neoproterozoic
parameter values to facilitate snowball conditions, the authors found that a
global ice cover was produced when using a mixed-layer ocean model that
parameterized heat transport through diffusion. In fully coupled experiments
with the ocean component, on the other hand, the ice margin would retreat to
high latitudes. Other studies have considered the Snowball Earth problem in
an EBM framework, with ocean heat transport typically parameterized by a
diffusion process . In particular,
examine the feasibility of a tropical thin-ice solution,
incorporating detailed treatment of optical properties of ice and a
nonlinear internal ice temperature profile, as well as a separate snow layer
and an evaporation minus precipitation term to facilitate surface
melt/accumulation.
GCMs that have more detailed ocean physics have also been used to study the
initiation of a Snowball Earth . A recent
study by uses the state-of-the-art atmosphere–ocean model
ECHAM5/MPI-OM to study the Snowball Earth scenario. They implement a Marinoan
(635 Ma) land mask in their coupled GCM simulations, as well as the lower
insolation of a younger, weaker sun. In addition to ocean dynamics, their
study also included sea ice dynamics (albeit with thin ice) and interactive
clouds. All three of these had previously been found to be essential for snowball
initiation .
were able to achieve snowball initiation, and also to
prevent snowball initiation in the same setting by doubling carbon dioxide
levels. Stability analysis of an EBM analog, based on the 0-D model of global
mean ocean temperature developed in , indicates an
insolation bifurcation point for Snowball Earth in the Marinoan setting of
about 95–96 % of pre-industrial levels, in agreement with their
computational results. In their experiments that resulted in partial ice
cover, the ice margin was around 30 to 40∘ latitude, with maximum
stable sea ice extent of 55 % of ocean cover observed in their
experiments.
Sea ice in a global ice cover can be very thick, to the extent that flow by
plastic deformation under its own weight should be considered. Thus, its
non-Newtonian fluid dynamics must be considered in addition to its
thermodynamics. (henceforth GP03) first considered these
flow effects in the Snowball Earth scenario. (The same framework was used in
and to transport dust to low latitudes in
the mudball scenario). Their model runs outside a global circulation model,
using FOAM output for forcing data, and it has neither an active ocean
component nor a parameterization for oceanic heat transport. They use the
term “sea glacier” to describe their modeled ice, to distinguish it from
present-day sea ice, which only grows to thicknesses on the order of meters,
and from land ice or ice shelves. The sea glacier is formed in the ocean, yet
it achieves the thickness of a land ice sheet without the land–ice interface,
and its non-Newtonian rheology is taken into account in the calculation of
its flow. They are able to achieve both partial glaciation and a full
Snowball Earth state through changes in the atmospheric forcing (surface
temperature and precipitation minus evaporation). They find that the
additional viscous flow term is highly effective at allowing the ice margin
to penetrate low-latitude regions of melting, thus encouraging Snowball Earth
initiation.
A recent study by found a dynamic ocean in a Snowball
Earth scenario with strong circulation, in contrast to a stagnant ocean
typically expected due to ice cover serving as an insulation layer to
atmospheric forcing. Their model was forced with geothermal heat, which was
spatially varying with a peak near the Equator, averaging to
0.1 W m-2. In their 2-D and 3-D ocean simulations coupled with a 1-D
ice model extending upon that of GP03, they found that the ocean plays a
larger role in determining ice thickness than the atmosphere, and that
geothermal heat forcing plays a dominant role in ice-covered ocean dynamics.
This was expanded upon in , where a dynamic ocean
with strong equatorial jets and a strong overturning circulation was found in
simulations of a steady-state globally glaciated Earth.
Atmospheric dynamics and cloud cover undoubtedly play a large role in such
climate systems, as demonstrated by . It is, however,
difficult to isolate the role played by oceanic transport in these coupled
simulations, due to the necessary inclusion of the complex dynamics of the
atmosphere and cloud distribution as well as sea ice dynamics. In fact, in
it was found that ocean heat transport has no effect on the
critical sea ice cover that leads to snowball initiation. This motivated us
to consider a simpler model that includes oceanic heat transport coupled to
ice dynamics. We aim to extend the framework laid out in GP03 to include
ocean heat transport effects, including under the ice layer. Realistic
oceanic transport undoubtedly leads to highly nonuniform heat distributions,
likely with local consequences on the global snowball scenario. However,
these local effects are beyond the scope of our study, and indeed beyond the
scope of any low-dimensional model. By omitting atmospheric effects, we aim
to get an assessment of the effects from oceanic heat transport alone.
The purpose of this paper is to investigate the interaction between sea ice
and the meridional overturning circulation. By now there are several studies
that have used complex circulation models to confirm that ocean transport is
an important component of any explanation of how sea ice recedes and grows
along with changes in forcing, albedo, biogeochemistry, etc. With a simple
model, however, it is possible to efficiently test our understanding and
propose questions critical to our being able to further understand the
complexities of radiation, ice cover, and oceanic/atmospheric transport, such
as the Snowball Earth hypothesis.
To this end we have combined a one-dimensional energy balance model with a
box model of the meridional overturning circulation and a dynamic ice
component. Our model is described in detail in Sect. , a key
element of which is the under-ice heat exchange with the ocean. In Sect. we present model results in different climate regimes. The
sensitivity of the model to key parameterizations is studied in
Sect. , and concluding remarks are in
Sect. .
Model description
Our model consists of a four-box ocean model with transport, similar to that
first proposed by , coupled to a one-dimensional EBM
similar to that of and a dynamic model for sea ice
coverage (GP03), and is depicted in Fig. . The ocean component
is a hemispheric model with thermohaline dynamics. While the ocean model uses
a traditional transport equation for the salinity, it differs from
traditional box models in the use of an energy conservation model to capture
the temperature dynamics. This allows us to couple the ice and radiation
components to the ocean dynamics. The ice layer is zonally averaged, so its
thickness is taken to depend only on latitude θ and time t. The ice
margin evolves dynamically, and we include non-Newtonian flow so that the
model can accommodate an ice layer thick enough to be appropriate to Snowball
Earth conditions. The surface absorbs incoming solar radiation, with an albedo
which takes into account whether the surface is open ocean or ice, and emits
longwave radiation with a specified emissivity. Where ice overlies the
ocean, heat conducts through the ice layer and is exchanged with the ocean at
the ocean–ice interface, where melting and freezing can occur, while also
supplying heat for redistribution through ocean circulation, a departure from
typical EBMs. We also account for geothermal heat forcing, found to be the
dominant forcing in a Snowball Earth ocean .
In addition to physical properties of ice and seawater, geometrical features
of the ocean basin, and the distribution of insolation, there are three
parameters which are important to our studies. In the energy balance model we
model the greenhouse effective by including an effective emissivity,
ε, in terms of modeling outgoing longwave radiation. The box
model requires a hydraulic coefficient, k, which relates the strength of
circulation to the densities of the water in the various boxes. The third
parameter quantifies the thermal coupling between the sea ice layer and the
water beneath; we express this as an effective thermal boundary layer
thickness, D, with the rate of heat transfer from the water to the ice
being proportional to the temperature difference between the box water and
the base of the ice, divided by the thickness D. Among the issues we will
investigate is the question of how, and indeed whether, the coupling relates
the bistability of the energy balance model and the independent bistability
of the box model.
Summary of results. Asterisk indicates
simulations with insolation at 94 % of present-day values. Negative
circulation values indicate surface poleward flow.
Simulation
Emissivity
Circ.
Heat transport
Ice-free
ε=0.5
-62.18 Sv
3.73 PW
Partial ice
ε=0.7
-44.44 Sv
1.83 PW
Near-global ice*
ε=0.82
-16.46 Sv
6.27 × 10-4 PW
Global ice*
ε=0.83
-17.46 Sv
3.34 × 10-2 PW
Physical parameters used in simulations. See text for details
on model initialization and settings of unconstrained parameters.
Parameter
Symbol
Units
Value
Hemispheric extent
ℓ
θ
π/2
Extent of Box 2 and 3
ζ
θ
π/4
Depth of Box 1 and 2
du
m
200
Depth of Box 3 and 4
dl
m
3000
Volume of Box 1
Vut
m3
2.83 × 1016
Volume of Box 2
Vup
m3
1.17 × 1016
Volume of Box 3
Vlp
m3
1.76 × 1017
Volume of Box 4
Vlt
m3
4.25 × 1017
Earth's radius
rE
m
6.371 × 107
Hydraulic constant
k0
m6 kg-1 s-1
7.8 × 107
Reference density
ρ0
kg m-3
1027
Reference salinity
S0
psu
35
Reference temperature
T0
K
283
Salinity exp. coefficient
βS
psu-1
7.61×10-4
Temperature exp. coefficient
βT
K-1
1.668×10-4
Ocean albedo
αw
–
0.32
Sea ice albedo
αi
–
0.62
Ocean water heat capacity
cw
J kg-1 K-1
3996
Sea ice heat capacity
ci
J kg-1 K-1
2100
Ocean water density
ρw
kg m-3
1027
Ice density
ρi
kg m-3
917
Ocean water conductivity
κw
W m-1 K-1
0.575
Sea ice conductivity
κi
W m-1 K-1
2.5
Sea ice latent heat
L
J kg-1
3.34×105
Freezing temperature
Tf
K
271.2
Stefan–Boltzmann constant
σ
J m-2 s K-4
5.6704 × 10-8
Ice/ocean boundary layer
D
m
0.05
Ice viscosity parameter
A0
Pa-3 s-1
3.61 × 10-13 T<263.15;
1.734 × 103
T>263.15
Ice viscosity parameter
Q
J mol-1
60 × 103 T<263.15;
139 × 103
T>263.15
Gas constant
R
J K-1 mol-1
8.31446
Acceleration due to gravity
g
m s-2
9.8
Glen's flow law exponent
n
–
3
Geothermal heat forcing
Fg
W m-2
0.05
Hemispheric four-box arrangement. Boxes 1 and 2 are
the surface ocean boxes of depth du, and Boxes 3 and 4 are the
deep ocean boxes of depth dl. The water in Box i has
(well-mixed) salinity Si and temperature Ti. The
boundary between polar and equatorial boxes is at latitude ζ. Ice cover
sits atop the surface boxes with height h and the ice margin at η. The
arrows between the boxes represent density-driven circulation f.
Ocean and energy components
The ocean component of our model is similar to the one proposed by
and . Four boxes are used to represent
the ocean in one hemisphere, from pole to Equator, with each box representing
a zonal average across longitude. Referring to Fig. , we define
Box 1 as the tropical surface ocean box, Box 2 its polar counterpart, Box 3
below Box 2, and Box 4 below Box 1. The depth of the upper boxes is du and the depth of the lower boxes is dl, with du≪dl. Boxes 2 and 3 extend from the pole at latitude
θ=90∘ to a fixed boundary at θ=ζ. Boxes 1 and 4
extend from θ=ζ to the Equator at θ=0∘. We
choose the latitude boundary ζ to be 45∘ simply because we are
interested in investigating climate regimes ranging from global ice cover to
zero ice cover, so there is no advantage in trying to confine the ice cover
to a polar box only. We have found that changing the location of this boundary
within midlatitudes does not qualitatively affect our results. The dynamic
ice margin is at η(t), the ice cover thickness is given by
h(θ,t), and its poleward meridional velocity is given by
v(θ,t). Model parameter values, including geometric quantities
pertaining to the box structure, are given in Table . For
clarity, model variables will be subscripted with letters rather than
numbers, using ut = upper (surface) tropic (Box 1), up = upper
(surface) polar (Box 2), lp = lower (deep) polar (Box 3), and
lt = lower (deep) tropic (Box 4).
Each box has a (well-mixed) temperature Tj(t) and salinity
Sj(t), where j = ut, up, lp, lt, which determine the density of each box by a linear
equation of state:
ρjTj,Sj=ρ01+βSSj-S0-βTTj-T0.
Here ρ0 is a reference density corresponding to a reference temperature
and salinity T0, S0. βS and βT are the expansion
coefficients associated with salinity and temperature. The density-driven
flow between the boxes is denoted by f, where we adopt the convention that
f<0 is surface poleward flow (from Box 1 to Box 2). As in
, the (buoyancy-driven) transport rate is
f=kdudlρut-ρup+ρlt-ρlp,
where k=k0 = 8 × 104/ρ0 Sv is the hydraulic
constant which governs the strength of the density-driven flow. In
Sect. we explore the model's sensitivity to this parameter.
The flux f is purely thermohaline-driven. We could modify this flux to
include the effect of wind stresses in a crude manner by an additive
correction to the flux, however, this has been omitted in this study.
The equations for each box's salinity and temperature will depend on the
direction of the mean meridional flow. The salinity equations when f<0,
corresponding to poleward surface flow, are
VutdSutdt=|f|Slt-Sut+Sut2π∫maxζ,ηζρwρiMθrE2cosθdθ,VupdSupdt=|f|Sut-Sup+Sup2π∫minζ,η90ρwρiMθrE2cosθdθ,VlpdSlpdt=|f|Sup-Slp,VltdSltdt=|f|Slp-Slt.
Here Vj is the volume of the j box and M(θ,t) is the total
production/melting rate of ice, with M > 0 corresponding to ice
production and M < 0 corresponding to melting, described in
Sect. . The terms involving the circulation rate f
correspond to fluxes across the box boundaries. We assume that ice sits atop
the surface ocean boxes, and that the mass of ice is much less than the total
mass of ocean water. The surface ocean box volumes Vj are kept constant,
and any changes to the deep ocean box volumes are negligible. An important
assumption is that ice that is formed is freshwater ice, which rejects
brine into the ocean. The integral term represents the change in salinity due
to net freshwater added/removed through ice melting/production. The bounds of
integration represent the portion of each box covered in ice, and
rE is the Earth's radius. The ice component of the model serves
as a saline forcing on the ocean box model component. There are similar
equations for when ocean circulation is in the reverse direction.
In contrast to traditional Stommel box models, rather than using transport
equations for the temperature, we opt instead for thermal balance equations.
The box temperatures change as heat transfers between boxes with the flow
f, and as it transfers into the box via net radiation or conduction through
overlying ice. Thus, for f<0 the temperature equations are
cwVutdρutTutdt=cw|f|ρltTlt-ρutTut+2π∫maxζ,ηπ/21-αwFsθ-εσTut4⋅rE2cosθdθ-κwDTut-Tf2π∫ζmaxζ,ηrE2cosθdθ,cwVupdρupTupdt=cw|f|ρutTut-ρupTup+2π∫minζ,ηζ1-αwFsθ-εσTup4⋅rE2cosθdθ-κwDTup-Tf2π∫0minζ,ηrE2cosθdθ,cwVlpdρlpTlpdt=cw|f|ρupTup-ρlpTlp+Fg2π∫0ζrE2cosθdθ,cwVltdρltTltdt=cw|f|ρlpTlp-ρltTlt+Fg2π∫ζπ/2rE2cosθdθ.
Here cw is the ocean water heat capacity. The first term in each
equation accounts for net energy accumulation due to fluxes across box
boundaries. The second term in Eqs. () and
() represents the radiative balance, in a similar form as
appears in EBMs dating back to . Here αw is
the ocean water albedo, and Fs(θ) is insolation, for which we
use the parameterization of :
Fsθ=342.951-e22π2∫02π⋅1-cosθsinβcosγ-sinθcosβ21/2dγ.
Here e is the eccentricity of the Earth's orbit (presently at 0.0167),
β is the obliquity (presently at 23.5∘), and γ is
longitude. This parameterization is annually averaged (no seasonal cycle),
but with time-dependent orbital parameters that allow for accounting for the
Milankovitch cycles. However, in our results we found that these were not strong
enough to qualitatively affect the resulting model state, regarding the
location of the ice margin, or Snowball Earth initiation or deglaciation, so
only present-day insolation values are used. The final terms in
Eqs. () and () represent a uniform
geothermal heating forcing, with default Fg=0.05 W m-2,
as in .
In Eqs. () and (), insolation is balanced
by outgoing longwave radiation, integrated over the exposed ocean portion of
the box, where we assume blackbody radiation from the surface using the full
Stefan–Boltzmann law, with σ the Stefan–Boltzmann constant. As
mentioned, ε is the effective emissivity, which is the ratio of
outgoing longwave radiation emitted at the top of the atmosphere to that emitted
at the Earth's surface, and therefore represents the greenhouse effect. Thus,
atmospheric effects are distilled into this single parameter, which we will
use as our control between climate states.
The last terms in Eqs. () and () represent
ice–ocean coupling by modeling heat transfer between the ocean box and its
ice cover, a key component of our model. This includes the parameter D,
having units of length, which parameterizes the under-ice exchange of energy
with the ocean and which we refer to as the effective thermal boundary layer. We
note that D is a key unconstrained parameter, which by default we set to
D=0.05 m, and explore the model's sensitivity to this parameter in
Sect. . Here κw is the seawater thermal
conductivity, which we take to be constant, neglecting dependence on salinity
and temperature.
The ocean heat transport in this model is then quantified as
Hocean=fcwρw(Tut-Tup).
As mentioned, the modeling approach taken herein does not include an active
atmospheric component for the sake of keeping the model as simple as possible
while focusing on the ice–ocean coupling in a reduced model sense. In this
spirit, the parameter ε plays the role of the net
forcing of the atmosphere on the ocean–ice system, including any
heat/moisture transport within the atmosphere, which we consider to be out
of our system.
Sea ice component
We largely follow the “sea glacier” treatment of GP03 and
later , with a few noted exceptions. The details of the
rheology of sea glaciers are left to Appendix . The
equation for the ice thickness, by conservation of mass, is
∂hθ,t∂t+∇⋅vθ,thθ,t=Mθ,t,
where h(θ,t) is ice thickness, v(θ,t) is meridional ice
velocity, and M(θ,t) is the ice melting/production term. The equation
for v is given by a Glen's flow law (GP03),
∇⋅vθ,t=μnhθ,tn,
where μ is a temperature dependent viscosity parameter accounting for the
non-Newtonian rheology of the ice, the details of which are left to
Appendix . The ice melting/accumulation term
Mθ,t is a departure from the treatment of GP03. Ice
melting or production can occur either from heat transferred through the ice
from the surface, or from heat transferred through the ocean through the
effective ice–ocean thermal boundary layer D, and is given by
LMθ,t=κiρiTf-Tsθ,thθ,t-κwρwT1,2θ,t-TfD,
where L is the latent heat of fusion of ice. When M(θ,t)>0, there is
net accumulation of ice, and when M(θ,t)<0, there is net melting. The
first term on the right side accounts for heat transfer through the ice,
assuming a linear temperature profile in the ice from the surface Ts(θ) to the base at freezing Tf. The second term accounts
for heat transfer with the ocean through the parameter D; an equivalent
term appears in the energy budget for the ocean box temperatures in
Eqs. () and (). Since only average ocean
box temperatures are computed by our model, to prevent an artificial and
arbitrary jump in temperature across the box boundary from influencing the
melting term, the step function surface temperature profile
Tut(t), Tup(t) is regularized to a smooth
T1,2(θ,t) for use in Eq. (). Note that our model only
accounts for melting and freezing at the base of the ice, and there are no
terms that model melting at the upper surface or accumulation due to
evaporation/precipitation forcing.
The ice surface temperature Ts(θ,t) is given by a primary
radiative balance, as well as a term accounting for heat transfer through the
ice. The (average annual) ice surface temperature Ts(θ) is
given by
ciρihθdTsθdt=Fsθ1-αi-εσTsθ4+κiTf-Tsθhθ,
where ci is the specific heat, αi the albedo, and
κi the thermal conductivity of ice. The last term of
Eq. () accounts for heat transfer through the ice, as in
Eq. ().
Model setup
Settings for the parameters are listed in Table . The
ocean component is run on a yearly time step, with the ice dynamics
subcycled on a monthly time step. (This does not imply a seasonal cycle;
rather we keep the insolation constant, as given in Eq. .)
For each ocean time step, we solve the set of differential equations for box
temperatures and salinities (Eqs. –), and
ice surface temperature Eq. () using a simple forward
Euler method. At each ice time step, we solve (Eq. ) using a
second-order upwind scheme after solving for the velocity in
Eq. (). We discretize our latitudinal domain with 100 points,
solving for ice thickness h and velocity v on staggered grids, and set
v=0 at the pole for the boundary condition in Eq. (). We
initialize the model in an ice-free state, so any ice is formed through the
model by Eq. (). Following the setup of ,
with initial box temperatures Tut=298 K,
Tup=Tlp=Tlt=273 K and
salinities Sut=36.5 psu, Sup=34.5 psu, and
Slp=Slt=35 psu. Models that equilibrate to a
climate state with no ice or a small ice cap typically reach an equilibrium
climate steady state after 10 000 model years; however, when the model
reaches a state of global ice cover, a longer period of about 100 000 years
is needed before equilibrium is reached.
Results
As mentioned, we use the effective emissivity ε as a control
between model climate states. We emphasize that, as we have no atmospheric
heat transport or other atmospheric effects in our model other than this
parameter to crudely account for greenhouse effects, we do not attribute
physical meaning to this parameter value. With a choice of ε=0.5, which is estimated to be a reasonable value for current climate
, the model remains in an ice-free planet with a thermally
driven poleward circulation of ≈ 62.1770 Sv and associated heat
transport ≈ 3.7332 PW. We note the circulation strength is in line
with results referenced in that give an approximate
meridional circulation strength of 20 Sv from the coupled ocean–atmosphere
model of NOAA's Geophysical Fluid Dynamics Laboratory (GFDL) model, and the
ocean heat transport is in line with estimates for the North Atlantic
(≈ 1 PW; ). The salinities of the
boxes quickly mix and converge to roughly the same value of 35 psu. The
equatorial surface box temperature settles to
Tut=304.0 K, whereas the other three boxes converge to
a closer temperature range with Tup=289.3, Tlp=289.3, and Tlt=289.4 K. Hence, we have a strong,
thermally dominated poleward circulation in this simulation. Depending on the
choice of the ε parameter (which controls the radiative balance
and parameterizes any atmospheric effects), the model can reproduce present-day partial ice cover conditions, as well as Snowball Earth global ice cover
conditions, as we will now describe.
Partial glaciation
Raising the effective emissivity from ε=0.5 to ε=0.7, we move to an equilibrium climate state with a small, stable ice cover.
To determine the role of oceanic heat transport, we run the model with the
circulation rate f set to zero for comparison against the full model run.
Figure a and b show the ice thickness and ice velocity
profiles with and without ocean circulation, and we see the ocean circulation
is effective at reducing ice thickness as well as pulling the ice margin
north. Without the additional heat from the equatorial region moving
poleward, the polar region remains cool, facilitating ice growth.
(a) Ice thickness h and
(b) meridional velocity v profiles in partial ice cover scenario
with effective emissivity ε=0.7, with and without ocean
circulation. Oceanic transport is seen to dramatically affect the ice
predictions. (c) Ice surface temperature Ts and
(d) ice basal accumulation rate in partial ice cover scenario with
effective emissivity ε=0.7, with and without ocean circulation.
The accumulation term stabilizes the ice margin, and the model produces
reasonable values of ice surface temperatures. M is positive over the
region of ice cover, indicating net accumulation of ice, and negative values
of M beyond the ice margin indicate any ice present would be melted.
The ice cover is approximately 700 m thick at the pole in the full model
run, much thicker than current sea ice cover; however, this is consistent
with partial glaciation results from GP03. It is because the ice is this
thick that viscous flow effects need to be considered. There is a strong
response in the ice velocity, largely due to the thicker ice cover when ocean
circulation is not included, resulting in stronger viscous flow. The ice
velocity reaches its maximum just before the ice margin (approximately 2
km yr-1 in the full simulation). We also note that the ice surface
temperature seen in Fig. c, calculated by
Eq. (), is in line with the air surface temperature
forcing used in the experiments of GP03. In Fig. d
we see the accumulation term becoming negative near the ice margin,
indicating a region of net melting that stabilizes the ice margin.
The steady-state ocean circulation strength in this partial ice cover
scenario is ≈44.44 Sv, which is closer to the numerical results of
the GFDL model referenced in than the ice-free run. As
with the ice-free runs, the box salinities quickly mix to the same value of
approximately 35.97 psu, while the surface equatorial box temperature
settles to Tut≈281.6 K, and the other boxes mix
to Tup≈271.63, Tlp≈271.65,
Tlt≈271.71 K. By increasing the effective
emissivity ε, the model steady-state ice profile moves smoothly
further equatorward, until a large ice cap instability threshold is reached.
When ice appears south of this instability threshold, the entire planet is
covered in ice and a Snowball Earth state is reached.
Global glaciation
To approximate the Neoproterozoic climate in our model, we lower insolation
to 94 % of its current value, accounting for a weaker, younger sun
, and raise the geothermal heat flux from 0.05 to
0.08 W m-2. Raising the effective emissivity from ε=0.7
to ε=0.83, we move from a climate state with a small, stable
ice cover to global ice cover, a Snowball Earth, as shown in
Fig. . Global glaciation is reached within the
first 10 000 model years, but then progresses slowly towards equilibrium
between ice thickness and geothermal heat, so simulations in this regime are
run for 500 000 years to reach steady state in the ice component. In the
comparative run with no ocean circulation, the system will not reach steady
state, as there is no geothermal heat flux reaching the ice layer, and as a
result there is a positive ice accumulation rate at all latitudes. In this
scenario, the ocean circulation weakens to ≈16.5 Sv, with an
associated heat transport of ≈ 6.27 × 10-4 PW. The
resulting ice thickness is relatively uniform with a polar maximum of 2 km,
down to 1300 m near the Equator. The back-pressure term in the boundary
condition for ice velocity in Eq. () brings the ice
velocity to zero at the Equator (Fig. b), the
effect of which is to move the maximum velocity (≈ 1 km yr-1)
to a location in the midlatitudes.
(a) Ice thickness h and
(b) velocity v profiles with 94 % insolation, Fgeo=0.08 W m-2, and effective emissivity ε=0.83, the
Snowball Earth scenario. (c) Ice surface temperature Ts,
which is seen to not be impacted by ocean circulation. In (d), the
simulation without ocean circulation cannot reach steady state due to
geothermal heat not reaching the ice layer.
Examining the model results between these very different climate states of a
small, stable ice cap at ε=0.7 and Snowball Earth at
ε=0.83, we find a threshold where the model transitions. With an
effective emissivity of ε=0.82, the equilibrium climate state
is near the large ice cap instability threshold, and we get a strong response
from the ocean circulation. In Fig. , we show the
results of a simulation with and without ocean circulation dynamics. We
observe in Fig. a that, without oceanic heat transport,
we get a Snowball Earth, but with oceanic heat transport, the ice line is
held back from the Equator. As with the previous Snowball Earth state, the
ocean circulation strength here, 17.5 Sv (poleward) is weaker than in the
small ice cap simulation. However, even this weakened circulation, and thus
weakened oceanic heat transport, is still enough to drive the climate into a
snowball state if turned off, demonstrating strong sensitivity in this
regime.
(a) Ice thickness h and
(b) velocity v profiles with 94 % insolation and
ε=0.82, with and without ocean circulation. We note neglecting
oceanic heat transport leads to drastically different global climate states.
(c) Ice surface temperature Ts and (d) ice basal accumulation M,
which without oceanic heat transport is positive everywhere.
Model sensitivity to ocean and atmosphere parameters
We have already seen the model sensitivity to the effective emissivity
parameter ε, which we used to drive the model through vastly
different climate states in Sect. . There are other key
sensitivities the model exhibits, which we now discuss, namely in the
parameterization of energy transfer between the ocean and ice components, as well as
the scaling strength of ocean circulation. By the nature of the large ice cap
instability, the threshold effective emissivity of ε=0.82 seen
in Sect. is sensitive to the parameter choices
representing these key processes. We also explore the model's bistability in
the small ice cap and large ice cap instabilities.
Effect of D parameter on equilibrium climate
state, small ice cap regime (ε=0.75, top), and large ice cap
regime (ε=0.82, 94 % insolation, Fgeo=0.08 W m-2 bottom). There is a particularly strong response in the
resulting melting/accumulation term which was used to constrain the value
of D. Furthermore, the instability near the large ice cap threshold is
reflected in nearby values of D to the default D=0.05, resulting in global
ice cover for ε=0.82. Note that the discontinuity in ice
accumulation rate for D=0.1 is an artifact of the regularization of
Eq. () as h approaches 0.
Ocean–ice energy transfer parameterization
The parameter D is responsible for parameterizing the transfer of energy
between the ocean and ice systems, a so-called effective thermal boundary
layer, and is the least constrained parameter in our model. It appears in
equations for the surface ocean box temperatures (Eq. ),
Eq. (), and in ice melting/production
(Eq. ). The role of D has competing effects in these two
equations. For the ocean box temperatures, D appears in the term
corresponding to energy loss due to the presence of the ice cover, and thus
increasing D cools the ocean. However, this energy loss is balanced in the
system by the ice melting/production term, where increasing D encourages
melting. In the limiting case of D=0, the ocean and basal ice would be
forced to the same temperature, and increasing D further insulates the
ocean from the ice, by slowing the heat transfer between the two component.s
There are also other feedbacks in the system, notably the indirect effect of
D on the strength of the ocean circulation, and thereby oceanic heat
transport, which we have already seen can strongly affect ice cover.
We explore the model's sensitivity to values of this parameter over two
orders of magnitude in Fig. in both the small ice cap regime
(ε=0.75) and near the large ice cap instability (ε=0.82, insolation at 94 % current values, Fgeo=0.08 W m-2), showing ice thickness profiles and ice
melting/accumulation rate at the end of a 50 000-year run to equilibrium.
In the small ice cap regime, increasing D steadily reduces ice thickness,
though the ice margin remains in a small high-latitude range across changes
in D, apart from the large end of the range, where the margin pushes
further equatorward. The ocean circulation (poleward) also steadily reduces
with increased D, from 32.9 Sv circulation with D=0.01 down to 20.5 Sv
circulation for D=1. There is, however, a strong response in the ice
melting/accumulation term, where large values of D yield small magnitude
melting terms. This proved to be critical in melting excess ice, which we saw
in our experiments with changing solar forcing discussed below in
Sect. .
Hydraulic constant
Effect of k0 parameter on small ice cap regime
(top) and large ice cap regime (bottom). Smaller circulation constants reduce
heat transport from the tropics to the poles, thereby facilitating ice
growth. The large ice cap instability is again reflected in the sensitivity
to changes to the hydraulic constant, with the shown deviations resulting in
Snowball Earth.
The default value of the hydraulic constant k=k0 in
Eq. () from results in a circulation
strength of ≈ 32.7 Sv, in line with estimates of present-day mean
meridional ocean circulation. This may not be appropriate for the
Neoproterozoic, warranting an examination of the sensitivity of the model to
changes to this parameter, explored in Fig. . We again show
results for the same experimental setups as in Sect. . Here we
see a stronger effect on the ice margin in the small ice cap regime, where
smaller hydraulic constants reduce heat transport from the tropics to the
poles, thereby facilitating ice growth (note that circulation always remains
poleward in this regime). In the large ice cap regime, we again see the
sensitivity of the large ice cap instability, as deviating from the default
circulation strength constant, in either direction moves the threshold
emissivity away from ε=0.82, and the resulting climate state is
a Snowball Earth.
Hysteresis experiments, radiative forcing. Top: small ice cap
regime – forcing changes in ε, warming first (ε=0.7↦0.5), then cooling (ε=0.5↦0.7). Resulting ice
margin (left) and circulation (right) are shown, demonstrating a hysteresis
loop in the ice margin. Bottom: large ice cap regime – with insolation set to
94 % present values, forcing changes in ε, cooling first to
achieve Snowball Earth (ε=0.6↦0.9), then subsequent
warming (ε=0.9↦0.6). A strong hysteresis loop is seen in
the ice margin, as the model is unable to escape Snowball Earth even as
radiative forcing is raised to levels associated with ice-free states, although
strong enough increases in radiative forcing will eventually melt ice away
from global ice cover. Circulation remains poleward, ever increasing through
the Snowball Earth state manual radiative forcing increase.
Model bistability
As discussed, a well-known feature of EBMs is hysteresis with respect to
radiative forcing. To study this in our model, we set up experiments to
investigate the hysteresis loop in the small ice cap instability and the
large ice cap instability. In the small ice cap case, we began with
conditions that resulted in a small ice cap, gradually increased radiative
forcing by decreasing the emissivity ε until the ice cap melted
away, and then increased ε to its starting value. With each
incremental change in ε, the model is run for 50 000 years to
ensure that the model is in equilibrium. A similar experiment was run for the
large ice cap case, except with the forcing changes in cooling first, until
Snowball Earth is reached, and then subsequent warming. The results of these
experiments are shown in Fig. .
The expected hysteresis loops in each scenario manifest themselves in the
left column of Fig. . In the small ice cap case, The ice
margin begins with ε=0.7 near 60∘ latitude, and
following the red path through decreasing ε, we see the ice cap
abruptly change from a margin at 80∘ latitude, completely
disappearing at ε=0.55. As the forcing is then cooled through the
blue path, the model remains ice-free until the appearance of a small ice cap
near ε=0.62, returning to its
starting location at ε=0.7. Looking at the response of the
ocean circulation through the experiment, we see that there is a small response in the
strength of circulation in the hysteresis loop. As alluded to in
Sect. , this behavior was actually used to constrain the value
of D=0.05, as larger values of D did not yield adequate melting rates to
melt ice once it appeared, resulting in very large hysteresis loops. It is
worth noting that the hysteresis loop in this model is present with only
ocean heat transport, in the absence of an explicit atmospheric heat
transport parameterization, which has previously been found to be necessary
for such bistability .
In Fig. c, we see the familiar large ice cap instability
hysteresis loop, where as the climate is cooled through increasing
ε, the ice margin gradually increases until the large ice cap
instability is reached near 20∘ latitude, beyond which Snowball
Earth is reached. As the climate is subsequently warmed, Snowball Earth is
maintained, but notably the ocean circulation remains poleward, continuing to
transport heat to the poles, and, in fact, with a strong increase in warming
(up to ε=0.4), Snowball Earth is escaped in the model.
Conclusions
We have presented a low-dimensional conceptual climate model consistent with
elements of classical low-dimensional models that is able to reproduce both
present-day, partial ice cover climate, and a Snowball Earth global
ice cover climate. The radiative balance terms similar to those in EBMs
produce states of ice cover consistent with classical EBM results of two
stable solutions, a small ice cover and a global ice cover, and an
unstable solution, a large but finite ice extent. A summary of these results
is given in Table .
Our primary interest is investigating the role of dynamic ocean circulation
in the initiation of Snowball Earth, particularly in the large ice cap
instability. We find in parameter regimes where there is no ice or a small
ice cap that the ocean circulation is expectedly thermally dominated and
poleward. Discounting ocean circulation altogether allows for easier
transition into global ice cover. We find, together with the effective
emissivity which parameterizes the atmospheric component, that the energy
transfer between the ice and ocean components plays a crucial role in
determining the model's resulting climate state.
The results from our ice component are largely in line with those of GP03, in
both ice thickness and position of the ice margin, despite some key
differences in modeling approach. There is a notable difference in the ice
velocity, however, in that our computed ice velocities are an order of
magnitude less those reported in GP03. One possible reason for this is that
the viscosity parameter μ, which GP03 are only required to calculate once
due to a static surface temperature forcing, is recalculated in our model in
response to the dynamic surface temperature in Eq. ().
Our steady-state surface temperature (shown in Fig. )
is cooler than the forcing used in GP03 partial glaciation case, and thus our
cooler surface temperature creates more viscous ice, slowing down ice flow.
The ice thickness profiles in our Snowball Earth experiments vary smoothly by a
few hundred meters from pole to Equator, in contrast to sea glacier models
that obtain a more uniform thickness in snowball state, as in .
Our results also qualitatively agree with that of (e.g.,
ice thickness, velocity, ice accumulation rates, surface temperature) in the
steady-state Snowball Earth scenario, with the notable exception of a sharper
decline in tropical ice thickness, and while we are confident that adopting
their more developed ice model would not significantly change the results of
our model presented here in the Snowball Earth scenario (apart from a thinner
tropical ice), it would certainly be of interest to study transient behavior
and parameter ranges that lead to climate regimes other than Snowball Earth.
The ocean in our Snowball Earth scenario has a considerably weaker circulation
strength of 16 Sv than present-day estimates, and while this is not
indicative of a stagnant ocean, it is not as strong as the circulation
strengths of approximately 35 Sv that were achieved in the study of
. Their 2-D simulations allowed for resolution of both
vertical mixing and horizontal eddies, and while they also did not have land
in their simulations, they did have an underwater ocean ridge that had a
highly localized and higher strength geothermal heat forcing corresponding to
a spreading center which, together with strong vertical salinity profiles,
drove the strong circulation.
The main conclusion we reach from this study is that ocean circulation and
its associated heat transport play a vital role in determining the global
climate state and ice cover. We have seen in the partial glaciation case that
the ocean circulation severely inhibits ice growth, and we have seen in the
near-global glaciation case that even in that state's severely weakened ocean
circulation, a lack of oceanic transport leads to a drastically different
Snowball Earth state. In particular, our study found that both heat flux
between the ice and ocean and the value of the circulation constant that
controls circulation strength (tuned to present-day conditions) played a
crucial role in determining the global climate state. Moreover, we find bistability in the small and large ice cap instability regimes with regard to radiative forcing.
Our inclusion of a simple parameterization of under-ice ocean heat transport,
mediating dynamic oceanic heat transfer not present in traditional EBM
diffusive heat transport models, is here found to be crucial in determining
the steady-state climate regime. This warrants further investigation in a GCM
setting in order to examine the role played by thermal processes in the ice and ocean
that account for heat transport in the sub-ice cover layer. To be more
direct, we find that oceanic heat transport is crucial to understanding
Snowball Earth initiation, that the ice cover affects it significantly, that the
results are sensitive to the water–ice thermal coupling and the factors
driving the circulation, and that it is therefore worthwhile using GCMs to
investigate these factors in detail.
While atmospheric effects were largely neglected for simplicity and because
our focus was on the role of oceanic heat transport in the Snowball Earth
setting, including an atmospheric component would be the natural progression
of this work, particularly a precipitation–evaporation component that would
facilitate ice surface melting/accumulation, though it is worth noting that
we were able to reproduce both present-day and Snowball Earth conditions
without a precipitation–evaporation forcing.