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A fully-implicit model of the global ocean circulation

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A fully-implicit model of the global ocean circulation
Wilbert Weijer
a,1
, Henk A. Dijkstra
a,*
, Hakan
€
OOks
€
uuzo
gglu
b,2
,Fred W. Wubs
b
, Arie C. de Niet
b
a
Institute for Marine and Atmospheric Research Utrecht, Department of Physics and Astronomy,Utrecht University, Utrecht, The Netherlands
b
Research Institute for Mathematics and Computer Science, University of Groningen, Groningen, The Netherlands
Received 25 November 2002; received in revised form 1 July 2003; accepted 16 July 2003
Abstract
With the recent developments in the solution methods for large-dimensional nonlinear algebraic systems, fully-implicit ocean circulation models are now becoming feasible. In this paper, the formulation of such a three-dimensionalglobal ocean model is presented. With this implicit model, the sensitivity of steady states to parameters can be in-vestigated eﬃciently using continuation methods. In addition, the implicit formulation allows for much larger timesteps than can be used with explicit models. To demonstrate current capabilities of the implicit global ocean model, weuse a relatively low-resolution (4
horizontally and 12 levels vertically) version. For this conﬁguration, we present: (i) anexplicit calculation of the bifurcation diagram associated with hysteresis behavior of the ocean circulation and (ii) thescaling behavior of the Atlantic meridional overturning versus the magnitude of the vertical mixing coeﬃcient of heatand salt.
2003 Elsevier B.V. All rights reserved.
PACS:
65P30; 65N25; 76E20
Keywords:
Stability of geophysical ﬂows; Numerical bifurcation theory; Hysteresis behavior; Iterative linear systems solver
1. Introduction
The last decade has witnessed an enormous increase of interest in the dynamics of the ocean circulationto understand its role in the climate system [1]. On time scales of a decade to thousands of years, oceanicprocesses are a dominant factor in controlling the patterns and amplitude of climate variability. A typical
Journal of Computational Physics 192 (2003) 452–470www.elsevier.com/locate/jcp
*
Corresponding author. Tel.: +31-30-253-3858; fax: +31-30-254-3163.
E-mail address:
dijkstra@phys.uu.nl (H.A. Dijkstra).
URL:
http://www.phys.uu.nl/~dijkstra.
1
Now at Scripps Institute of Oceanography, UCSD, La Jolla, CA, USA.
2
Now at Institute for Marine and Atmospheric Research Utrecht.0021-9991/$ - see front matter
2003 Elsevier B.V. All rights reserved.doi:10.1016/j.jcp.2003.07.017
example is the interdecadal variability as observed in North Atlantic sea-surface temperature records [2,3]which is likely caused by changes in the thermohaline (i.e. density driven) ocean circulation [4].As both the instrumental record of observations and the current paleodatabase is by far inadequate toassess the role of the ocean on these long time scales, there is an enormous eﬀort to model the oceancirculation, both in isolation as well as coupled to the atmosphere and cryosphere. For studies of thevariability of the ocean circulation on decades or less, high-resolution eddy-resolving models are used. Low-resolution models are typically used in climate related problems since long time intervals of integration canbe achieved [5].There are several models available which can be used to study the global ocean circulation. One of thewell-known models is the Modular Ocean Model (MOM, [6 a much used climate conﬁguration has ahorizontal and vertical resolution of 4
and 12 layers [7]. Other types of models are the Miami IsopycnalOcean Model (MICOM, [8]) and the Hamburg Large-Scale Geostrophic (LSG) model [9]. While MOM andMICOM are fully explicit [10] and use time steps of typically one hour, the LSG model is semi-implicit andcan run with time-steps up to one month.A problem that has been much studied in the literature is the sensitivity of the global thermohalinecirculation to anomalies in the freshwater ﬂux. Many models have a parameter regime where multipleequilibria of the thermohaline circulation occur. A typical transient simulation demonstrating the existenceof the multiple equilibria is a quasi-equilibrium run where the freshwater input is changed very slowly withrespect to the equilibration time scale of the ﬂow. The multiple equilibria show up as a hysteresis curvewhen the anomalous forcing is ﬁrst increased and then decreased [7,11]. The jumps in the hysteresis curveare related to a transition from one stable state to another one.Models such as MOM, MICOM and LSG are capable of forward time integration only and hence canonly determine (linearly) stable states. However, in a typical bifurcation diagram associated with thehysteresis there are also unstable states [12]. While these states remain hidden for the forward models, theyplay an important role in the transient behavior of the ﬂow, for example in the response to a temporaryinput of freshwater [13,14]. One would like to have models that determine these unstable equilibrium statesdirectly and that are able to follow these states eﬃciently in parameter space.It is here that there is a role for fully-implicit ocean models. The
common knowledge
advantage of theformulation of these models is that the time step is not limited by numerical stability but by accuracy, thelatter related to changes in the solution. On the other hand, the time steps are much more expensive becausea system of nonlinear algebraic equations has to be solved within each time step. Whether the implicitmodel is computationally more eﬃcient than an explicit model depends on a lot of factors and a com-parison is in most cases diﬃcult. However, when the linear systems solver performs well for large time steps,one is also able to compute solutions to the steady state equations directly. As the solver for the nonlinearequations determines isolated solutions, the steady states found may be either linearly stable or unstable.When an implicit model is combined with a parameter continuation, the steady states can be eﬃcientlyfollowed in parameter space.When the Newton–Raphson method is used, several large-dimensional linear algebraic systems have tobe solved. Each of these involve sparse nonsymmetric ill-conditioned and usually banded matrices. Directsolvers limit the dimension of these systems to several thousands. However, over the last decades a numberof iterative solvers have been developed which enable one to tackle systems having dimensions up to a fewhundred thousand; here application to global ocean models comes within reach.A ﬁrst step in the development of implicit ocean models was presented in [15]. For a single-hemisphericsector domain, it was shown that three-dimensional ocean ﬂows could be computed using much larger timesteps – in the approach to equilibrium, time steps of 50 years could be taken – than with explicit models. Inaddition, it was demonstrated that with the implicit formulation, steady states could be followed in pa-rameter space without computing any transient behavior; this is eﬃcient to determine parameter sensitivityof the ﬂows. Finally, it was shown that the linear stability of steady states could be determined explicitly.
W. Weijer et al. / Journal of Computational Physics 192 (2003) 452–470
453
Although several limitations of the fully-implicit model were presented in [15], it was stated that theimplicit approach would be applicable to a global ocean model with full continental geometry and bottomtopography. In this paper, we present results computed with such a fully-implicit global ocean model usinga relatively low horizontal resolution. Although there remain limitations with respect to representing theglobal ocean ﬂow as it is observed today, the results shown here are a major step forward in the devel-opment of implicit ocean models. The advantages of the implicit approach are demonstrated by computingthe sensitivity of the ocean circulation versus changes in the spatial pattern of the North Atlantic freshwaterﬂux and versus the strength of the vertical mixing of heat and salt.
2. The global implicit ocean model
The global model here extends the model of [15] by applying the equations below to a global domainwith the inclusion of realistic bathymetry, wind forcing and thermohaline forcing. The ocean velocities ineastward and northward directions are indicated by
u
and
v
, the vertical velocity is indicated by
w
, thepressure by
p
and the potential temperature and salinity by
T
and
S
, respectively. The governing equationsin coordinates (
k
;
/
;
z
Þ
areD
u
d
t
uv
tan
/
r
0
2
X
v
sin
/
¼
1
q
0
r
0
cos
/
o
p
o
k
þ
Q
ks
þ
F
u
;
ð
1a
Þ
D
v
d
t
þ
u
2
tan
/
r
0
þ
2
X
u
sin
/
¼
1
q
0
r
0
o
p
o
/
þ
Q
/s
þ
F
v
;
ð
1b
Þ
o
p
o
z
¼
q
g
;
ð
1c
Þ
0
¼
o
w
o
z
þ
1
r
0
cos
/
o
u
o
k
þ
o
ð
v
cos
/
Þ
o
/
;
ð
1d
Þ
D
T
d
t
¼r
h
K
H
r
h
T
ð Þþ
oo
z K
V
o
T
o
z
þ
Q
T
;
ð
1e
Þ
D
S
d
t
¼r
h
K
H
r
h
S
ð Þþ
oo
z K
V
o
S
o
z
þ
Q
S
;
ð
1f
Þ
q
¼
q
ð
T
;
S
Þ ð
1g
Þ
withD
F
d
t
¼
o
F
o
t
þ
ur
0
cos
/
o
F
o
k
þ
vr
0
o
F
o
/
þ
w
o
F
o
z
;
r
h
F
¼
1
r
0
cos
/
o
F
o
k
;
1
r
0
o
F
o
/
T
;
r
h
F
¼
1
r
0
cos
/
oo
k
F
k
þ
1
r
0
cos
/
oo
/
ð
F
/
cos
/
Þ
;
454
W. Weijer et al. / Journal of Computational Physics 192 (2003) 452–470
where
F
,
F
¼ð
F
k
;
F
/
Þ
T
are an arbitrary scalar and vector, respectively and the superscript T indicatestranspose.In the equations above, vertical and horizontal mixing of heat and salt is represented by eddy diﬀu-sivities, with constant horizontal and vertical diﬀusivities
K
H
and
K
V
for both heat and salt. Laplacianfriction is taken for the mixing of momentum (for the terms (
F
u
,
F
v
), see [15]) with constant mixing coef-ﬁcients
A
H
and
A
V
in the horizontal and vertical, respectively. Density
q
is related to potential temperatureand salinity through an equation of state that is based on the polynomial expression used by [16]
q
ð
T
;
S
Þ¼
q
0
1
þ
a
1
S
b
1
T
b
2
T
2
þ
b
3
T
3
:
ð
2
Þ
The ocean circulation is driven by a wind stress
~
ss
ð
k
;
/
Þ¼
s
0
ð
s
k
;
s
/
Þ
T
, where
s
0
is the amplitude, andwhere
s
k
ð
k
;
/
Þ
and
s
/
ð
k
;
/
Þ
provide the spatial patterns of the zonal and meridional winds. The transfer of momentum from the surface downwards occurs in a thin boundary layer, i.e., the Ekman layer. Althoughthis may be explicitly resolved [17], we follow the methodology applied in many low-resolution oceangeneral circulation models. The surface forcing is distributed as a body forcing over a certain depth of theupper ocean using a vertical proﬁle function
g
ð
z
Þ
Q
ks
¼
g
ð
z
Þ
s
0
q
0
H
m
s
k
;
Q
/s
¼
g
ð
z
Þ
s
0
q
0
H
m
s
/
;
ð
3a
Þ
where
H
m
is a typical vertical scale of variation of the proﬁle function
g
ð
z
Þ
. The function
g
ð
z
Þ
is taken unityin the upper layer and zero below, so that
H
m
is the depth of the surface ocean layer.The upper ocean is coupled to a simple energy-balance atmospheric model, in which only the heattransport is modelled (no moisture transport). The atmospheric model used is one of the simplest versionswithin the class of energy-balance models [18]. The equation for the atmospheric surface temperature
T
a
onthe global domain is given by
q
a
H
a
C
pa
o
T
a
o
t
¼
q
a
H
a
C
pa
D
0
r
h
ð
D
ð
/
Þr
h
T
a
Þð
A
þ
BT
a
Þþ
R
0
4
S
ð
/
Þð
1
a
Þð
1
C
0
Þþ
l
oa
ð
1
L
Þð
T
T
a
Þþ
l
la
L
ð
T
l
T
a
Þ
;
ð
4
Þ
where
q
a
¼
1
:
25 kg m
3
is the atmospheric density,
C
pa
¼
10
3
J (kg K)
1
the heat capacity,
a
¼
0
:
3 theconstant albedo,
H
a
¼
8
:
4
10
3
m an atmospheric scale height,
R
0
¼
1
:
36
10
3
W m
2
the solar constant,
D
0
¼
3
:
1
10
6
m
2
s
1
a constant eddy diﬀusivity and 1
C
0
¼
0
:
57 is the atmospheric absorption coeﬃ-cient. The functions
D
ð
/
Þ
and
S
ð
/
Þ
give the latitudinal dependence of the eddy diﬀusivity and the short-wave radiative heat ﬂux with
D
ð
/
Þ¼
0
:
9
þ
1
:
5exp
12
/
2
p
;
S
ð
/
Þ¼
1
12
ð
0
:
482
ð
3sin
/
Þ
2
1
Þ
:
ð
5
Þ
The constants
A
¼
216 W m
2
and
B
¼
1
:
5 W m
2
K
1
control the magnitude of the long-wave radiativeﬂux.In (4),
T
is the sea-surface temperature,
T
l
the temperature of the land surface points and the coeﬃcient
L
indicates whether the underlying surface is ocean (
L
¼
0) or land (
L
¼
1). The exchange of heat betweenatmosphere and ocean and between atmosphere and the land surface is modelled by constant exchangecoeﬃcients. We assume here for simplicity that both are equal, with
l
la
¼
l
oa
¼
q
a
C
pa
C
H
j
V
a
j
l
;
ð
6
Þ
where
C
H
¼
1
:
22
10
3
and
j
V
a
j¼
8
:
5 ms
1
is a mean atmospheric surface wind speed; it follows that
l
13 W m
2
K
1
.
W. Weijer et al. / Journal of Computational Physics 192 (2003) 452–470
455
Boundary conditions for the atmosphere are periodic in zonal direction and no-ﬂux conditions at thenorth–south boundaries, i.e.
T
a
ð
k
¼
k
W
Þ¼
T
a
ð
k
¼
k
E
Þ
;
o
T
a
o
k
ð
k
¼
k
W
Þ¼
o
T
a
o
k
ð
k
¼
k
E
Þ
;
ð
7a
Þ
/
¼
/
S
;
/
N
:
o
T
a
o
/
¼
0
:
ð
7b
Þ
The net downward heat ﬂux into the ocean and land is given by
Q
oa
¼
R
0
4
S
ð
/
Þð
1
a
Þ
C
0
l
ð
T
T
a
Þ
;
ð
8a
Þ
Q
la
¼
R
0
4
S
ð
/
Þð
1
a
Þ
C
0
l
ð
T
l
T
a
Þ
:
ð
8b
Þ
When the heat capacity of the land is assumed zero, then
Q
la
¼
0 and the land temperature
T
l
is com-puted directly from
T
l
¼
T
a
þ
R
0
4
l
S
ð
/
Þð
1
a
Þ
C
0
:
ð
9
Þ
The freshwater ﬂux is prescribed in each of the results below and indicated by an amplitude
F
0
and aspatial pattern
F
S
. Hence, the expressions for
Q
T
in (1e) and
Q
S
in (1f) become
Q
T
¼
g
ð
z
Þ
Q
oa
q
0
C
p
H
m
;
ð
10a
Þ
Q
S
¼
g
ð
z
Þ
F
0
F
S
H
m
ð
10b
Þ
and this forcing is again represented as a body forcing over the upper layer. On the continental boundaries,no-slip conditions are prescribed and heat- and salt ﬂuxes are zero. At the bottom of the ocean, both theheat and salt ﬂuxes vanish and slip conditions are assumed.Note that the model formulated here does not guarantee stably stratiﬁed solutions. As in all other large-scale ocean models the eﬀect of small-scale convection, which occurs when the stratiﬁcation is not staticallystable, must be explicitly parameterized. Such a parameterization is usually referred to as
convective ad- justment
. Two parameterizations have been implemented in the implicit model: the locally-enhanced dif-fusion parameterization [19] and the Global Adjustment Procedure [20]. In the former method, the mixingcoeﬃcient is increased locally in regions of unstable stratiﬁcation. The Global Adjustment Proceduregenerates a stably stratiﬁed solution by increasing vertical diﬀusivities in an iterative procedure thatremoves all static instabilities.
3. Numerical methods
The equations are discretized in space using a second-order control-volume discretization method on astaggered (Marker and Cell or Arakawa C-) grid, that places the
p
,
T
and
S
points in the center of a gridcell, and the
u
,
v
, and
w
points on its boundaries. The spatially discretized model equations can be written inthe form
456
W. Weijer et al. / Journal of Computational Physics 192 (2003) 452–470

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