One of the important problems in modern oceanology is estimating
the heat and mass flows in both particular regions and the ocean as a whole.
These estimates are necessary for some practical problems, first of all for
calculations of the heat and moisture balance on which the medium- and
long-term weather forecasts and climatic scenarios are based. It is impossible
to understand these processes without their pictorial presentation with the
help of modern visualization facilities. In addition, such values are important
to determine the water, chemical pollution, and biogenic elements’ transport in
the ocean and understand the fauna cyclic genesis in the ocean.
It is very difficult to estimate the sea water characteristics
and its enthalpy in the deep ocean. Unlike the ocean surface, for which the
satellite observations describe relatively reliably the sea surface temperature
and ocean level, there are not long series of observations for deep sea layers,
except for some relatively small volume of data obtained in large-scale
international projects [1—2], as well as the data from the Argo drifters [3],
collected during the past decade. As for the near-bottom layers, especially in
the South Atlantic and in the south part of the Pacific Ocean, there are only
separate uncoordinated observation data obtained in oceanographic expeditions,
in particular, in the course of the WOCE (World Ocean Circulation Experiment) [4].
Therefore, the main tool used to obtain such estimates
is model calculations. In the past decade, a considerable progress is made in
mathematical modeling of the processes in the ocean and joint geophysical
processes in the ocean and atmosphere. The state-of-the-art models of
circulation have appeared and are being developed; they include not only the
ocean and sea dynamics, but also describe the joint ocean/atmosphere/Earth/ice/carbon/solar-radiation
exchange processes, i.e., all physical processes that influence directly or
indirectly on the medium- and long-term ocean dynamics. Let us note the most
significant recent studies concerning the pure ocean circulation [5] and
modeling with the use of joint ocean-atmosphere models [6]. In these studies
the problems of describing the processes of ocean and ocean-atmosphere dynamics
are solved at a high scientific level by using supercomputers and parallel
programming methods.
Nevertheless, computer modeling cannot provide in full
measure accurate estimates for such important characteristics as the heat and
mass transport in the ocean. This is conditioned by the fact that the models
applied for the calculations are not completely adequate to the reality, since
they are based on different approximations and hypotheses; the approximations
and numerical methods for solving the differential equations of the proposed
models add errors to calculated parameters. In addition, there is a lack of
real data used in the models as the initial and boundary conditions; these data
are often not completely reliable, there are gaps in the series of
observations, instrumental errors, etc.
It is possible to increase the modeling accuracy in
the problems under consideration by using data assimilation methods, i.e., by
correcting the results of model calculations with the help of observed data.
In data assimilation, the fields of calculated
parameters is corrected so as to make it, on one hand, satisfy the model
equations, i.e., the principal conservation laws, and, on the other hand,
approximate the observed values with a sufficient accuracy. As a result, this
compensates both the incompleteness of the database of observations and the
error of the model itself. Even if the assimilation methods do not solve all
problems, their application gives better results as compared with the results
with no assimilation, which is shown, for example, in [7].
Data assimilation methods are widely used in modern
oceanology for both the prompt forecasts [8] and climate modeling [9]. A great
contribution to the development of the theory and methods of data assimilation
was made by the school of thought under the leadership of G. I. Marchuk, the
founder of this research domain in the USSR (Russia), Academician of the
Russian Academy of Sciences [10]. The modern versions of these assimilation
schemes, named as the 4D-VAR method, use the ideas of the theory of inverse
problems and conjugated equations proposed by Marchuk still in the 1970s. For
example, the 4D-VAR version of the assimilation method is described in detail
in [11].
Another approach, alternative in many respects to that
above, is the so-called dynamic-stochastic assimilation schemes. Their modern
version got the name of the ensemble Kalman filter (EnKF) [12]. It should be
noted that a great contribution to the development of this theory and its
application was made by Soviet (Russian) researchers under the leadership of
the Academician A. S. Sarkisyan [13]. As for the recent publications in this
domain, we should note [14]. On the whole, this research field is actively and
successfully developed in Russia and abroad. There are other approaches to data
assimilation, different from these two mainstream approaches; we will not
consider them, since this study is not devoted to data assimilation itself; we
use these methods only for concrete calculations.
In recent years, the hybrid data assimilation methods
combining these two approaches have become especially topical. One of such
methods used in this study is the generalized Kalman filter (GKF) method developed
by the authors. Its detailed description can be found in [15, 16]. Below, we
present a short description of this method, necessary to understand the matter
of this paper.
This study is devoted to searching for the model
estimates for the heat and mass transport in the South Ocean. This theme is
topical and there are a series of studies devoted to it. It has to do with the
necessity to understand the genesis of the water flows that appear, in
particular, as a result of the Antarctica ice melting process and propagate up
to the Mediterranean and describe them quantitatively [17].
The aims of this study are as follows: (a) modeling of
the heat and mass transport in the South Atlantic by using the joint MPI-ESM
(Max Planck Institute ‒ Earth System Model) model, developed in the Plank
Institute of Meteorology (MPIMET, Germany) with application of the GKF
(Generalized Kalman Filter) authors’ hybrid data assimilation method; (b) analyzing
these results and comparing them with the control results obtained by means of
model calculations with no assimilation; (c) quantitative and qualitative
estimating the effect of data assimilation on the final result, especially for
the World Ocean zones distant from the data assimilation region, in particular,
for the Arctic; (d) pictorial presentation of the results of modeling by means
of modern visualization facilities.
The databases of the Institute of Oceanology of the
Russian Academy of Sciences (the sections from the WOCE A6-A11 program) were
used for data assimilation. The observation data were provided by Prof. E.G.
Morozov.
The calculations were performed on the supercomputers "Mistral"
in DKRZ (Deutsches
Klimarechenzentrum, Hamburg, Germany) and "Lomonosov-2" in the
Lomonosov Moscow State University.
The visualization of the numerical experiments was
made by using the Panoply Data Viewer (Panoply 4.9.4, released 2018-08-30,
NASA, developed by Dr. Robert B. Schmunk) to treat the netCDF, HDF, GRIB
geophysical data sets.
The MPI-ESM global joint model [6] was used as the
mathematical model. Since this model is necessary only as a tool of
investigation and is not an object of analysis; the peculiarities of its
structure and detailed description are not presented here; only necessary
information on its configuration and the main parameters is provided.
In our study, we used its high-resolution version with
a grid pitch of 0.25° in the latitudinal direction and with approximately the
same, however, varying with latitude, resolution in the meridional direction.
There were 40 levels in the vertical direction from the sea surface to the
bottom. At that, the first 1000 m had a higher resolution and contained 20
levels. The model comprises a special block for calculating the ice dynamics
which is important to describe the Antarctic circulation, in particular, the
circumpolar current; however, it was not analyzed in our study. The investigated
region in our study was bounded by 75° south latitude and 55° north latitude in
the Atlantic Ocean and from the Drake Passage (75° west longitude) to the Cape
of Good Hope (15° east longitude).
A principal scheme of the experiment was as follows:
first, the model free run was conducted with forcing over 150 years. For the
ocean part, the average “historical” values of temperature and salinity from
the atlases of observations [18] were used as the initial data. For the
atmospheric part, the corresponding values (heat flows, wind velocities, etc.)
were taken from the NCEP archives [19]. Conventionally, these data were related
to 1800. With these initial conditions, the model was integrated with
calculation intervals from one month to one year to accumulate the values of
the ensemble statistics. In such a way, the model was prepared to calculate the
current values, while the values of restarts (the ocean and atmospheric
characteristics) were archived and used to form the arrays for statistical
treatment.
Data assimilation was carried out by
using the GKF method, according to the formulae
,
(1)
,
(2)
,
(3)
where 
are the model
fields (written in the form of a column bit vector in each grid point) after
and before correction, i.e., the analysis and background fields, respectively; Y
is the vector of observations, in our case, it is the temperature and salinity
vector at the point of observation; K is the Kalman gain (the weighting
matrix) whose meaning is to render information about the difference between
the observed and model values, i.e., the errors of modeling, to the model
itself including unobserved model parameters; Q is the model error covariance
matrix determined only for the observed parameters (structurally, this is a
positively definite symmetrical matrix); Λ and Ñ are temporal trends, i.e., the time derivatives of model and
observational parameters, respectively, written as column bit vectors at the grid
points; H is the matrix of the linear projection operator from the phase
space (set of values) of the model onto the phase space of observations, which
interpolates the values of the model to the points of observations and exclude
from consideration the unobserved model parameters; the upper index T designates
the transposition of a vector and/or a matrix.
The GKF method (1)-(3) is described
in detail and theoretically substantiated in [15, 16]. It is also shown in
these papers that this method generalizes the known Kalman algorithm EnKF [12].
The values of the vector Ñ and matrix Q are considered to be known and are determined,
in particular, by the methods described in [15]. The advantages of this
algorithm are as follows: it considers not only the difference between the
model and observations, but also the temporal trend (the time derivative) in
both the model and observation data. At that, this method does not require a
preliminary filtration of the model bias, since the consideration of the linear
trend as the filter basis eliminates automatically the model bias.
In this study, we have assimilated 5
sections of the temperature and salinity profiles from the sea surface (0-6 m)
to the bottom (to 5720 m) performed by boats of different countries in the
course of the WOCE international experiment at about 650 stations. The charts
of the sections are shown in Fig. 1 (provided by Prof. E. G. Morozov).
Fig. 1. Scheme of the stations’ location
The observed data are the temperature
and salinity profiles with a pressure interval of 2 dbar. These values were
recalculated to the standard depth levels of the model and subjected to a
quality control. In particular, we verified the stratification condition 
, where 
is the density at
the 
-th depth level. If the stratification condition was
violated, then, at the depth levels where this violation was observed, the
temperature and salinity data were transposed and the procedure was repeated
until the stratification condition was strictly satisfied. In addition, the
temperature in situ in the observations was converted to the potential
temperature, according to [20]. It should be noted that the difference between
the potential and in situ temperatures proved to be about 0.8°Ñ at the depths that exceeded 3000 m. This
conversion significantly reduced the difference between the model and observed
temperatures, as compared with the initial values.
As was noted above, the model free
run was conducted with atmosphere forcing over 150 years, beginning from the
conventional 1800 with zero initial velocities. In the process of integrating,
the calculation data were recorded and archived for each month. In such a way,
the ensemble of values was formed. These data were used to calculate by the
Monte-Carlo method the vector Ñ as an ensemble average of the difference between two
parameters in two successive model calculations divided by the time pitch and
the model error covariance matrix Q.
It was established in previous investigations [14] that
it is sufficient for data assimilation to have an ensemble of about 50 values
for each ocean characteristic (temperature, salinity, horizontal and vertical
velocities, ocean level, etc.). Therefore, the annual averages for the
conventional 1900‒1950 were used as the ensemble’s components. Further,
the calculations were performed from the conventional 1950 to 1993 without and
with data assimilation, according to formulae (1) ‒ (2) at one instant of
time, namely, the conventional 1993, however, the real measurements were
carried out during a few months. The parameters obtained in the calculation
were compared with the results of calculation without data assimilation (the
control calculation).
Figure 2 shows the temperature fields at a depth level
of 150 m before their assimilation (control, Fig. 2a), after data assimilation
(analysis, Fig. 2b), and their difference (analysis minus control, Fig. 2c). The
level of 150 m was chosen, since the structure of currents typical of the
South Atlantic Ocean is well seen at it.
(a)
(b)
(c)
Fig. 2. Temperature fields at a depth level of 150 m:
(a) control calculation; (b) analysis (calculation with data assimilation); (c)
difference between the analysis and control calculations
It is seen in these figures that the control field is
quite smooth, nearly without synoptic structures, without well-marked changes
in both latitudinal and meridional directions. On the contrary, the analysis
field is more dynamic; on the whole, it is warmer than the pure model field,
especially in the equatorial region with a marked synoptic variability in the
south part, near the coasts of the Latin America in the Brazil-Malvinas
confluence zone and in the Gulf of Guinea, in Africa. These two differences are
seen in Fig. 2b, which represents the difference between the fields. The zones
of data assimilation are well-marked; however, the structure of this difference
is strongly anisotropic and inhomogeneous. The value of the difference has the
maximum amplitude (both with plus and minus signs) about 2°Ñ and considerably varies with latitude. In
the equatorial region, the model significantly underestimates the real data,
while in the region of the Brazil current the real temperature is considerably
lower than the model value.
Figure 3 shows the temperature fields at a depth level
of 3770 m. For illustration, this depth level is chosen because the Antarctic
deep water (AADW), which is of great interest for climatic research, manifests
itself at this level. The fields at this depth level vary insignificantly;
however, there are some small differences. The conclusion that the model
somewhat underestimates the deep water temperature remains; the field after the
correction (data assimilation) is approximately by 0.5°Ñ warmer than the control field; however, in
some regions, the difference between them reaches 1°C. This difference is most remarked
in the Brazil-Malvinas confluence zone and in the Drake Passage region. This
underestimate significantly influences on the correction of the velocity and
especially on the correction of the heat flows, since the latter is a
multiplier of two variables: the velocity and temperature. òåìïåðàòóðû.
(a)
(b)
(c)
Fig. 3. Potential temperature at a depth level of 3770 m:
(a) control calculation; (b) analysis, (c) difference between
the fields of analysis and control calculations
Let us analyze the meridional
component of the velocity of currents. Figure 4 shows the meridional component
of the velocity of current at a depth level of 150 m before and after data
assimilation and the difference between them; Fig. 5, the same parameters at a
depth of 3770 m.
(a)
(b)
(c)
Fig. 4. Meridional component of the velocity of current at a
depth level of 150 m: (a) control calculation; (b) analysis; (c) difference
between the analysis and control fields
Like in the case of the temperature
field, one can notice that the velocity field becomes more chaotic, with a well-marked
synoptic variability. At that, the structure of the Brazil and Malvinas
currents along the coasts of Brazil and Argentina is well seen. In Fig. 4b, a
synoptic structure of the equatorial counter-current flow manifests itself,
while it is almost unnoticeable in Fig. 4a, since it has a pronounced zonal direction
rather than the meridional direction. The amplitude corrections are not large;
they do not exceed 0.05 m/s. However, the model velocity does not have any pronounced
tendency to increasing or decreasing. In the regions of the Brazil-Malvinas
confluence, one can notice the maximum of amplitudes of both positive and
negative corrections. In addition, one can notice the “tongues” of these quantities
in the zones of the Equatorial and (North (South) Equatorial currents. On the
whole, the water dynamics in both cases with and without data assimilation
corresponds to the universally recognized pattern of currents known from other
studies [5].
At a depth of 3770 m, the charts of fields do not have
a pronounced structure. Figure 5a shows the meridional component of velocity
for the control field; Fig. 5b, for the corrected field; Fig. 5c, the
corrections for the deep-water current in the meridional direction. It should
be noted that the amplitudes of these corrections are not lower than those at a
depth of 150 m and even exceed them in some cases. In Fig. 5c, a track of about
0.06 m/s is noticeable in the west-east direction in the region of the 45°
south latitude, as well as a small negative difference between the meridional
components of velocity after and before data assimilation for the currents
along the America continent in the central Atlantic.
(a)
(b)
(c)
Fig. 5. Structure of the meridional component of the velocity
of currents at a depth level of 3770 m: (a) control calculation;
(b) analysis; (c) difference between the analysis and control fields
Figure 6 shows the meridional mass transport in the
South Atlantic. We have calculated both the total meridional water mass transport
and the transport of the AADW mass, which was separated according to the
criterion PT <2°C [23], where PT designates the water potential temperature.
Figure 6a shows the total water transport in the meridional direction; Fig. 6b,
only the AADW transport. The mass transport was calculated according to the
formula
,
where M is the total mass transport
at a fixed latitude; v is the meridional component of the velocity of
flow in a current point at the fixed latitude and depth level. The integration
is made over each latitude and vertical, from the surface to the bottom. In the
case of the AADW transport calculation, the corresponding integral is taken
only over such a region (not fixed beforehand) where the potential temperature
is lower than 2°Ñ. The unit of flow is a
Sverdrup (1 Sv = 106 ì3/ñ).
Let us make the following remark. The
detailed analysis of the AADW transport requires a special consideration, as
well as its comparison with the analogous calculations performed earlier by
using other methods and models, for example, in [21]. In this study, only some
estimates are made. This problem will be studied more in detail in another
paper.
(a)
(b)
Fig.
6. Mass
transport in the South Atlantic: (a) integral; (b) only for the AADW
In Fig. 6a, the solid line indicates the mass transport
for the model calculation (control); the dotted line, the results of
calculation of the mass transport by using the corrected (assimilated)
velocities. It is seen that these curves are significantly different. At that,
the values of the corrected mass transport are considerably, sometimes by two
times, larger than those of the control mass transport; even the direction of
the mass transport can be different. In fact, there is nothing astonishing in
this: as is seen in the above-presented figures, in the upper levels, there is
a well-remarked synoptic component of the meridional velocity that can make a
significant contribution to the transport, which is different from the control one.
Even a small discrepancy between the control and assimilated velocities after
integrating over a considerable area can give and really gives large
discrepancies, both quantitative and qualitative, between the control and
assimilated values of the mass transport M. However, the maximum and
minimum values of the control and corrected mass transport are nearly identical
and are equal to +60 Sv and ‒40 Sv, respectively; at that, the
regions of their maxima and minima do not coincide. We can come to conclusion
that the total (integral) and average (normalized by the area) control and corrected
mass transports are close in their values, however, being locally different.
Figure 6b shows the curves of the AADW transport. Like
in Fig. 6a, the control flow is shown by the solid line; the corrected flow, by
the dotted line. One can notice that the AADW transport occurs strictly in the
north direction up to 46° south latitude and in the south direction below this
boundary. At that, the flows in the south direction for the control calculation
and for that with assimilation nearly coincide with each other, which can be
explained by the absence of the observed data for the regions located southward
of 46° south latitude, while in the northern direction, they are noticeably
different both quantitatively and qualitatively. The maximum value of the
control flow is approximately 8 Sv, while with data assimilation, this value
increases twice, up to 15 Sv. The maxima of flows are shifted in their
latitude: for the control flow, it occurs at the 40° south latitude, while for
the assimilated flow, it occurs at the 42° south latitude. At that, as is seen
from Fig. 6, there is no considerable difference between their velocities;
those are the flows that differ from each other.
In addition, we have
calculated the meridional heat flows: the integral flow and the flow only for
the AADW. The calculation of the heat flows was performed according to the
formula
,
where Q is the total heat flow
for each latitude; S,
are the sectional area, specific heat
capacity with a value of 4220 J/(kg×grad) [20], density with a value of 103(kg/m3), meridional
component of velocity (m/s), and temperature (°Ñ). The integral is taken over the latitude and over the
entire vertical, from the sea surface to the bottom. Correspondingly, for the AADW,
the region is chosen where Ò < 2°C. The flow is measured in W/m2.
Figure 7a shows the
integral heat flow; Fig. 7b, the heat flow in the AADW.
(a)
(b)
Fig.
7.
Meridional heat flow in the South Atlantic: (a) integral; (b) for the AADW
We can notice that in the integral heat flow, the difference
between the corrected and pure model curves can be both positive and negative;
at that, their average values (the areas under the curves) are close to each
other. It means that the flows considerably differ locally but not summarily.
It is reasonable that the assimilated values of the meridional heat flow are
considerably larger in their moduli than the control values; sometimes the
difference between them can reach one order of magnitude, since the assimilated
values of heat flow are the product of two corrected quantities – the velocity
and temperature and, thus, as a result, we have a multiplicative effect.
However, the general integral heat balance is conserved, which proves indirectly
the correctness of our calculations. Figure 7b shows that the heat transport
for the AADW corresponds to the mass transport (Fig. 6b), it also has
well-marked flows to the north from the 46° south latitude and to the south
from this boundary. However, there is a positive value of the heat transport in
the region of the 62° south latitude. It can be explained by the presence of
negative values of temperature at a depth of about 4500 m and deeper. Even in
the case of the general mass flow passing in the south direction, the heat
transport occurs in the north direction.
Since the experiments are carried out with the joint
ocean-atmosphere model, the local changes in particular regions of the ocean
will cause a global effect through the changes in the fields in both the
atmosphere and ocean. At that, in the case of the changes in the initial values
of the model parameters due to data assimilation, we can assert that the fields
of parameters in the subsequent integration will be balanced and the
conservation laws will hold true. This is a distinct point of our study in
comparison with the model sensitivity experiments [22], in which the initial
conditions are given quite arbitrary and the model behavior is investigated
with no consideration of the balance of all model parameters.
In our study, the initial control state without data
assimilation and another case with the initial state obtained as a result of
data assimilation by the GKF method were considered. The model was integrated
forward for 11 months starting from these two initial states independently of
each other. Figure 8a shows the difference between the assimilated and control sea
surface temperature fields after 11 months of integration; Fig. 8b, the
difference between the assimilated and control meridional velocity fields.
(a) (b)
Fig. 8. Difference between the fields of parameters after 11
months of integration from the initial control filed without data
assimilation and from the initial field with data assimilation:
(a) difference between the surface
temperature fields; (b) difference between the meridional velocity fields
It is seen that the changes have a
global character and they are relatively small, except for some particular
regions. On the whole, the difference between the sea surface temperature fields
does not exceed 1°Ñ, except for a region in
the Kuroshio Current zone in the Pacific Ocean and to the south from the Gulf
of Alaska, where the changes are more considerable, which is related above all
to the stronger interactions between the ocean and atmosphere in these regions
(so-called the energy-active zones). One can see the intensification of the
meridional currents in these regions; the differences between the assimilated
and control meridional velocity fields are larger than 1 m/s. In addition, it
is seen from Fig. 8b that as a result of data assimilation, the Gulf Stream (the
North Atlantic Current) is intensified to the south from the Greenland, which
is well confirmed by independent observations. In other regions, the changes in
the meridional velocity are relatively small and are most likely caused by the
changes in the atmospheric circulation due to the initial field correction.
The changes in the initial filed have effects on such a
characteristic as the sea ice area fraction in the Arctic. Figure 9 shows the
difference between the analysis and control ice area fraction fields. It is
seen that this zone as a whole is reduced, except for a relatively narrow
region near the Spitsbergen and the adjacent region. This also corresponds to the
observed data and the results obtained in earlier investigations [23].
Fig.
9.
Difference between the analysis and control sea ice area fraction fields
Figure 10 shows the change in the ocean surface
temperature (OST) before and after the initial field assimilation after 11
month of integration in the region of the Barents Sea and the Kara Sea in
Russia. It is seen that in general, there is a rise in temperature (the
difference between the corresponding temperatures is negative); however, in the
northern zone of the Barents Sea, we can see a local decrease in temperature,
apparently related to the reduction of the warm water inflow from the Atlantic
Ocean. This result corresponds to different independent investigations, for
example, [24].
Fig. 10. Difference between the surface temperature fields
after 11 months of integration starting from the initial control field without
data assimilation and from the initial field with data assimilation.
Figure 11 demonstrates the change in the meridional
current velocity before and after assimilation after 11 months of integration
in this region of the Russian Arctic. One can remark a moderate increase of
about 10 cm/s in the flow velocity; however, in the region of Novaya Zemplya,
the velocities, on the contrary, decrease.
Fig. 11. Difference between the meridional velocity fields
after 11 month of integration starting from the initial control field without
data assimilation and from the initial field with data assimilation.
On the whole, the water dynamics in the Russian zone
of the Arctic is a consequence of global tendencies in the World Ocean.
The numerical experiments performed in our study allow
us to come to conclusion that the model climatic fields after assimilation of
the observed data considerably change not only the measured and assimilated
quantities, in particular, temperature and salinity, but also the quantities
that are not directly measured and are derived from the balance equations, in
particular, the velocities of currents. Using visualization facilities, it is
shown that the application of data assimilation methods significantly corrects
qualitatively and quantitatively the characteristics of the model physical
quantities, namely, the heat and mass transport. The corrections obtained as a
result of application of the data assimilation method can be significant and their
values can reach 100% of the values of the model calculations with no data
assimilation.
The changes in the calculation parameters obtained in
the integration with the control and assimilated initial fields have a global
character and agree with the observed data and natural tendencies.
The calculated characteristics of the heat and mass flows
are in agreement with known estimates; at that, they are more detailed.
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