The development of a homogeneous numerical ocean model for the Arctic Ocean

ShTcal report sectioM

NAVAL POSTGRADUATE SCHOOL
MONTEREY. CAL.FORN.A 93940

NPS-58G172071A

NAVAL POSTGRADUATE SCHOOL

//

Monterey, California

THE DEVELOPMENT OF A HOMOGENEOUS NUMERICAL

OCEAN MODEL FOR THE ARCTIC OCEAN

by

J. A. Gait

July 1972

Approved for public release; distribution unlimited

FEDDOCS

D 208.14/2:NPS-58GL72071A

NAVAL POSTGRADUATE SCHOOL
Monterey, California

Rear Admiral A. S. Goodfellow, USN Milton U . Clauser

Superintendent

ABSTRACT:

A numerical ocean model driven by surface stress and a source-sink
distribution is developed for a homogeneous ocean. Non-linearities,
lateral friction and bottom friction are included. The basin shape can be
varied to accommodate a large variety of configurations. Variable bathy-
metry and sources/sinks around the perimeter are included. The numerical
scheme is conditionally stable and has second order accuracy in space
and time .

A number of test cases are run to explore the dynamic significances
of the various processes represented. The possible influence of these
processes on the circulation of the Arctic ocean are discussed.

TABLE OF CONTENTS

Section I - Introduction and Report Summary 4

Section II - Development of the First Stage Numerical Model . 8

Section III- Preliminary Results 21

Section IV -Future Model Plans 32

References 34

Appendix - Use of Computer Program 36

LIST OF FIGURES

Figure 1. Chart of the Arctic Basin with the 100, 1000 and 2000 fathom
depth contours drawn. Cross indicates location of the North
Pole.

Figure 2. Grid pattern used in finite difference scheme and numbering
system used for calculational molecule.

Figure 3 . Configuration of the model used to check out the relaxation
of the stream function.

Figure 4. Streamlines of source-sink driven flow with zero vorticity in
the interior and uniform depth

Figure 5. Streamlines of source-sink driven flow with uniform negative
vorticity in the interior and uniform depth .

Figure 6. Streamlines of flow driven by uniform stress curl in an irregu-
lar shaped ocean with a central ridge. (The ridge runs from
top to bottom in the figure . )

Figure 7. Streamlines of flow driven by a simple source-sink distribution
for an irregular shaped ocean with a central ridge. (The ridge
runs from the top to the bottom in the figure.)

Section I - Introduction and Report Summary

The general equations that describe the flow of both the atmosphere
and the ocean are well known and are available to geophysicists for the
investigation of a large variety of circulation problems. For many cases
of interest analytic solutions to these general equations are not possible
because of significant non-linearities in the equations, or because of
irregular geophysical boundary configurations, or both. Although analytic
solutions are not generally available it is often possible to obtain use-
ful approximate solutions using numerical techniques. This numerical
modeling has become a powerful tool for the investigation of both the
ocean and the atmosphere.

Large scale numerical models available for the description of many
features of the atmosphere and large portions of the oceans can be divided
into roughly two groups. The first group is exploratory in nature and is
used to investigate the significance of various physical processes. The
prime emphasis is to understand just why a particular system responds as
it does and what effects variations in forcing have on the outcome. The
results from this type modeling may, or may not, be physically realistic
but in all cases should elucidate the characteristics of the flow to be
expected, its sensitivity to various inputs and significant correlations
that are likely to exist. The second group of numerical models is generally
more advanced and is designed for forecasting geophysical fluid flow on
a more or less real time basis. The relative stage of refinement and the

large amounts of high quality data required for initial conditions has in-
hibited the development of these type models for the ocean and essen-
tially all of the forecasting models routinely used deal with the atmosphere

The object of this research was to begin a numerical exploration on
the large scale circulation of the Arctic Ocean.

In the past numerical models of the Arctic Ocean have concentrated
their attention on the sea ice that overlies most of the Ocean (Campbell,
1965) and tried to answer questions concerning the drift and climatological
permanence of the pack ice (Maykut and Untersteiner, 1971). With the
exception of Campbell's use of a simplified ocean model under an ice-
layer model no significant effort has been directed towards a numerical
study of the actual flow of the Arctic Ocean's waters.

At the onset of this research it must be admitted that very little
is actually known about Arctic Ocean dynamics and that difficulties can
be anticipated in deciding on appropriate boundary conditions and input
parameters for the model. In many cases field data from the Arctic is
lacking, or inadequate to give the required stress fields or inflow-outflow
conditions with sufficient accuracy. This then demands that best esti-
mates of boundary conditions serve as a tentative guide and that wide
ranges of parametric inputs actually be investigated. The resulting
numerical exploration yields considerable insight into the relative signi-
ficance of various oceanographic parameters.

In addition to the uncertainties related to the lack of actual input
data the development of a new numerical model has potential difficulties

inherent in the finite difference mathematic's that must be used. Both
these problems can best be addressed by the careful development of the
model from simple to more complex cases in a stepwise progression and
with each step being checked against any available data (from the field
or analytic considerations). This model is carried through this sort of
genesis.

The first step in the model to be used for the Arctic assumes homo-
genous water and variable depth which is in some way similar to a model
used by Holland (1976). The grid system used is based on a triangular
plan similar to some systems used by Williamson (1968) and Sadourny,
Arakawa and Mintz (1967). Both lateral and bottom friction are included
in the model. The flow is driven by stress applied at the surface (simu-
lating the wind or ice stress) and by source-sink distributions around
the edge (that simulates major channels between the Arctic and other
oceans). The details of the development for this first step in the model
are given in the next section. The initial check out and experimentation
with this first stage in the development of an Arctic Ocean circulation
model has led to some interesting results. In particular: 1) negative
curl introduced into the flow results in current patterns resembling the
Beaufort Gyre; and 2) the Lomonosov Ridge (Fig. 1) acts like a dynamic
block which may greatly increase the significance of the circulation caused
by the source-sink distribution. There is some indication from field data
reported by Muench (1970) and Thorndike (1971) that the processes indi-
cated in the model have some counterpart in actual circulation observed

CANADA

Figure 1. Chart of the Arctic Basin with the 100, 1000 and 2000 fathom
depth contours drawn. Cross indicates location of the North
Pole.

in the Arctic Ocean. A more detailed discussion of these results are
presented in section three of this report.

Future extension and development of the model will be continued in
stages. In the immediate future the present model will run with greatly
increased resolution to delineate more of the complicated bathymetry of
the Arctic Basin. After those results have been carefully explored the
stratification appropriate to the Arctic Ocean must be introduced into the
modeling. The results for the stratified ocean model will indicate how
the next step (the inclusion of more complicate thermodynamic exchanges)
can best be approached. A more detailed discussion of future modeling
plans is given in section four of this report.

Section II - Development of the First Stage Numerical Model

When one considers the actual complexity of the circulation in the
Arctic Ocean and the wide variety of numerical modeling techniques
available for its study it is by no means obvious which path will lead to
maximum returns. Each of the presently used numerical models has its
advantages and limitations. In an attempt to address this question in a
rational way it was decided to start with the simplest numerical model that
could simulate what are thought to be the dominant forcing and geo-
morphology in the Arctic.

There can be little doubt that much of the large scale circulation of
the Arctic Ocean is wind driven either directly, or indirectly through an
ice cover that acts as some sort of coupling element (Campbell, 1965).

The details of how the ice cover couples the atmosphere to the ocean and
to what extent it filters the time and space variations are unknown. There
is however, a substantial effort directed towards this problem (Untersteiner,
and Fletcher, 1971) and some progress can be anticipated. For the pre-
sent model these interesting questions can not be treated in detail and
thus the stress on the top of the ocean will be considered a known field,
externally specified.

Studies by Coachman and Barnes (1961, 1963), Aagaard (1966) all
indicate that there may be dynamically significant exchange between the
Arctic Basins and the adjacent portions of the world ocean. For this
reason the numerical model incoporates sources and sinks of water around
its perimeter to simulate the major channels between the Arctic and the
adjacent parts of the ocean.

Looking at Figure 1 . , it is seen that a large fraction of the Arctic is
covered by the relative shallow Siberian Shelf, while the deeper portion
of the Arctic Ocean is divided into two basins (both over 4000 meters
deep) by the Lomonosov Ridge. In an attempt to retain some of the dynami-
cal effects caused by these large bathymetric variations the model includes
variable depth.

It is likely that the density variations in the Arctic Ocean effect the
flow. For example a full treatment of the movement in the Atlantic layer
(Coachman and Barnes 1963) will certainly require a baroclinic model. On
the other hand there are some actual current measurements from the Arctic
(Nikitin and Demyanov, 1965) (Gait, 1967) (Coachman, 1969) that indicate

9

a substantial barotropic, or depth independent component to the currents.
This coupled with a consideration of the great increase in complexity re-
quired for variable density models suggests that for the initial numerical
exploration a homogeneous or barotropic formulation be used.

The circulation in the Arctic is not well enough known to come up
with an accurate appraisal of the significance of frictional forces. It
seems likely that near source-sink points lateral friction could effect the
flow. Over the large area covered by the Siberian Shelf it is quite possible
that bottom friction might also be significant. Accordingly both lateral
and bottom friction were included in the model and it was anticipated
that some range of frictional parameters would be investigated to test
for significance.

To develop a model with the characteristics described above the
following integrated form of the equations of motion are used:

Du , 1 BP . „ 2 Ru TX ,.v

— - - fv = --—-+ Kv u- — + — (1)

Dt p Bx h ph

Dv . 1 dP „ 2 Rv TY ...

— +fu = --7-+K7 v- — + — (2)

Dt p dy h ph

Where the dependent variables u and v are the horizontal components of

velocity which are independent of depth. The density, p, is a constant.

r and r the components of the wind stress, h, the depth, and f, the
x y

coriolis parameter, are functions of position. K and R are constants
that specify the effectiveness of the horizontal and vertical frictional
forces respectively.

10

In addition to these equations of motion we have the continuity
equation:

k (hu) + k (hv) = ° (3)

A transport stream function is introduced such that:

Sy (4)

bib

hv=-^
bx

The pressure can be eliminated from equations (1) and (2) and after some
manipulation the following vorticity equation is obtained:

br h

= Kv2^ -|[4 + V0-7(^)] + VX (^) (5)

where:

i

_ dy_ d_u
bx by

V

_ r* i_ + ^ A.
bx J a y

T

= T 1 + r 7

x y

and i, j, k are the unit vectors in the right handed x, y, z coordinate
system.

From the above the following relationship between ^ and 0 is obtained;

v(^70)=i (6)

Equations (5) and (6) can now be solved for the vorticity and stream

11

function provided that the proper initial and boundary conditions are
given. In particular the following must be specified:

a) ij) - given within the region of interest at t = 0

b) ij) - given on the boundary of the region for all time

c) £ - given within the region of interest at t = 0

Note that condition b) is equivalent to specifying the source-sink distri-
bution around the edge of the model.

Equations (5) and (6) can be non-dimensionalized by introducing the
following new variables:

t = t'(^) f = f'F

x = x' A y = y' u

h = h'D (7)

r = r'(— -£
A

Where the primed quantities are all non-dimensional, F is the average value
of the Coriolis parameter, A is the finite difference grid spacing, D is
the average depth of the model and 0 is equal to the max range of stream
function on the boundary of the model, or the maximum value expected for
the wind driven portion of the flow, which ever is the most convenient.
Using the new variables defined above equations (5) and (6) become;

— »

12

(8)

vlj,^')*!' (9)

Where:

ot = 72

■ o

4fDF

_ _R_
y DF

These three non-dimensional parameters govern the character of
the solution. For example setting a = 0 removes the non-linear advective
terms from the model. The size of ft determines how important lateral
friction is in the solution and y scales the importance of bottom friction.

The finite difference grid that equations (8) and (9) are solved on is
made up of a one dimensional array of N x M points. They are arranged
as shown in figure 2. Grid points are numbered sequentially, left to right
starting in the top row. This means that grid point in the neighborhood
of the point L are given as shown in figure 2.

Two additional one dimensional arrays are used to specify the extent
of the interior domain. These are integer arrays labeled IA and JA of
dimension (M-2) and are defined so that a sweep of interior model points
is obtained via the following algorithm.

DO 2 K=l, M-2

I = IA(K)

I = JA(K)

DO 1 L = I, J

1 Statement on Interior Point (L)

2 Continue

13

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The depth in the model is specified in terms of an average depth and
an array of deviations from this average. The amount of bathymetry to
be used in any particular run is specified by a depth factor which is in-
put at run time. The following depth variables are thus defined.

HM = average water depth

HFAC = depth factor

H (DIMENSIONED TO N x M) = array of depth deviations.
The actual depth used in any particular run is calculated with the following
algorithm

DO 3 I = 1, NM
3 H(I) = HM + HFAC * H(I)
Obviously HFAC = 0 gives a constant depth model and HFAC = 1.0 includes
all of the bathymetry. Any intermediate fraction is also possible if ex-
perimentation suggests such a case would be of interest.

To integrate equation (8) the three level Adams - Bashforth method
is used. That is, writing equations (8) as:
9€' _ ,

at " g

we use:

4'(t' + At') = i'(V) + [f g'(t') -j g'(t'-At')]At' (10)

To use this equation we must write g1 in finite difference form. This will
be done term by term starting with g' written as:

g' = (vx0'). V(°^ht f') +£v2£'

-jj([* + V0' '*<£.)] + VX(^) (11)

15

The first term on the right hand side of equation (11) represents the net
rate of potential vorticity advected into a unit area. To approximate this
for the hexagon centered on the point L we write: (all quantities are

non dimensional and primes have been omitted for simplicity)

-L* 0 PVT AT , -. + PVT ,

(7x*k) • ?(— ) - ^Jj [(^L_N+1 " 0L+1)( 2 >

+ (*L-N " *L-N+1) 2 + ^L-l " W 2

. ,(pVn-i+pvl-i) ;; , .v^n-i1

(PVT , + PVT )
n , \- L+1 L":"N ho,

Where:

and

L+l ^L+N

pVj = (ci)(4z) +fj

o

& DF

The second term on the right hand side of equation (11) represents the
diffusion of vorticity horizontally and in finite difference form becomes:

Sv2C=(c2)[4L+1 + 4L_N+1 + 4L_N + 4L.1

+ Wi + «l+n " (6)iL] (13)

where

c2 = 21 = _2K

OZ 3 3^27

the third term on the right hand side of equation (11) represents the

16

dissipation of vorticity through bottom friction and is written as:
+ (2)(*L-N+1 ~ <Wl - *L-N + W]

1 H^t^-T1-^^)]

hL+l hL-l 2 hL-N+l hL+N-l hL-N hL+N

+ (!)[*L-N+1 " Wl + *L-N A+N] (14)

]}

r 1

1 , 1

1

Lh

L-N+l

~ h h
L+N-l L-N

\+l

C3

~7 DF

where:

The last term on the right hand side of equation (11) is the torque
added per unit time by wind stress. This is assumed constant in time and
calculated only once at the beginning of the program using the following
finite difference form .

** I> 4 «fV+l " <f >L-1 ^

(15)

_ / _L ) r (01 \ . (01 ) , (Ql) _ (01 ) ]

V3 lvhi-N+l h yL+N v h yL-N v h ;L+N-1

Where Gl is the component of the wind stress in the direction from L to
L + 1 and G2 is the component of the wind stress in a direction 90 degrees
to the left of Gl.

Equations (12) through (15) are used in equation (11) which in turn is
used in the time integration scheme defined in equation (10). Once the
vorticity has been obtained for a new time step equation (9) is solved by a

17

successive overrelaxation technique.

The algorithm used is a slight modification of the scheme suggested
by Winslow (1961). The scheme used is as follows:

A residual is calculated:

RES = 7((t- 70) - i

= (1)( hfk + 0L-N+1 + ^L-N

3 (WV (hL-N+l+V (hL-N+V

(16)

*L-1 ^L+N-l ^L+N , , x t

where:

hfacl ■ 'drr' + (hT * 1+hT > + t^rr »

L+l L L-N+l L L-N L

(hT +hT ) + (hT AT n + hT * + (hT +hT '
L-l L L+N-l L L+N L

The residual is then normalized using the coefficient of \ji in equation

J_i

(16), i.e. ,

d* = (f)(i?k)RES

J-i

and the relaxation is then given by:

</>L = </>L + (R)d^L (17)

A number of tests were run to estimate the optimal value for R and over a
wide range of cases a value of 1.48 was indicated. This value was then
used throughout the modeling efforts reported here.

The typical experiment anticipated for the model will be to apply some

1

stress field to a static ocean. The model ocean will then spin-up, or
develop a circulation pattern that will generally approach a steady state
providing an appropriate balance of torque is possible and that the
applied stress field is steady. The time dependent development of the
steady state circulation and to some extent the final flow pattern will
depend on the characteristics of the planetary waves that make up the
transients in the model. For this reason it is of some interest to look
at the propagation characteristics of these waves within the model and
to estimate the errors introduced by the finite difference scheme.

To estimate how the model transients will respond a simple, free,
linear Rossby wave will be considered. For this case the vorticity
equation (5) reduces to:

r"(^V20) = -hv • V(f ) (18)

3t h r h

A wave form of the solution is assumed and near by values used in the

computational molecule (Fig. 2) taken as follows:

i(kx + 4y -tut)
*L = *o e

*L+1 " f *L
*L-N= C *L

*L-l=<~*Nl (19)

19

*£i-k-JTi)

^L+N-l

= € 0L

y^(kV^e)

^L+N

= C' ^L

Where x and y refer to position in a right handed co-ordinate system
superimposed on the finite difference grid with the positive x-axis in
the direction form L to L + 1 . It can also be assumed with no loss of
generality that the x and y axes are directed east and north respectively-
This then gives:

^=0; ^=** (20)

dx dy

To obtain a reasonable estimate of the phase velocity for the Rossby

waves in the model the time differencing can be assumed exact and the

depth assumed constant. Using equations (19) and (20) with the finite

difference forms given in equations (12) and (13), the vorticity equation

given in (18) becomes, after some simplification.

4j/j ___

-iw ^ZTZ tcos kA+ 2 cos ( ^ — ) cos ( — )-3]
3 h A 2 2

-2i8* r . , A . ,kA v (JMA n
= — - — [sin kA+ sin (-r— ) cos {"-z )J

Which gives

,kA< J3JA

8*A
a; = ~r; —

sin (kA) + sin ( — ) cos r~r — )

" ~ " ,kA\ r^1^- I -a
cos (kA) + 2 cos (— ) cos f^ ) -3

For small A the trig functions can be expanded using small angle approxi-
mations.

20

This gives:

U>= -/?'

. l(k3 + k^2)(A2)

K 8

4 2 2

And from this dispersion relation it can easily be seen that the x com-
ponent of the phase velocity is:

C = ~= -£*

x k

12 2 2
l-77(k +0(£ )

4 2 2

•* +k +(T8 + -i- )(A)

(21)

Clearly as ^goes to zero this goes to the exact form

*

Cx ,2 2

k +i

and the error term is second order in A. This then suggests that the
transient wave solutions that develop in the model will be accurately
represented to second order.

The actual fortran program used for the model and an explanation of
the input data required is given in an appendix.

Section III - Preliminary Results

The characteristics of the model described in the last section have
been investigated in a series of tests designed to check out the numerical
properties of the solutions. In a number of cases it was also possible to
illustrate the physical significance that the processes represented might
have on the circulation in the Arctic Ocean.

The first test was to check the part of the program that handled the

21

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24

relaxation of the stream function. A basin shape was chosen that
roughly covered the deeper portion of the Arctic Ocean (Fig. 3). Repre-
sentative source-sink distributions were specified as follows:
2 x 10 m sec flow in across the Chukchi Sea, 7 x 10 m sec

CO _ -I

flow in spread out on either side of Franz Josef Land, 1 x 10 m sec

CO _ i

flow out through M'Clure Strait, 1x10 m sec flow out into the

CO _ 1

Lincoln Sea, and 7x10 m sec flow out into the East Greenland
current. With a flat bottom the equations governing the flow in the in-
terior reduce to:

V2)j) = 0

Solving this for the specified boundary conditions gives the flow indi-
cated in Figure 4 .

The next test case introduced some finite vorticity into the above
equation. For the initial testing this was simply specified as a constant.
Thus the governing equation becomes:

v2ib = h4

r o

This obviously corresponds to the artifical case where equation 8 has
gone to a steady-uniform solution for the vorticity. Although this is not
very realistic the results are interesting and shown in Figure 5 for the
case where the vorticity in the flow is constant and negative as one might
expect to get from the stress field developed by the polar easterlies.
Circulation resembling the Beaufort Gyre develops. Qualitatively the
results show a striking resemblance to the very viscous solution

25

for ice drift presented by Campbell (1965, Fig. 7c ,d).

The next set of experiments with the model were to check out the
time dependent vorticity equation (8) in conjunction with relaxation of
the stream function (Eq. 9). To do this a basin roughly the shape of the
Arctic was specified. Non-linear terms and bottom friction were not
included (a =y = 0). A uniform negative stress curl was applied to the
water that was initially at rest. The Coriolis parameter was assumed
constant.

For the first series of runs made on this configuration the depth was
assumed constant. Under these conditions the model would spin-up
forming a symmetric clock-wise circulation pattern. The steady state
solution represented a balance between the torque added by the wind and
that which was lost through lateral friction. The magnitude of the
steady state solution and the spin-up time of the model depended on the
magnitude of the frictional coefficient {8 ) that was used. It is interesting
to note that for this series the vorticity equation reduces to:

|f = ?v2^ +[(VXT) (22)

at h

With a steady wind stress this equation has no wave solutions and the

appropriate von Neumann type stability condition (Richtmyer and Morton,

1967) will be of the form:

St < constant m — (23)

(Assuming the Adams -Bash forth scheme is used for time differencing.)

26

Figure 6. Streamlines of flow driven by uniform stress curl in
an irregular shaped ocean with a central ridge. (The
ridge runs from top to bottom in the figure . )

27

Where t is the non-dimensional time step in half pendulum days. This
is relatively weak restriction in that with more or less realistic geo-
physical parameters time steps of a number of days are possible. This
was confirmed with the model. •

The second series run on this configuration was designed to test the
models response to variable depth. In this series a ridge of smooth
profile was placed across the basin. The appropriate form of the
vorticity equation was

^7-f(vx 0k) -v(Tr) =3v24 + 7xf) (24)

dt h h

Under these conditions topographic Rossby waves are possible (Veronis,

1966) and the von Neumann stability criterion takes on the more restrictive

form: (Once again assuming an Adams-Bashforth scheme for the time

differences)

— < constant ?» — (2 5)

Af 2

Where C is the phase speed of the topographic Rossby waves. This

general behavior was again confirmed by the model.

In this series the spin-up of the model begins as before, but the

variable depth acts in many ways analogously to a variable F in a flat

bottom ocean and western boundary type currents can develop. These

stronger boundary currents develop not on the western side of the ocean

basin as in Munk's model (Munk, 1950) but rather to the left of an observer

looking from deep to shallow water. Figure 6 shows a typical solution

28

for this series. The ridge is seen to divide the flow into two cells
with regions of intensified currents in each half.

During the initial spin-up for this series transient wave patterns
were seen to propagate along the ridge. It is quite likely that the real
time dependent stress fields applied to the Arctic ocean will result in
transients of this form being propagated along the Lomonosov Ridge
(Fig. 1).

An additional series of tests were run to investigate the interaction
of source-sink terms with the bathymetry and the formulation used to
represent bottom friction. For this series the stress applied at the
surface was zero. The source-sink distribution was simply to have flow
in through the Bering Straits and out through the East Greenland area.
The bathymetry was again the smooth ridge described in the previous
series of tests.

The first test in this series assumed a =0 = y = 0. Thus there was
no friction, or non-linear terms. The appropriate form of the vorticity
equation that was:

~- (U^k) • v(£) = 0 (26)

This equation shows the interesting result that the only steady flow
possible is when the stream lines are parallel to lines of constant f/h.
Thus the steady- state flow can not cross the ridge and any source-sink
distribution that demands cross ridge flow will not lead to a steady
circulation. The numerical model shows these characteristics with

29

circulation continuing to increase with time. A strong negative (clock-
wise circulation builds up on the in flow side of the ridge and a
corresponding positive circulation develops on the outflow side of the
ridge. This is similar to the effect one might get if the water simply
piled up on the in-flow side and drained out on the out-flow side
satisfying the overall continuity requirements for the ocean but with
minimum cross-ridge flow.

This has an important implication for the circulation in the Arctic.
If the source-sink component of the flow is to be baratropic and geostrophic
then flow can not cross the Lomonosov ridge. This means that continuity
must be satisfied separately for both basins in the Arctic. There will
be a strong tendency for flow that enters on the Canadian side of the Arctic
to exit on the same side. Field data from the Arctic suggest that this may
in fact be the case (Coachman, 1970). Thus one might expect that flow
through the Bering Straits and the Canadian Archipelago should be strongly
correllated.

The next runs in this series included the non-linear terms and
lateral friction which gave the following governing vorticity equation:

^ - (vx0k) • v(Q^Lf) =ev24 (27)

For this case both frictional and non-linear boundary layers are present
with cross ridge flow taking place and a steady-state solution is possible.
For all reasonable values of a and 3 the interior of the flow is still essen-
tially geostrophic and the cross ridge flow takes place in boundary layers

30

Figure 7. Streamlines of flow driven by a simple source-sink

distribution for an irregular shaped ocean with a central
ridge . (The ridge runs from the top to the bottom in the
figure .

31

where the ridge intersects the sides of the basin. (Fig. 7)

A final case run in this series included the bottom friction terms. It
can be anticipated that bottom friction would be effective in causing
cross bathymetry flow. In particular the

£*# • v(i)

term will have a tendency to turn flow towards shallow water. This
is in fact what the model showed. Steady state flows resembled figure 7
for very small values of y and showed a smooth transition to stream line
patterns resembling simple hydraulic flow (similar to Fig. 4) as the value
of y was increased.

In the deeper interior of the Arctic Ocean the flow must be essentially
geostrophic, but it seems likely that frictional effects must be significant
over large portions of the Siberian shelf. In the vicinity of the Lomonosov
Ridge both friction and non-linear effects may be significant.

Section IV - Future Model Plans

The next stages in the development and use of this model are well
under way at the time of this writing.

The first extension is to use the model with greatly increased reso-
lution and the actual bathymetric variation of the Arctic basins. With
this configuration realistic source-sink distributions are investigated.
A wind stress field obtained from Felezenbaum's pressure data
(Felezenbaum , 1958) is used to drive the flow and a number of runs are

32

anticipated to investigate the effects of variations in a, 3 and y .

The next series of tests will use the same basic model configuration
but the stress field will be obtained by applying Felezenbaum's pressure
data to Campbell and Rasmussen's (1971) ice model and then using the
stress field from the bottom of the ice to drive the ocean model. Again a
series of runs are anticipated to investigate variations in parameters.

Another series of runs is planned for the model that will have even
higher spacial resolution and definitions of the bathymetry. In this
series the ocean will be driven by a time dependent wind stress field that
is made up of random frequency components. The dependent variables at
a number of locations will be spectral analyzed with the intension of
obtaining normal mode frequencies for the Arctic basins.

The next stage in the development of model exploration for the Arctic
will be the formulation of a baroclinic model. The first set of experiments
anticipated for this formulation will be the investigation of the dynamics
associated with the circulation of the Atlantic Layer (Coachman and Barnes,
1963).

33

References

Aagaard, Knut, (1966). The East Greenland Current North of the Denmark
Strait. Ph.D. Thesis, Dept. of Ocean. Univ. of Wash. 83 pg .

Campbell, W. J. , (1955). The Wind Driven Circulation of Ice and Water
in a Polar Ocean. Jour. Geophy. Res. 70(14): 3279-3301.

Campbell, W. J. and L. A. Rasmussen, (1971). Personal communication.

Coachman, L. K. , (1969). Physical oceanography in the Arctic Ocean:
1968. Arctic 22(3): 214-224.

(1970). Towards an operational technology for the Arctic Ocean.

Arctic Inst. N-Am. workshop Oct. 12, 13 and 14. Arlington, Va.

, and C. A. Barnes (1961). The contribution of the Bering Sea water

to the Arctic Ocean. Arctic 14(3): 147-161.

, and C. A. Barnes (1963). The movement of Atlantic water in the

Arctic Ocean. Arctic 16(1):9-16.

Felzenbaum, A. I. (19 58). The theory of the steady drift of ice and the
calculation of the long period mean drift in the central part of the
Arctic basin. Problems of the North, 2, 5-15, 13-44.

Gait, J. A. (1967). Current measurements in the Canadian Basin of the
Arctic Ocean. Summer 1955. Univ. of Wash., Dept of Ocean.,
Tech. Report No. 184.

Holland, William R. (1967). On the wind-driven circulation in an ocean
with bottom topography. Tellus 19(4): 582-599.

Maykut, Gary A. and Norbert Untersteiner. (1971). Some results from a
time dependent thermodynamic model of sea ice. Jour. Geophy.
Res. 76(6): 1550-1575.

Muench, R. D. (1971). The physical oceanography of the Northern Baffin
Bay region. Arctic Inst. N-Amer. Baffin Bay - North Water Project
Sc. Rep. No. 1 .

Munk, W. H. (1950). On the wind-driven ocean circulation. Jour, of
Met. 7(2): 79-93.

Niktin, M. M. and N. I. Dem'yanov (1965) O glubinnikh techeniyakh
Arkticheskogo Passeina. Okeanologiya 5(2):261-263 .

34

Richtmyer, R. D. and K. W. Morton (1957). Difference methods for
initial-value problems. Interscience pub. Inc., New York. 237
pages.

Sadourny, Robert, Akio Atakawa and Yale Mintz (1967). Integration of
the non-divergent barotropic vorticity equation with an icosahedral-
hexagonal grid for the sphere. Dept. of Met., U.C.L.A., Num.
Simulation of Weather and Climate, Tech. Rept. No. 2.

Thorndike, A. (1971) personal communication AIDJEX planning conference,
June 1971 Lake Wilderness Conference Center, Washington.

Untersteiner, N. and J. O. Fletcher (1971). AIDJEX planning conference
introduction AIDJEX Bulletin No. 9. Arctic-Ice-Dyn-Joint Experiment,
Div. of Marine Res. Univ. of Wash., Seattle.

Veronis, George. (1966). Rossby waves with bottom topography. Jour.
Mar. Res. 24(3): 388-394.

Williamson, David L. (1970). Integration of the barotropic vorticity
equation on a spherical geodesic grid. Tellus, 20(4): 624-653.

Winslow, Alan M. (1967). Numerical solution of the quasilinear Poisson
equation in a non-uniform triangular mesh. Jour. Comp. Phy.
1(2):149-172.

35

Appendix - Use of Computer Program

The ocean model program has been written in FORTRAN - IV and a
listing of the program is included in the end of this appendix. To use
the program an appropriate data deck must be placed after the program.
The composition of the data deck must be as follows:

1) N, M, NB FORMAT (313) - this is a single card where N and M are
the dimensions of the grid system (ref . Fig. 2) and NB is the number
of boundary points.

2) IA FORMAT (2013) - this is a grid number list of the first interior
point of each line down the left hand side of the model.

3) JA FORMAT (2013) - this is a grid number list of the last interior
point of each line down the right hand side of the model.

4) LV, BLANK FORMAT (21A1) - this is a single card with the code to be
used in the graphical out put of the program. (Ref. subroutine GRAOUT)

5) HM, FAC FORMAT (2F6.1) - this is a single card where HM is the
reference depth and FAC is the fraction of the bathymetry that is
desired in the model.

6) H FORMAT (12F6.4) - this is a set of cards that has the normalized
depth variation for each grid point.

7) F FORMAT (12F6.4) - this is a set of cards that has the normalized
values of the Coriolis parameter for each grid point.

36

8) I FORMAT (13) - this is a single card with the exponential part of
the values to be used for the wind stress.

9) CI FORMAT (12F5.3) - this is a set of cards with the significant
figures part of the wind stress in the direction of the positive axis,
i.e. from point L to L + 1 (Ref. Fig. 2).

10) G2 FORMAT (12F5.3) - this is a set of cards with the significant
figures part of the wind stress in the direction 90 to the left of the
positive axis .

11) IBC FORMAT (6(611, 13)) - this is a set of cards that contains the
information required to calculate boundary values of vorticity. For
each broundary point a six digit code and the grid number are given.
The six digit code refers to the neighboring points in a counter clock-
wise order starting with L + 1. The following code is used:

0 - exterior point, 1 - interior point (free slip B.C.), 2 - interior
point (no slip B.C), 3 - boundary point along streamline, 4 - boundary
point across a streamline. Boundary points at sources are not in-
cluded in the IBC list.

12) CI, C2, C3, DT, TOUT, TMAX, R, CON, 1PUNCH FORMAT(6E10 .3 ,
F4.2, E10.3, 12) - this is a single card that contains the run para-
meters. CI, C2 and C3 are defined in section two of this report. DT
is the time step. TOUT is the time interval at which output of the
dependent variables is desired. TMAX is the maximum time the model
is to run. R is the SOR parameter to be used for the relaxation of the

37

stream function. CON is the absolute convergence limit to be used
for the relaxation of the stream function. IPUNCH is a flag that
should be non-blank if the final values for the dependent variables
are required as a source deck for future runs.

13) S FORMAT (6E12.5) - this is a set of cards that contain the initial
values for the stream function at each grid point.

14) V FORMAT(6E12.5) - this is a set of cards that contain the initial
values for the vorticity at each grid point.

15) Gl FORMAT (6E12. 6) - this is a set of cards that contain the initial
values of the local time rate of chance of the vorticity at time -(-DT)

38

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Unclassified

Security Classification

DOCUMENT CONTROL DATA -R&D

[Security classification of title, body of abstract and indexing annotation must be entered when the overall report is classified)

I originating activity (Corporate author)

Naval Postgraduate School
Monterey, California 93940

2a. REPORT SECURITY CLASSIFICATION

Unclassified

26. GROUP

3 REPORT TITLE

The Development of a Homogeneous Numerical Ocean Model for the Arctic Ocean

4. descriptive NOT ES (Type of report and.inc lus ive dates)

5 au THORISI (First name, middle initial, last name)

J. A. Gait

6 REPORT DATE

July 1972

7a. TOTAL NO. OF PAGES

59

7b. NO. OF REFS

21

■a. CONTRACT OR GRANT NO.

b. PROJEC T NO

9a. ORIGINATOR'S REPORT NUMBER(S)

NPS-58G172071A

9b. OTHER REPORT NOISI (Any other numbers that may be assigned
this report)

10 DISTRIBUTION STATEMENT

This document has been approved for public release and sale; its distribution
is unlimited.

11 SUPPLEMENTARY NOTES

12. SPONSORING MILI TAR Y ACTIVITY

Naval Postgraduate School
Monterey, California 93940

13. ABSTR AC T

A numerical ocean model driven by surface stress and a source-sink distribution
is developed for a homogeneous ocean. Non-linearities, lateral friction and bottom
friction are included. The basin shape can be varied to accommodate a large
variety of configurations. Variable bathymetry and sources/sinks around the
perimeter are included. The numerical scheme is conditionally stable and has
second order accuracy in space and time.

A number of test cases are run to explore the dynamic significances of the
various processes represented. The possible influence of these processes on
the circulation of the Arctic ocean are discussed.

DD.Fr:..1473 (PAGE"

S/N 0101 -807-681 1

58

Unclassified

Security Classification

A- 3 1408

Unclassified

Security Classification

key wo R DS

Ocean Circulation
Numerical Modeling
Arctic Ocean

DD ,T,"..1473 back,

S/N 0101-807-68?)

59

Unclassified

Security Classification

U146264

DUDLEY KNOX LIBRARY - RESEARCH REPORTS

5 6853 01068096 0