A numerical model of coastal upwelling. - Center for Ocean ...

14
1. Introduction
)OURNAL OF PHYSICAL OCEANOGRAPHY
A Numerical Model of Coastal Upwellingl
JAKES J. O'BRIEN' AND H. E. HURLBURT'
National CmIer for Almospheric Research BOIJdv, Clio. 8OJQZ
(Manuscript received 17 September 1971)
ABSTRACT
A wind-driven model of coastal upwelling induced into a stratified, rotating ocean is, solved numerically.
The circulation is on an fplane and longshore variations are neglected. A multilevel model is derived, but
only solutions for a two-layer model are discussed. A longshore baroclinic surface jet is discovered. The
time-dependent geostrophic jet is dynamically explained by conservation of potential vorticity. The existtence
of the jet depends critically on stratification and non-zero wind stress at the coast. Coastal upwelling
is confined to within 30 km of the shore. The model exhibits no deep countercurrent during active coastal
upwelling. A time scale of the order of 10 days or longer is required for a pycnocline at 50 m depth to penetrate
the surface. Solutions for a \\ide (>300 km).coastal shelf, an irregular shallow shelf, and a continental
slope region are illustrated. A secondary upwelling region is found offshore at sharp breaks in the shelf
~aphy. In all cases, the offshore flow is a simple Ekman drift and downwelling offshore is created by
Ekman pumping caused by negative wind-stress curl.
The phenomenon of oceanic upwelling off the west
coasts of continents is important from physical and
ecological viewpoints. The occurrence of strong upwelling
of cold water in a narrow coastal strip contributes
to increased productivity of the sea as well as to
climate modification of the adjacent land. Smith (1968)
has reviewed the physical process of upwelling. We
will not review upwelling here, but we will cite various
observational and theoretical papers as they are pertinent
for support or comparison.
It is our intent to describe a simple theoretical model
of the onset of upwelling by the use of numericalintegration
and scale analysit. We are interested in the
physical description of the near-shore circulation induced
into a stratified rotating ocean by the surface
winds. A general multi-layer model is developed, but
specific attention is confined to a two-layer ocean. The
Bow is on an f plane and longshore variations are
neglected. The winds are steady but may vary offshore.
The effects of wind-stress curl, bottom topography, and
stratification are explored. Both barotropic and baroclinic
modes are present in the model.
Following the derivation of -t4e multi-layer model,
we concentrate on describing a two-layer model of
depth 200 m on a wide Bat shelf. The numerical solution
for an equatorward wind stress of 1 dyn an~ is presented
to demonstrate the response of the system for
-
1 Contn"bution No. SS of the Geophysi(:al Fluid Dynamics
Institute, Florida State University, T.~assee.
t Pennanent affiliation: Departments of Meteorology and
Oceanography, Florida State University.
a The National Center for Atmospheric Research is sponsored
by the National Science Foundation.
VOLUME 2
this simplest case. Using the numerical solution as a
guide for the scale analysis, we scale the model equations
and the physics of the modelaie deduced. A time-
dependent longshore jet is discovered. Finally, the twolayer
model is solved for shelf configurations containing
bottom topography.
2. Formulation of a general multi-layer model
Consider a stably stratified, rotating, incompressible
fluid on a continental shelf-slope crbSS section near a
north-south coast line. Suppose the fluid consists of m
incompressible layers which have initial thickness 1I;(x)
and densities Ph j = 1 (l)m, counting down from the
surface layer. We will assume in this model that northsouth
variations are neglected (a/a,=O); the traditional
Coriolis approximations apply; molecular transport
of momentum is assumed to be unimportant compared
to turbulent transport by eddy stresses; the
fluid is hydrostatically balanced in the vertical; and
the atmospheric pressure is uniform at the sea surface.
The appropriate {;oordinate system is a right-handed
Cartesian coordinate system with x increasing eastward
and s upward. The origin will be in the coastal cross
section such that distances offshore are negative. In
these initial studies, we shall neglect thennodynamic
effects, i.e., the exchange of latent heat or sensible heat
with the atmosphere is excluded as well as radiation.
The latter may play an important role in the shallowest
near-shore region. Also, no mixing of heat and ,salt
between the ocean layers is considered. This is easily
added if desired [see O'Brien (1967) for the detailsJ.
For upwelling induced by wind over several days, we
are explicitly assuming that the time scale for vertical

JANUARY 1972
J I
mixing is much slower than that for vertical transport
of fluid.
We are particularly interested in the vertically averaged
velocity within each layer and the vertical velocity
at the interface of each layer. If the pertinent
hydrodynamic equations of motion are integrated
vertically over the depth of each layer and the vertically
averaged horizontal velocity components I'i, 'OJ are
assumed independent of depth within eaCh layer, we
obtain
aUi
at
aui '1'.
+ur-+Pi- ~s+[(... .
8:% i
avi
~
at
av;
ax
Bs
'1', )/pilJ+A
,.. B.
--~(.,. -.,. )/ph;]+A
I ;
O'BRIEN AND H E HURLBURT
a2Ui
-,
ax'
a~;
-, ar
(1)
(2)
alii allJUi
I -0, (3)
at axwhere
P; is the pressure integral for the layer and T r
and TIJ are the stresses at the top and bottom of each
layer. When j=l, ~r =Tf~+iTfY, where these components
are the wind stresses at the sea surface. They will
be specified as functions of x and t. The vertical velocity
is implicit in (3) and may be obtained a posteriori from
the h; and 11; fields. Both barotropic and baroclinic
modes have been retained. The barotropic mode will
be shown to be fundamentally important for the winddriven
upwelling problem. It has been neglected by
previous investigators.
o. The pressure integrals
From the hydrostatic assumption, it follows that the
pressure p, at any point inSide ,a layer k is given by
layer 1
: layer 21
1-1 ~.
Pt,=P.4+ #1 PJthl+p.,[/=J ~I+D(x)-.],
layer
(-i)
where m is the number of discrete layers, P A the uniform
atmospheric pressure, and D(x) the elevation of
the bottom above the reference level s=O. The influence
of the torque of the atmospheric pressure
gradient may be included if desired (e.g., Galt, 1971).
The x derivatives of these pressures are
apt "-1 ah; {
-- y psg-:-+p
-. all--i a
j
~ -+- .
ax 1=1 ax ~ ax as
(5)
1$
Integrating these over the appropriate layer, we find
that
1 12' at'.
PJli= - -=-ds
p. B a~
h. ~l ani - { -. all-; a
- I:; pJt-:-+h ~ -+- ~
p. 1-1 a~ ~ ax ax
(6)
These values of P. are used in (1). Thus, for an appropriate
specification of th~ wind stress, a functional
form of the interior stresses, and valid initial and
boundary conditions, the formal statement of the
problem will be complete.
The form of P. can be simplified if either 1) the
bottom layer (or layers) is at rest or 2) there is no
bottom topography.
b. Initial conditions and boundary conditions
There are several allowable initial conditions. First,
if the interface surfaces are horizontal, then the appropriate
initial condition is Ui=lIi=O. If Pi does not
vanish initially in any layer, the initial conditions may
be Ui=O, flli=Pi, i.e., the layer is in geostrophic
equilibrium.
The velocity components vanish at the coast, i.e., at
x=O. At x= - ~, the fluid velocity takes on its initial
value. We anticipate specifying a wind-stress distribution
that approaches zero far offshore. In such a case,
a radiational condition (Shapiro and O'Brien, 1970) or
a viscous far field (Galt, 1971) may be used to close
the solution at a computational outer boundary. We
will demonstrate that the solution in the upwelling
zone is independent of the~forcing outside the upwelling
zone and thus apply far-field boundary conditions
which cannot alter the solution near the coast in any
significant way.
c. The interior stresses
The equations of motion (1)-(3) are coupled for
adjacent layers through the preSsure gradients P J and
the interior stresses. To complete the statement of the
problem, we must specify a functional form of these
stresses.
The stresses must be continuous across the interfaces,
i.e., "':.="'~1 and .,.:--.,.1;1' They act in opposition
to one another on the interface and thus are
balanced. We wish to choose a functional form of
~j=.,.~+i.,.~ de~ndent upon the velocity q;=uJ+illj,
but not dependent upon aq,Ios. The logical physical
choice is a quadratic form from a dimensional argument.
To arrive at an appropriate form, let us consider the
energy equation.
In general, for a continuously stratified system
(A =0), one obtains an integral energy equation of the

16
fOrD}
aE
JOURNAL OF PHYSICAL OCEANOGRAPHY
1"
-+VgoJat
B
a'f
q.-ds=I,
as
(7)
where E is the swn of the kinetic and potential energy
densities and .J is an energy fiu..'t density. H we con.
centrate our attention on the integral on the right (I),
we can arrive at an integrated expression for I and
eventually at a form for 'C. The approach outlined
below was suggested by Robert O. Reid (personal
communication) for a somewhat different problem.
Integrating I by parts, we obta.in
I-[..'C]r-[q.'C]B'
.r
,1.
If the velocity q or the stress ~ vanishes at the bottom,
the second term vanishes. The first term is the rate of
supply of kinetic energy from the winds. The integral
on the right represents collectively the rate of decrease
of energy of the organized motion, i.e., the rate of
conversion to turbulence and ultimately to thermal
energy. Moreover, from the second law of thermodynamics,
we expect that the integrand
aq
>0.
a.
-
e
o
-
d
z
.
aq
az
(8)
(9)
VOLUME 2
3. The two-layer model
-
We will explore the details of a two-layer model.
Consider a two-layer fluid governed by the dynamics
derived in the previous section. The equations are
aU1 aU1
-+u1--c-+g(h1+it+D).
at ax
- JVt+ ("aa_.,.,8)/(piIs) +A
a~
+.r ~+("s- - ,.,-)/ (plIv + A-,
at a%
,\...t
Ot/l
a.i
attll
-,
ax'
ah1
,-
ax
8tu2
- fts+("z8-orB8)/(p/I,)+A-,
8r
a"2

JANUARY 1972 J J
O'BRIEN AND H. E. HURLBURT
FIG. 1. Typical geometry for a two-layer model with a free surface and one interface.
The bottom topography is D(z), and"l and"t are the thicknesses of each layer of density
PI and Pt. The velocities are depth-averaged.
For a nonlinear problem, we must resort to numerical
methods of solution. The techniques used are new and
extremely efficient, and they are explained in the next
section.
Boundary conditions are required to close the problem.
At the coast, Ui=Vl=Ut=v,=O and hl and h2 are
determined from the continuity equation using a onesided
finite difference. At x= - ~, all of the velocities
vanish and hl and ht are independent of time. In the
computer model, we should utilize a radiational condition
or the method 9f characteristics at some large
but finite distance from the coast. However, empirical
experimentation demonstrated that infinite channel
(no slip) boundary conditions at a considerable distance
(~300 kIn) from the coast do n9t affect the solution in
the coastal region. . . .
500 ~a KAt 100
0
2
N
I~
U
f/)
~
0%
~
%-
-I t-
FIG. 2. Wmd stress components as a function of distance offshore;
, is 1 dyn cm-t Dear the coast; 7"'- is zero everywhere.
-2
It is practically impossible to obtain oceanographic
estimates of -r(x,t) in upwelling regions. The data just
do not exist. For this simple model, we choose a wind
stress which is independent of time and space for
XE[O, -200 kIn] and drops off rapidly in magnitude
200 kIn off the coast (Fig. 2).
The parameters for the standard case are f = lij-4
sec-i, g=lO' an sec-'!, D=O, T8s=O, p=l, A=lO' ant
sec-i, g'=2, c=3Xlij-4. We have altered these for
various computer solutions. The importance of the
parameter values will become clearer in the section on
scale aD;alYSis.
4. Semi-implicit method
We have designed an extremely efficient numerical
scheme following the example of K wizak and Robert
(1971). For the present one-dimensionallayered model,
the scheme is sufficiently different from Kwizak and
Robert that ,we shall present the technique in some
detail.
Since the equations contain external gravity waves
as solutions (as :w-ell as internal gravity waves), the
garden variety explicit finite-difference schemes require
a tinle step At bounded above by 4%/(gB)t, where H is
the total depth of the fluid. The fine resolution
[4%=0(1 km)] and the deep fluid [H=0(1 kIn)],
envisioned for realistic simulation of coastal upwelling,
dictate a time-step < 1 min, whereas we anticipate
integrating our model for days. Kwizak and Robert
demonstrate that if we choose an implicit scheme for
the terms which govern the physics of the fastest
moving waves, a much larger time step may be utilized.
We elect to treat both the external gravity wave and

~
18 JOURNAL OF PHYSICAL OCEANOGRAPHY
VOLUllE2
the internal gravity wave modes implicitly. If the
pressure gradient terms in the momentum equations
and the divergence term in the continuity equations
are treated implicitly, then we may use a much larger
time step, say on the order of 1 hr. Application of the
following scheme resulted in a factor of 20 savings in
computer time for the present problem on a CDC~;
i.e., a 20-min integration time using an explicit finitedifference
scheme is reduced to less than 1 min by
using the semi-implicit method. Identical results were
obtained using both methods.
Consider the model equations in the form
~I+
-2Al~+ilTl
.- ~H1- 8II1J_l
. 8s i
~1-:-(h1 a + it+ D) .,.. 4tg'ailJe+l
a~ 8s i
-LIt (25)
-2.6JU~+~1
8 8il J -'
~(iJ+it+D)-~t'-
-Lt. (26)
0..1 a
-+t-:-(.h1+.h,+D)-Ul,
at as
ail 0..1 0..1 ail
-+H~- -hI' -"I--Dt,
at as as
(19)
(20)
cJscJsiAi(H.+D)hI
] -+-1
hti-+l+
Bsl
ht J_I
At(H.+D)-
-
ax
8M, a a~l
-+r(~l+ht+D) -g'-=
at ~ a%
Ut,
(21)
=2AJDI/'+itrl-
-L.. (27)
as I
All spatial derivatives are replaced by second-order
finite differences. The subscripts j and n imply for any
ait
-+(B.+D)-at
8M,
a~
-11.'
aus ail
-~+D--DI,
as as
~
as
(22)
scalar q that
qj--q(-j~, 1IAI). (28)
where B 1 is the mean depth of the upper layer and
Bt+D the mean height of the interface between the
two layers. No approximations have been made; Ul
and U t are all the remaining tenns in the %-directed
momentum equations. The linear part of the divergence
term in the continuity equation is retained on the lefthand
side. The primed quantities are defined by
All of the right-hand side terms, defined now as L,
(i= 1, 2, 3, 4), are "known" at any time level in the
calculations. We wish to find (lIl,Ut,"I,"I)j_+l for all j
at time level (11+ 1) AI. K wi.7.ak and Robert recommend
elimination of the velocities and the solution of coupled
Helmholtz equations for "1 and "1. Since we have
homogeneous bQundary conditions for III and Ut, we
elect to p-liminate "1 and "1 fIml the above and obtain
it-il'+BI
} .
II-I,'+B,
J ~1 l
+(C-(J) [~
tI~l-b[~ as' "
]"'l
{23)
.
It is important to ~ iliat Hi and Iit+D are constants.
We will treat the left-hand side of (19)-(22)
implicitly and the right-hand side explicitly. (We have
-4, . 8,si I
--V""l+NtJ-+ t+C[ ~ ] '" t
-..
8,si J
(29)
(.'to)
also made the horizontal diffusive terDlS implicit using where 4=tH1~, b=t(Hs+D) ~, c-iH1~, and d
the Crank-Nicholson method. This is not included and e are a linear combination of the L,. These equa-
below to simplify the discussion.) The tendencies are
evaluated over 2411 and the pressure gradient terms
and the divergence terms are averaged between time
levels (ft-1) 411 and (ft+ 1) 411.
The time-differenced equations are
tions are coupled one-dimensional, Helmholtz equations
in the unknowns "1-+1 and fIt-+1 lor all space
points j. When the spatial derivatives are replaced by
standard second-order finite differences, the resulting
algebraic equations are tridiagonal and are easily solved
by the special "up-down" variant of Gaussian elimination.
In practice, we solve (29) and (30) iteratively by
.~.+
solving (30) for "iJ'-+l using "up-down" and ~en (29)
for flti-+l. Only a few scans are needed for convergence.
The )'-directed momentum equations are solved using
-2AIU~+-rl
4l~(hl+il+D)
8~
]_l
-Lit
I
(24)
well known techniques-leap-frog for time differences,
a quadratic-averaging method known as Scheme F from
Grammeitvedt (1969) for the advective terms, Crank-
Nicholson for the diffusive terms, and centered-in-time

~
JANUARY 1972
Coriolis terms. We believe it unnecessary to repeat
these well-known finite-difIerence schemes here.
5. Flat shelf case
A simple flat shelf case is used to e.xplore the basic
physics of our modd of the onset of upwelling. This
standard model is integrated numerically until hi approaches
zero near the wall. The solutions are shown
in Figs. 3-10. The fluid is initially at rest with hi = 50 m,
h, = 150 m for all x.
In Fig. 3, the height anomaly of the interface and
free surface are shown after 4 days. Near the shore in
a width of 20 km or so, the interface height anomaly
is order 10 m. The free surface anomaly is order - 20
cm. Offshore, where .,.8 approaches zero, a negative
height anomaly is calculated. We shall refer to positive
height anomalies as upwelling and negative as downwelling.
We shall demonstrate that the downwelling is
simple Ekman pumping which occurs when curl ~ 8 is
,
J
J. O'BRIEN AND H. E HURLBURT
50
.5
~G
~cn 0:
50~
~~
20~
t5 Zto
E-o
5:I:
02
-5t&1
:I:
-to
-t5
~O 200- KM tOO D -20
FIG. 3. Interface height anomaly after 4 days as a function of %.
Upwelling is indicated by positive height anomaly near the shore.
Downwelling is found centered at 210 kID. The dashed line is the
free surface anomaly. ~
~
IS
.0
S
~
~
20
15
-20
5OC 200 KM 100
0
10
5
0
-5
'1'
19
FIG. 5. Interface height anomaly after 10 days. The interface
bas almost surfaced near the shore. The downwelling is centered
near 210 km. The dashed line is the free surface anomaly.
-100
~O 200 KM tOO
~
I' aUtaJ
-I' f/)
"-
-21 :s
-,~ U
~O ~
-so >-
E-
~~ U 0
-7' ~
~
-Be >
FIG. 6. Velocity profiles as a function of s afw 10 days. Dashed
lines are lower layer velocity components. A strong surface jet
occurs near s=O. The longshore velocities are barotropic between
50-150 kin.
~O 200 KM 100 0 .
\
-~
10
9
8
rU1
>ct;
60
5Z -
rn
~
~
~
ze--
:I:
to)
&3
:I:
.r:1 2
.-
3f-o
2
FIG. 7. Contours of fll as a function of time and ~ The
inertial oscillations are visible. The minimum #1 is O( -3 CD!
sec-1). The inertial oscillations decay in time in the forced region.
The oountdur interval is 1 cm sec-1.

~
20
JOURNAL OF PHYSICAL OCEANOGRAPHY
~
e
VOLUMB2
40
cn
~~
5GCaJ
Z5~
202
15 Z-
10 E-
5:1:
0,9
-5CaJ
:I:
-II
-15
- 200 KAt I" . -21
FIG. 10. Interface ~t anomaly for each day as a function of ~.
After 10 days, ..- 2 m at the w.n aDd 6Sm near 210 km.
negative. The dynamics of the upwelling at the wall
the height anomaly is alm~t 50 ro, i.e., hi is approaching
zero. The downwelling OCCUR 200 km offshore.
The free surface anomaly is order -50 an. It is im-
are more complicated.
In Fig. 4, the four velocity components are shown
after 4 days as a function of space. The upper layer
Bow is offshore (111

i ;. JANUARY 1972
j. j. O'BRIEN ANP H E HURLBURT
21
tive, downwdling occurs through Ekman pun1ping.
Except for the presence of the baroclinic jet and the
narrowness of the upwelling region, this description is
s [ HSOUs - au, a
RsAs ~ at +Fr-

~
22
JOURNAL OF PHYSICAL OCEANOGRAPHY V~UME2
the multiple balance. The horizontal friction tenn is hI/hIe> 1. Therefore, "1 must be a nllnimum at the
required to bring the longshore flow to zero at the wall.
Therefore, on the shore side of the jet, 0_' will be
coast and fJz must be a maximum. This s&}'S that the
jet and the upwelling region have the same width
important. The acceleration tenn must be important scale! Near the shore, the barotropic pressure gradient
in the vicinity of the jeL In the steady state, there must increase since 81 is geostrophic. This demonstrates
cannot be any jet. (This has been found numerica.l1y the real importance of the free surface in the problem.
for large c, but not shown here.)
When horizontal friction is included, the minimum in
The presence of the jet depends on two important 81 will be displaced away from the shore a distance
physical processe.s which are included in this model,
but not in previous models of coastal upwelling. First,
the wind stress may not vanish near the coast as in
ffidaka (1954) and Hsueh and Kenney (19.72). Secondly,
AV. I .
L.-(jU;) ~S km.
the upwelling model must be truly time-dependent and
not steady state as in Yoshida (1967) and Hidaka.
This estimate is obtained by setting EHA1-1=1. Since
A 1 decreases in the jet region, the estimate for Lv is a
Since the jet is observed in nature (Mooers, 1970;
Mooers et 01., 1972), it is a real feature of coastal
lower bound. This scaling predicts that the jet must
vanish as we approach the equator. If A is large enough
upwelling.
The scaling thus far has not clearly supported the
dynamical basis for the jet; however, a potential vorticity
argwnent does. Consider the equations
at/l av. ,..'
-+-t-+ f"l - -,
(43)
at ax HJP
ani a
-+-(111"1) =0,
(M)
at aoS
~ att
-+~-+ J",=o,
(45)
c)1 ! h
(10' an' sec-i) to pennit the viscous boundary layer
to envelop the entire upwelling region, no jet will
develop. These conclusions have been verified numerically.
It is our thesis that horizontal eddy ~ity
must not be allowed to playa dominant role in more
complicated models of coastal upwelling, since real
physical features of the circulation will be smeared
away.
For the v, momentum equation, the outer balance is
HI at.
R.-- = -tit-
HI at
The onshore flow is balanced by the tendency tenJ:\
ail, a
I (/I.-s) -0.
at h
(46)
when HI> H 1. Since the E~!r'1Ln drift offshore will
always be confined to a thin upper layer even in a
continuously stratified ocean, this balance must be
valid. The absence of the longshore pressure gradient
to balance tit confuses our physical argument. Garvine
(1971) allows a constant (but small) longshore pressure
gradient for his steady-state model. However, a con-
(47)
sequence of his model is that the vertical integral of
the longshore flow must vanish. This is rarely obserVed.
He does obtain a deep poleward flowing countercurrent
which does not appear in our model. From our model,
we conclude that the commonly observed deep counter-
-0,
(48)
current is produced by the large-scale circulation
(Pedlosky, 1969; Durance and Johnson, 1970) and not
by the local wind-induced upwelling. This is, of course,
where the curl of the wind stress near the coast has sheer speculation which must be tested in fully threebeen
neglected. Since the fluid is at rest initially, we ~imensional models of coastal upwelling.
may integrate once and obtain
Near the shore, i.e., where (I~I

JANUARY 1972 1.1 O'BRIEN AND H E HU_RL'BURT
UI
U.
VI
V.
HI
Ht
H
L T
TABLE 1. Values of P8;rametcrs and scaling
variables.
2 cm sec-l
i cm sec-l
50 cm sec-1
17 cm sec-1
SXIO1 cm
lSXIO1 cm
IScm
JOkm
10 days
f
}
At'
P
c
As
41
G
1&-4 sec-l
1~ cm sec-'
2cmsec-'
10' cml sec-1
1 dyn cm-t
1 gIn cm-t
3XIO"""
lkm
3Omin
10-1
in the dynamics of the model. The time scale T = L/ U 1
is order 10 days or larger. This means that the interface,
originally at 50 m, will surface in about 10 da}-s.
This implies an average vertical velocity of 3 X 10-' cm
sec-1. In the real ocean, the decrease of temperature
with depth will allow appearance of cold upwelled
water almost immediately after the onset of upwelling.
However, our solution implies that it will take several
days for a subsurface thermocline or pycnocline to
appear at the surface as an oceanic front.
The assumed values of the parameters and the
-.C- '-'-'-~"'" ,-- _.,_i -} .' 0
20
40 (/)
~
6OCzJ
Z-
eot
:2
100
120 :c
e-
1'0 Co
~
160 0
180
ZaG
300 200 KM 100 C
2.3
FIG. 11. Geometry for the ~lf case. The upper layer is
SO m thick initially. The lower layer is 14 m thick near the coast
and 150 m thick offshore.
layer is only 14 m thick, whereas seaward of 100 kIn it
is 150 m thick. The wind stress profile is identical to
the standard case (Fig. 2).
The numerical solutions are shown in Figs. 12-16.
In Fig. 12, the velocity profiles after 10 days' integration
are shown. The effect of the shelf is clearly seen.
The upper layer flow offshore (UI) is unchanged. The
compensating onshore flow in the lower layer increases
as the sea bottom rises near 100 km. The longshore
velocities are barotropic between 50-150 kIn. A weak
deduced values of the scaling variables are given for
convenience in Table 1. The reader should realize that
if a larger value of the bottom drag coefficien t c is us"ed,
bottom friction will be important in the tit momentum
equation. We have tried to reduce the dynamic role
of bottom friction in our upwelling model by keeping
c small. The kinematic role of bottom friction has been neaIShore jet in VI is still evident. At 210 kIn, the wind
retained. In the numerical solutions given in this paper, stress curl creates a secondary baroclinic region. In
the interface surfaces (and the computations are termi- Fig. 13, the height anomaly for the upper layer is
nated) before bottom or internal stresses play any shown as a function of time. A weak secondary up-
appreciable role in the dynamics of the problem. welling region is found at 100 km. The potential
The downwelling occurs through Ekman pumping. vorticity argument, (49), easily explains the dynamics
Far offshore, the balance in (36) can only be
of the secondary upwelling region. After a few days,
U.lll=TS., (53)
this actually decreases, but at the shore the upwelling
is still occurring. Comparing Figs. 10 and 13, we ob-
whose curl is
serve that the presence of the bottom topography
ax at
(54)
Since curl 'C

~
~
24
JOURNAL OF PHYSICAL OCEANOGRAPHY
, ":-.' "\ ~
I ~
j "
--
I cn
~==:::;: ~
~
~ ..~
- 200 KM '10
~~i~
c
~ --
~
j~D:
:E '28 :s
t! Z-
21
t.
t
-f,
-28 2
U
-,.
I... ~
I
~
f::
... u
-"I 9
C&J
.. >
..
FIG. 14. Profiles of \'I e8ch day for the ~ cue. The ~ven
spacing of the lines indicates the iriertial oscillations. 8. Oregon coast case
~~1O-C-
II .
, .'-'
,I ~.
~~
.~
,~~
c:::a.
rc--=~
.
q
-f"
,
~
~~~-.I.
~O . 200 KW 100 .
t: ~
. DC-' -
I of' ~~ :I:
-'!
-21
.: at KY 'H 0
Flc. 13. IntedKe betcht anomaly ~ for e8Ch day. After 10
days, "1-19 m at the cout.
., 9
e
T~
6~
sZ-
. ~ 2
3E=
2
.
FIG. 15. CCR1tourI of "-WI u a fuDCtioD. of time aDd ~ for
the ~ caa. The C8tour intecva1 is 5 CID eec:-t. The am-
1)Jiblde of the iDertial C*iUation at 150 km is 0(2 CID wc-t).
'!'he IOIki line at the coast indicates the_baroclink.Jet.
delays the upwelling at the coast. In the standard case,
111=2 m after 10 days, ~ut 111=19 m in the sharp-shelf
case after 10 days. The downwelling in this ~ al5O
u
ru
cn
> -
- -
-
> -
- - ~
~
--- --
50. 200 KM \00 . .
..
Vm.UKE2
9
6
7m >-
. <
60
5Z-
.CaJ
:2
;&::
,
FIG. 16. Contours of III as a function of time and ~ for the
lhaIp-sbeU case. The contour interval is 1 cm sec-1. The inertial
.-,.ll8.donl near 150 kIn have au amplitude of 2 cm.c-t.
occurs offshore at 210 km in response to the negative
wind stress curl.
The evolution of III is shown in Fig. 14. The uneven
spacing of the lines indicates the presence of the inertial
oscillations. These are seen more clearly in Fig. 15
where the baroclinic velocity VI-III is contoured as a
function of time. It is of interest to observe that the
amplitude of the inertial oscillation is damped over the
shelf. In Fig. 16, tit is contoured on the x, t plane. The
strong inertial oscillations are primarily observed between
100-200 kin. At 6 days (when the secondary
upwelling has diminished at the edge of the she1f), the
pattern of the inertial oscillations is affected by a
shoreward propagating internal wave.
There is some value in integrating a case with
bottom topography similar to the Newport, Oregon
region, since considerable amounts of direct current
observations have been acquired in this area by Oregon
State University. In Fig. 17, the simulated topography
,
.,.
2M rn
~, E
lH ~
eo, Z-
I:~
-
;, , .., Q
. . . . .
;:::::::::~...
:::::::::::::: ..
;,.;:::::::;:::::. . .
~. at
KM
100 0'00'
Fro. 17. Bottom topography for the simulated Oreaon Coat
caR. The ~ layer is 15 m thkk initially. The lower layer it
1 kin tbkk ofisOOR.

JANUARY 1972
""-_"c.C\_~.,'\I
J
-:--:-:-:-:-:-:-: ::---
J. O'BRIEN AND H.K. HURLBURT
~
.u raJ
oScn
I.I~~
: .15 U
-2~ ~
.-25 >-
E-
-55 U 0
-~ 1
-41 ~
FIG. 18. Veloci~y profiles after 3 days for the Oregon Coast case.
The lower layer velocity profiles are dashed. A narro\v jet in VI
is centered at 5 km offshore.
9. Critique
25
We have idealized coastal upwelling in this paper.
Even within the framework of our layered model, it
is possible to vary many parameters. We have run
numerous solutions, 0(100), varying bottom topography,
wind stress, stratification, latitude, frictional
constants, etc. Space does not permit a full disclosure
of these results. There are some comments, however,
that are appropriate.
Our simple two-layer model has no realistic steadystate
solution. Examination of (14) and (17) will con-
~ 200- -KM- 100
-4S
C ~
vince the reader that the only steady~state solution is
Ul=U2=O everywhere. The longshore velocities are
geostrophic and the three y-directed vertical and horizontal
stresses must balance. Clearly, the role of the
north-south pressure' gradient must be included to
balance non-zero x-directed motion in a realistic steady
state. Since steady-state motion never occurs in the
real ocean and we are concerned with the onset of
for this case is shown. The shelf drops off to 200 m by upwelling over a few days, we do not regard this as a
30 kIn offshore and to 1 kIn depth at 100 kIn offshore. serious problem. It is our intention to develop fully
In this example, we ch()OSe the upper layer to be only three-dimensional models in the futUre.
In real eastern boundary currents, a deep poleward
flowing countercurrent is almost always present
(Wooster and Reid, 1963). Our model has no mechanism
for producing a countercurrent. Huyer (1971) has
shown that during a period of strong upwelling off the
Oregon Coast the countercurrent is greatly diminished.
Hidaka (1954) reported that upwelling is a maximum,
when the wind is equatorward but with an angle 21°
offshore. We do not concur. In our stratified model,
the ma.'timum rate of upwelling occurs when the wind
is only slightly offshore [0 (5°)J. As f decreases and
we approach the equator, this angle increases; at 4N
the angle is 0(30°) (solutions not shown). The wind
angle for maximum upwelling is an unknown function
of latitUde, strauncation, depth of the upper mixed
layer, and time.
15 m thick. This is necessary to avoid a major reprogramming
effort. Sielecki and Wurtele (1970) have
shown how to integrate the shallow water equations
in a basin wi th sloping sides. Their ideas are being left
for future work.
We adopt the same wind stress profile (Fig. 2) with
0.5 amplitude in .,.8r. The velocity profiles after 3 days
are shown in Fig. 18. The longshore jet is narrow,
0(15 kIn). The solution is essentially barotropic from
50-150 km. In Fig. 19, the height anomaly is shown as
a function of time. Since the upper layer is quite thin
(15 m) initially, tIle interface has almost surfaced in
3 da)"S. The downwelling occurs, as predicted, at 210
km offshore. . .
.,,0 2"" ' ~
'" .. KM tOO I
IS
cn
10 ~ raJ
f-o
raJ
2
5~
f-o
:I:
t-'
.~ :J:
FIG. 19. Interface height anomaly for the Oregon Coast case
after 3 days. The interface bas risen at the coast to ~-ithin 1 m of
the surface. Downwelling occurs offshore at 210 km.
10. Summary
A two-layer numerical model of coastal upwelling in
a stratified ocean on an f plane has been solved. A
geostrophic baroclinic surface jet has been discovered
and explained dynamically by a potential vorticity
argument. The coastal upwelling and the jet are shown
to be confined to within 30 km of the coast.
Considerable experimentation can be done with the
model described here. We have pedormed many numerical
experiments varying the parameters of the
model. Solutions with several layers and time-dependent
winds have been obtained, but these must be
reported elsewhere. It seems essential that the next
generation of numerical models of coastal upwelling
mtlSt be three-dimensional to explore the dynamic role
of the north-50uth pressure gradient and the northsouth
divergence.

26
JOURNAL OF PHYSICAL OCEANOGRAPHY
VOLU)IE 2
Acknowledgments. This paper is the basis ofamaster's
thesis for Mr. Hurlburt in the Department of Meteorology,
Florida State University. Mr. Hurlburt has
ffidaka, K., 1954: A contribution to the theory of upwelling and
coastal currents. T,./JftS. A_. Geophys. Union, 35, 431-444.
Hsueh, Y., and R. N. Kenney ill, 1972: Steady coastal upwelling
in a continuously stratified ocean. J. P",s. Oceonoft'., 2,
been partially supported by a NASA fellowship. J. J.
O'Brien haS been supported by NCAR during the
summers of 1970 and 1971. The Computer Facility at
NCAR has provided CDC 6600 time for the computa-
27-33.
Huyer, A., 1971: A study of the relationship between local winds
and currents over the continental shell off Oregon. M. S.
thesis, Oregon State University, Corvallis.
Kwizak, M., and A. Robert, 1971: A semi-implicit scheme for
tions. The Florida State University Computing Center grid point atmospheric models of the primitive equations.
has provided some computer time on its CDC 6400.
The bulk of the research costs was supplied by the
Office of Naval Research under Contract NONR-
N14-67-A-235-(xx)2 at Florida State. Partial support
Mon. Wea. Rev., 99, 32-36.
Mooers, C. N. K., 1970: The interaction of an internal tide with
the frontal zone in a coastal upwelling region. Ph.D. dissertation,
Oregon State University, Corvallis.
-, R. L. Smith, C. A. Collins and J. G. Pattullo, 1972: The
has been derived from National Science Foundation dynamic structure of the frontal zone in the coastal
under Grant GA-29734.
We wish to acknowledge the encouragement, insight
upwelling region off Oregon. Deep-Sea Res. ("m press).
O'Brien, J. J., 1965: The non-linear response of a two-layer,
baroclinic ocean to a stationary, axially-symmetric hurricane.
and support of Phil Hsueh, Christopher Mooers, John Tech. Rept., Ref. 6S-34T, Texas A & M University, 99 pp.
Allen, Dana Thompson and Richard McNider during
the evolution of this research.
-, 1967: The non-linear response of a two-layer, baroclinic
ocean to a stationary, axially-symmetric hurricane: Part II.
Upwelling and mixing induced by momentum transfer.
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