Lesson 11 – Modeling the Gulf Stream current
In the previous section, we obtained a solution for the horizontal currents in the
ocean as a function of depth due to a balance between turbulent and Coriolis forces. By
integrating over the water column, we were then able to deduce some basic properties of
the horizontal transport of the ocean at the surface. Assume the ocean is incompressible
in this case with density  o . The horizontally and vertically isotropic equations of motion
for an incompressible medium are:
  2u  2u 
  2u 
Du
1 p
 fv  
  h  2  2    v  2 
Dt
 o x
  x y 
 z 
  2v  2v 
  2v 
Dv
1 p
 fu  
  h  2  2    v  2 
Dt
 o y
  x y 
 z 
(1)
 2w 2w 
 2w 
Dw
1 p

 g   h  2  2    v  2 
Dt
 o z
  x y 
 z 
u v w


 0.
x y z
I. Physical assumptions: Let us systematically examine the physical assumptions
needed to obtain a simple model of the horizontal current flow at the surface near a
continental boundary.
1. Neglect inertial terms:
As in the previous chapter, we will assume a steady state solution and neglect the non
linear advection terms. This is the same as assuming that the Coriolis and frictional
forces are much greater than any inertial effects in the equations of motion.
2. Shallow water assumption:
Examine a vertical profile of the ocean domain as shown in figure 1. We assume
a flat ocean bottom at a depth, H, from the free surface. In addition we superimpose a
small displacement,  t, x, y  , to the free surface so total ocean depth = H   where

 1 .
H
1
H
Figure 1 – Vertical profile of the ocean domain showing a small displacement surface
superimposed over a constant depth.
The overall volumetric domain of our fluid is then defined as an approximate rectangular
basin:
0  x   ,0  y  b,0  z  H  
H 
 1 . From the incompressible
b, 
continuity equation, we know that the vertical currents are much smaller than the
horizontal currents under the shallow water assumption and therefore the vertical
component of the equation of motion is in approximate hydrostatic balance:
We assume that the domain is shallow so
p
 o g
z
(2)
We can derive the pressure dependence with ocean depth in equation (2) by integrating
from the ocean surface at H   to an arbitrary ocean depth z:
px, y, z   po  g o H   x, y, t   z 
The horizontal pressure gradients are then related to the surface displacements by the
following relationship:
 h p  g o  h
(3)
3.  -plane model approximation:
Given that we are interested in oceanic flows along the entire coastline of the
continent, it is important to include the Coriolis vector’s dependence on latitude in
^
equations (1). Recall under the  -plane CS that f  2 sin   k where  is the
latitude from the equator. We can expand the Coriolis vector about a central latitude
value  o using a Taylor series. To first order,
2
^
f  k  2 sin o  d   2 sin o   2 cos o  d . But from spherical geometry we
know that dy  Rearth d  where Rearth is the radius of the earth. Therefore we can
approximate the Coriolis vector as a linear function of y:
^
f   fo   y  k
where f o  2 sin  o  and  
(4)
cos o 
df
. The approximate linear dependence
 2
dy
Rearth
of the Coriolis vector on y is called a  -plane model. It shows us that f  f  y  and
therefore we will not be able to assume that velocity field can be split into Geostrophic
and frictional components as we did last chapter.
4. Frictional forces are proportional to the velocity field:
We will also continue to assume that the mixed layer is horizontally homogeneous so that
 h  0 and  v is spatially constant in equations (1). Further, we will assume that the
vertical friction components depend linearly on the velocity field in the following form1
Therefore:
  2u 

 u
2 
H
 z 
v 
Where  is a constant proportional to our eddy viscosity and H is ocean depth as defined
in assumption (2). Although this assumption is questionable, we can appreciate its simple
form. Further, its physical justification is clear as it is of a similar form as that of
Hooke’s law where the resistance of the system increases as the velocity increases.
5. Horizontal current is independent with depth (2-D case):
Our interests are in the horizontal dependence of the surface currents in an
idealized ocean basin. Therefore we should not confuse matters by considering the
dependence of our horizontal ocean currents on depth for now. Therefore we can
assume that the fluid medium is homogeneous in the vertical and that u  ux, y  and
v  vx, y  . The equations of motion are now a 2-D problem where the currents on the
ocean surface specify the whole fluid domain of the problem. This provides some
interesting consequences to the equations of motion as we will see with our next physical
assumption.
A more acceptable approach that leads to this assumption is by examining mass
transport instead of horizontal currents. Recall that at the end of the last chapter, that the
1
The assumption that vertical friction depends linearly on the velocity field can be a simple consequence of
taking the Laplace transform of the equations of motion . Our main goal is an even simpler case, however,
as we wish for our currents to be independent of depth altogether as will be shown in assumption 5.
3
horizontal mass transport is defined by integrating the horizontal current over the vertical
fluid column.
M H  x, y  
0
 u  x, y, z  dz
H
h
The above integration eliminates the vertical dependence of the problem. Thus if we
consider mass transport instead of current flows, we only have a 2-D problem. This
approach is actually more commonly accepted as can be seen by the literature. For the
sake of consistency with our derived equations of motion at the beginning of the chapter,
however, we will examine current flows instead of mass transport and just assume the
flow is 2-D.
6. Wind stress on ocean surface:
Last chapter, we assumed that the wind stress was constant in the east-west
direction. In this case, we will assume a latitude-dependent wind stress,
^
 
  y    cos y  , acting in the same east-west or i direction on the free surface of the
b 
ocean. This positional dependence accounts for the shift from easterly winds at low
latitudes in the northern hemisphere,  0   and a westerly wind component in the
higher latitudes  b   .
Notice that in section 4 we approximated the flow as 2-D, therefore the above
wind stress boundary condition now acts over the whole domain of the problem. This
means that it is no longer a boundary condition but a stress term in the equation of
motion. We must modify the equations of motion to account for this additional stress
term over the fluid domain.
 fv   g
 


 u  cos
x H
H
b

y

(5)
 
fu   g
 v
y H
7. No slip conditions at the basin boundary:
This is just the boundary condition requirement that ocean currents must be 0 at the basin
boundaries of our problem. This means that for horizontal ocean currents, ux, y  and
vx, y  :
ux,0  ux, b  u0, y   u, y   0
and
vx,0  vx, b  v0, y   v, y   0
4
II. Derivation of the equations of motion in scalar form:
This again is a set of coupled equations. Taking

of the north-south component and
x

of the east-west component of equation (5) we can eliminate the
y
dependence on the ocean surface displacement:
subtracting it from
 u v 

f     v  
H
 x y 
 v u  

   
sin 
 x y  Hb  b

y

By the continuity equation and shallow water assumption, the first term is zero,
f  h  u  0 and the resultant equation of motion is
v  
  v
u  

   
sin 
H  x y  Hb  b

y

(6)
We can simplify the above expression even further by noticing that the flow is solenoidal
(due to incompressibility) and approximately 2-D due to the shallow water assumption.
Therefore we can express the horizontal currents as a function of a stream function ,
 x, y  :

y

v
x
u
We can then eliminate the first term in the above equation due to the commutability of
the differential operator:

  2


 
sin 
x
H
Hb  b

y

or
 2  
H


 
 A sin  y  where  
and A 
x

b
b 
5
(7)
III - Solution to the equations of motion:
Equation (7) is a non-homegenous Poission’s equation. We will assume the
solution consists of a linear combination of a homogenous and particular solution.
  x, y    h  x, y    p  x, y 
i) Particular solution:
 
The forcing term, A sin  y  has only y-dependence and so we can assume that the
b 
particular solution also has only y-dependence,  p x, y    p  y  . Substitution of the
particular solution into equation (7) leads to a simple 1-D Differential equation
d 2 p
dy
2
 
 A sin  y 
b 
Which has a solution of
p 
 
A sin  y 

b 
b2
(8)
2
ii) Homogeneous solution:
The homogenous form of equation (7) is,
 2  

0
x
To solve this equation, consider a homogenous solution of separable form
 h x, y   f x g  y 
Then equation (7) takes the form:
f ' ' g  fg ' 'f ' g  0
Divide both sides by fg and notice that we can equate all terms as function of x to all
values as a function of y:
f ''
f'
g''
  
f
f
g
6
The only way to have an equality such that one side depends entirely on x and the other
side depends entirely on y is for both sides to equal a constant. We will set both sides of
the above equation equal to the constant  2 :
f ''
f'

 2
f
f
g''
  2
g
The above two equations are a simple set of linear second order differential equations
with solutions:
f x   c1e m1x  c2 e m2 x
and
g  y   c3 cosy   c4 sin y 
Where m1  
   2  4 2
2
and m2  
   2  4 2
2
So the general form of our homogenous solution is
 h x, y   c1e m x  c 2 e m x c3 cosy   c 4 sin y 
1
2
And our total solution, including the particular contribution, is
 x, y   
 
A sin  y   c1e m1x  c2 e m2 x c3 cosy   c4 sin y 

b 

b2
2

IV - Application of the no-slip conditions:
In order for the currents to be constant at the boundaries of the ocean basin, the stream
function must be constant at these locations. Without loss of generality, we can choose
that constant to be 0 so our no slip conditions then take the form:
 x,0   x, b   0, y    , y   0
Application of  x,0 in equation 10 leads to
 x,0  c3 c1e m x  c 2 e m x   0
1
2
7
Indicating that c3  0
Application of  x, b  0 leads to
 x, b  c4 sin bc1e m x  c1e m x 
1
2
The choice of setting c4  0 is trivial and will eliminate all latitude dependence in the
problem so we must instead require that sin b  0 which leads to the following
restriction on the constant  :
n

n  1,2,3....
b
Our solution now has the form:
2
 n   b
 
y    2 A sin  y 
 b  
b 

 x, y   d1e m x  d 2 e m x  sin 
1
2

Where we have absorbed the coefficient c4 into d1 and d 2 .
Application of  0, y   0 leads to
d1  d 2  sin  n

2
 b
 y 
y    2 A sin    0
 b  
b 
The above expression is simply a linear combination of two sin functions. If we set
b2
1  d1  d 2  and  2   2 the above equation then has the less tedious form:


 n  
 
y     2 sin  y   0
 b 
b 
1  sin 

Differentiation of the above equation with respect to y and examination at the boundary
y=0 leads to the following relationship between the two constant coefficients:
n1   2  0
(11)
Further differentiations leads to the following additional relationship at the boundary:
n 3 1   2  0
(12)
8
Equations (11) and (12) are only possible if n=1 so our boundary condition takes the
form:
d1  d 2  sin  
2
b2
 b
 
y    2 A sin  y   0  d1  d 2  2 A

 b  
b 

And our solution takes the form:

    b2
 b2

 
 x, y    d1e m x   2 A  d1 e m x  sin  y   2 A sin  y 
b 



 b  
1
2
Our final no-slip condition  , y   0 leads to the equation
 m1  b 2
 b2

 d 1 e   2 A  d 1 e m2    2 A  0



 




Solving the above equation for d1 we find that
1  e m2 
 2 e m1  e m2
And a little algebra shows us that
d1 
b2
A
 b2
 b2
e m1  1
 2 A  d1   2 A m 
e 1  e m2

 
And our solution is of the final form:
 1  e m2
e m1  1 m2 x    
m1 x
 x, y   2 A m1
e  m1
e  1 sin  y 
  e  e m2
e  e m2
 b 
Where
b2
  2 4
m1  
A
2
2
b2
  2 4
, m2  
2

b
2
b2
,
cos o 
H
,   2
and

Rearth
As a final note, recall that this solution has been defined over the fluid domain:
0  x   ,0  y  b,0  z  H   .
9
(13)
Interpretation of Stommel’s solution:
In his paper, Stommel assumed the frictional proportionality constant to have a value of
dyne
  0.02 . He also assumed a maximum wind stress of 1 2 . The ocean basin can be
cm
defined by the boundaries
b  6249km
  10000km
H  200m
The above physical parameters provide us with sufficient information to calculate the
streamlines for an idealized ocean basin. The result is shown in figure 1 for an assumed
  2  10 13 cm 1 sec 1 .
8
Contour lines of stream function
x 10
6
5
distance (cm)
4
3
2
1
0
0
1
2
3
4
5
distance (cm)
6
7
8
9
10
8
x 10
Figure 1 – Contour lines of a stream function of an idealized ocean basin surface currents
with latitude-dependent Coriolis force and wind forcing terms.
10
Figure 1 shows the resultant stream function as described by equation (13) with
the above assumed physical parameters. Clearly there is a longitudinal asymmetry that
must be a consequence of the function form of  . This asymmetry leads to a southerly
flow that is significantly stronger along the western boundary of the ocean basin as
observed along the Eastern Coast of the United States. This westward intensification is
related to the longitudinal dependence of the Coriolis force as derived in the  -plane
approximation. To verify this westward intensification is a consequence of the latitude
dependence of the Corilis force, let us assume that   0 in equation (13) and then


m1   , m 2   . Figure 2 shows that the longitudinal asymmetry or westward
b
b
intensification disappears when we make this simplification.
8
Contour lines for a stream function with a coriolis force with no latitude dependence
x 10
6
5
distance (cm)
4
3
2
1
0
0
1
2
3
4
5
distance (cm)
6
7
8
9
10
8
x 10
Figure 2 – Stream function of and idealized ocean basin surface currents with   0
As a final note to this section, the reader might be overwhelmed by all the
physical assumptions needed to obtain the results given by figure 1. In particular, the
frictional assumptions given in the problem had little basis on empirical observation
rather than simple additions needed to ensure a steady state form of the solution.
Stommel concludes his appear by stating “The artificial nature of this theoretical model
should be emphasized, particularly the form of the dissipative term. The writer thinks of
this work as suggestive, certainly not conclusive. The many features of actual ocean
structure omitted in this model should be evident2” Even given the weak physical basis
2
Stommel, W. H. 1948. The westward intensification of wind-driven ocean currents. Trans. Amer.
Geophys. Union, 29, 202-206
11
for Stommel’s solutions, his qualitative results are impressive in that he is able to
analytically show the dependence of latitude on driving the currents along merdianal
coastlines.
In 1950 Walter Monk extended the work of Stommel’s westward intensification
by assuming a more physically plausible form of the wind field and frictional effects in
the ocean basin. The resulting linear solutions indicate a set of similar flow patterns as
seen in figure 1 but as a series of “gyres” which include not only the westward
intensification effects of Stommel but also other countercurrent flows as shown in the
figure below3.
These gyre’s are split up into regions which are defined by vanishing of the forcing term
(frictional forces) in the equations of motion.
3
Munk, W. H. 1950. On the wind-driven ocean circulation. J. Meteorol., 7, 79--93.
12