1 On the Cooling of a Buoyant Boundary Current ∗ HSIEN-WANG OU Lamont-Doherty Earth Observatory Columbia University, Palisades, New York January 27, 2003 In press, J. Phys. Oceanogr. ∗ Lamont-Doherty Earth Observatory Contribution Number xxxx. Corresponding author’s address: Dr. Hsien-Wang Ou, Lamont-Doherty Earth Observatory, Columbia Uni- versity, Route 9W, Palisades, NY 10964 E-mail: [email protected]
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On the Cooling of a Buoyant Boundary Current∗∗∗∗
HSIEN-WANG OU
Lamont-Doherty Earth Observatory
Columbia University, Palisades, New York
January 27, 2003
In press, J. Phys. Oceanogr.
∗ Lamont-Doherty Earth Observatory Contribution Number xxxx. Corresponding author’s address: Dr. Hsien-Wang Ou, Lamont-Doherty Earth Observatory, Columbia Uni-versity, Route 9W, Palisades, NY 10964 E-mail: [email protected]
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ABSTRACT
Through a steady-state reduced-gravity model, we examine the downstream evo-
lution of a buoyant boundary current as it is subjected to surface cooling. It is found that
the adverse pressure gradient associated with the diminishing buoyancy is countered by
falling pressure head, so the overall strength of the current --- as measured by the (trans-
port-weighted) mean square velocity --- remains unchanged. This constancy also applies
to the cross-stream difference of the square velocity because of the vorticity constraint,
which leads to the general deduction that the net current shear is enhanced regardless of
its upstream sign. As a consequence, if the upstream flow contains comparable near-
shore and offshore branches, this parity would persist downstream; but if the near-shore
branch is weaker to begin with, it may be stagnated by cooling, with the ensuing genera-
tion of anti-cyclonic eddies. On account of the geostrophic balance, the buoyant layer
narrows as the square root of the buoyancy --- the same rate as the falling pressure head,
but more rapid than that of the local deformation radius. Some of the model predictions
are compared with observations from the Tsushima Current in the Japan/East Sea.
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1. Introduction
In the ocean, a warm buoyant current may move poleward along the eastern
boundary, being confined there by the Coriolis force. Examples of such boundary cur-
rents abound, including the Tsushima Current in the Japan/East Sea. As such currents are
separated from the ambient water by a sharpened density gradient across which property
exchanges are curtailed, their buoyancy is more susceptible to depletion by the atmos-
pheric cooling. For the case of the Tsushima Current, for example, frequent cold-air out-
breaks in winter may extract a heat of several hundred 2−⋅ mW from the surface, rapidly
eroding its buoyancy as the current moves north. Because of the heat capacity of the wa-
ter, such poleward decrease of the buoyancy persists into summer even when the surface
is heated.
Questions of obvious dynamical interest include: how does the current respond to
such buoyancy decrease? What might be its eventual fate? The addressing of these ques-
tions is also of practical importance since it may aid our understanding of how the heat
and salt carried by the current may be dispersed.
Surprisingly, there does not seem to be dynamical studies of the problem in the
literature; the closest is perhaps that of Nof (1983), who considered the cooling effect on
the path of a free jet. His study however has little relevance to the present problem since
the path of our buoyant current is affixed to the boundary by the Coriolis force. On the
other hand, also in contrast to a mid-ocean jet, there is a well-defined boundary separat-
4
ing our buoyant current from the ambient, so the conservation laws impose a stronger
constraint on flow properties. As we shall see, such dynamical constraints lead to current
behaviors that are not obvious at the outset, which nonetheless are robust and possibly
testable by observations.
One recognizes of course that other external changes, such as that of the Coriolis
parameter, boundary curvature or bottom topography, may all elicit a response from the
current. Some these responses have been discussed in the literature, and need to be taken
into account when assessing the total behavior of the flow (see discussion in section 5).
The intention here however is to examine the narrow effect of cooling of which our dy-
namical understanding is particularly lacking. To facilitate such understanding, we con-
struct a highly idealized model by removing all non-essential elements, including the
complications mentioned above.
The model is formulated in section 2, which reveals some strong constraints im-
posed by conservation laws. In section 3, the general behavior of the solution is dis-
cussed based on its non-dimensional form. Some model predictions are compared in sec-
tion 4 with observations from the Tsushima Current, and the paper is concluded in sec-
tion 5 by a summary of the main findings and additional discussion.
2. Model
5
Let us consider a model configuration sketched in Fig. 1, which is placed in the
northern hemisphere for convenience. The coastal boundary is taken to be straight and
vertical, so that a right-handed Cartesian coordinate system can be used with x, y and z di-
rected to the east, north and upwards, respectively. On account of the Coriolis force, the
buoyant current is pressed against the eastern boundary and separated from the ambient
by a density interface. The origin of the coordinate system is set at the eastern boundary
at some upstream point where the flow conditions are specified, and z=0 is aligned with
the unperturbed ocean surface outside the buoyant layer. The Coriolis parameter is taken
to be a positive constant for simplicity.
We assume that the vertical mixing has erased the vertical shear within the layer,
which exchanges no mass with the ambient. In accordance, the continuity equation for
the time-mean fields can be integrated vertically through the layer depth h to yield
( ) 0=⋅∇ vh � , (2.1)
which allows the definition of a transport streamfunction ψ as
ψ∇×= kvh ˆ� . (2.2)
The y-component of (2.2) may be integrated in x to yield
∫−=x
ldxhvψ , (2.3)
where l is the width of the buoyant layer, an unknown function of the downstream dis-
tance. This streamfunction spans a constant range [0,Q] across the buoyant layer, with Q
being the volume transport, an external parameter. As seen later, this streamfunction
provides a more convenient independent variable than the cross-stream distance in the
derivation of the model solution.
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We assume the ambient water outside the buoyant layer to be homogeneous and
motionless, and denoting its variables by the subscript “r” (as in “reference”), the buoy-
ancy b of the moving layer is then defined as
)( ρρρ
−= rr
gb , (2.4)
where g is the gravitational acceleration and ρ , the density. For simplicity, we assume
this buoyancy to be uniform crossing the layer (that is, in both z and x), which can be jus-
tified by its upstream source and/or cooling-induced convection and mixing. It dimin-
ishes however over the longer downstream distance due to surface cooling, and the object
of the study is to examine the change of the flow structure in response. Given the cross-
stream homogenization of the buoyancy, it is coupled to the dynamics only through the
volume transport, independent of the current structure. Combined with the fact that it is
not our intention to explain the buoyancy field, the latter thus may be regarded as external
to the model. Moreover, as it turns out, the buoyancy enters the solution only parametri-
cally, which thus can be used as an independent variable, in place of the downstream dis-
tance. This direct linkage of the flow field to the local buoyancy without reference to the
downstream coordinate allows a broader application of the model --- for example, even in
summer when the ocean is heated so long as the other model approximations remain
valid.
To derive the pressure-gradient force in the presence of a varying density (see
also Nof 1983), one first notes that hydrostatic balance implies that the lower layer has a
pressure of
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)( hzghgp rr −−+= ηρρ , (2.5)
where η is the surface displacement, and hence is subjected to a pressure gradient force
of
)()(1 hghgpr
rr
ρρ
ηρ
∇−−∇−=∇− . (2.6)
As the lower layer is motionless, one sets this force to zero to derive, using the definition
(2.4),
)(bhg ∇=∇ η , (2.7)
which can be integrated once to yield
bhg =η , (2.8)
since both variables vanish outside the buoyant layer. The surface height thus is propor-
tional to the layer depth --- albeit with a decreasing proportion constant, and (2.8) will be
referred as the pressure head. Similarly, the pressure in the upper layer is given by
)( zgp −= ηρ , (2.9)
which, using (2.8) and the Boussinesq approximation, exerts a force of
)()(1 bhbzp ∇−∇−=∇− ηρ
. (2.10)
Although this force varies vertically, only its vertical average enters the momentum equa-
tion since the velocity is vertically uniform within the layer. Averaging (2.10) vertically,
one derives
∫ −∇−∇=∇−
η
η ρhbhbhpdz
h)(
211 , (2.11)
so the momentum equation is given by
8
)(2
ˆ bhbhvkfdtvd ∇−∇=×+ �
�
, (2.12)
with f being the Coriolis parameter, a constant. It is seen that the spatial variations of
both buoyancy and surface displacement (the two terms on the rhs) may exert a pressure
gradient force. The former in particular has a definite sign --- adverse to the flow --- as
the buoyancy decreases downstream, but since it can be countered by the falling surface,
the flow need not slow down.
Since the buoyancy is constant in x, the x-component of (2.12) states simply that
the time-mean flow satisfies the geostrophic balance
xbhfv = , (2.13)
where the derivative has been short-handed by a subscript. Substituting (2.13) into (2.3),
one derives
2
21
bhf
=ψ , (2.14)
which can be evaluated at the eastern boundary (denoted by the index “0”) to yield
2/10 )/2( bfQh = . (2.15)
It is seen that as a direct consequence of the mass conservation and geostrophy, the layer
depth along the eastern boundary increases downstream. But since this increase is slower
than the inverse of buoyancy, the surface height (2.8) still decreases downstream --- at a
rate of 2/1b . This stretching of the layer would induce a positive shear in the current, a
tendency that is seen later however to be countered by other effects on the vorticity.
9
Taking the curl of the momentum equation (2.12), one derives the vorticity equa-
tion
bhkdtdqh ∇×∇⋅= ˆ
21 , (2.16)
where
)ˆ(1 vkfhq �×∇⋅+= − (2.17)
is the potential vorticity (PV). It is seen that, since the buoyant layer thickens toward the
coastal boundary and the buoyancy decreases downstream, the solenoidal term (rhs in
[2.16]) acts to reduce PV. Physically, this is because the adverse pressure gradient in
(2.12) is greater over the thicker part of the layer, thus tending to induce a negative shear.
Drawing the similar conservation --- and hence mixing --- of PV by the turbulent motion
as the buoyancy, we assume the time-mean PV to be homogenized across the moving
layer, just like the buoyancy. With this approximation, the time-mean of (2.16) can be
integrated cross-stream to yield
yy bhQq 021)( = , (2.18)
where q is henceforth understood to represent the time-mean, and hence given by (from
[2.17])
)(1xvfhq += − . (2.19)
Substituting (2.15) into (2.18), the latter can be integrated in y to yield
)()/2( 2/12/12/1 bbQfqq uu −−= , (2.20)
where the subscript “u” is used hereafter to indicate upstream values --- external parame-
ters of the model. As expected from earlier discussion, the downstream erosion of the
10
buoyancy causes a reduction of PV, which thus may counter the stretching of the layer in
affecting the current shear, as seen later.
The Bernoulli function of course is not conserved with the turbulent motion,
hence its time-mean not mixed as PV. There however is a well-known relation linking
the two on account of the geostrophic balance. With B denoting the (time-mean) Ber-
noulli function
bhvB += 2
21 , (2.21)
it is readily seen from (2.3) and (2.13) that
qddB =ψ
, (2.22)
so its span-wise distribution is known. In particular, with q being uniform cross-stream,
one has then
)2/( QqBB −+= ψ , (2.23)
where overbar is used henceforth to denote the transport-weighted mean so that, for ex-
ample,
∫−≡
QdBQB
0
1 ψ . (2.24)
Since we deal only with time-mean rather than turbulent variables from this point on, this
transport-weighted mean will sometimes be referred simply as “mean” without causing
undue confusion. To examine the downstream variation of B, one takes the dot product
of v� with the time-mean of (2.12) and uses the continuity equation (2.1) to derive the
Bernoulli equation
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bvhBvh ∇⋅−=⋅∇ �� 2
21)( . (2.25)
As an extension of the previous discussion, the adverse pressure gradient associated with
the decreasing buoyancy acts to diminish the flux of Bernoulli function. As remarked
earlier, this does not necessarily slow down the flow since the decrease can be fully ac-
commodated by a falling surface. Integrating (2.25) in x, we have
∫−=0 2
21)(
lyy dxvhbBQ , (2.26)
or, using (2.13) and (2.15),
yy bbfQB 2/1)/2(31= . (2.27)
Integrating this equation in y, one derives
)()2(32 2/12/12/1 bbfQBB uu −−= , (2.28)
where the upstream value uB is an external parameter. With this, B is now fully deter-
mined as a function of ψ (2.23), so is the velocity from (2.21) given the layer depth.
To translate this dependence on ψ to x, one rewrites the geostrophic balance
(2.13) as
v
dhfbdx = . (2.29)
Expressing v in terms of ψ (through [2.21] and [2.23]) and using (2.14) that relates ψ to
h, (2.29) can be integrated once to yield
−+Γ−+Γ
=
qfhqfh
qfbx
//ln
02/1
0
2/12/1
, (2.30)
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with
( )qQBqbfh
qfh −+−≡Γ 222 . (2.31)
It is seen that given the values of Q, ub , uq and uB , one now has a full determination of
the flow as a function of the downstream buoyancy b, which will be discussed next.
3. Solution
For a more general discussion of the solution, we non-dimensionalize the vari-
ables according to the following scaling rules (indicated by brackets): ubb =][ , Q=][ψ ,