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Journal ArticleDOI

A String Function for Describing the Propagation of Baroclinic Anomalies in the Ocean

01 Mar 2001-Journal of Physical Oceanography (American Meteorological Society)-Vol. 31, Iss: 3, pp 765-776
TL;DR: In this article, the authors derived a string function that describes the propagation of large-scale, potentially large amplitude, baroclinic energy anomalies in a two-layer ocean with variable topography and rotation parameter.
Abstract: The authors derive a string function that describes the propagation of large-scale, potentially large amplitude, baroclinic energy anomalies in a two-layer ocean with variable topography and rotation parameter. The generality of the two-layer results allows results for the 1-layer, 1.5-layer, inverted 1.5-layer, lens, and dome models to be produced as limiting-cases. The string function is a scalar field that acts as a streamfunction for the propagation velocity. In the linear case the string function is simply c2o/f, where co is the background baroclinic shallow water wave speed, and typically describes propagation poleward on the eastern boundaries, westward (with some topographic steering) over the middle ocean, and equatorward on the western boundaries. In the more general nonlinear case, the string function is locally distorted by the anomaly. In the fully nonlinear examples of a lens or dome, there is no rest or background string function; the string function is generated entirely by the disturbance and propagation is due to asymmetric distribution of the anomalous mass over the string function contours. It is shown that conventional beta/topographic propagation results (e.g., beta drift of eddies, the Nof speed of cold domes) can be obtained as limiting cases of the string function. The string function provides, however, more general propagation velocities that are also usually simpler to derive. The first baroclinic mode string function for the global oceans is calculated from hydrographic data. The westward propagation speeds in the ocean basins as derived from the meridional gradient of the string function are typically two to five times faster than those expected from standard theory and agree well with the propagation speeds observed for long baroclinic Rossby waves in the TOPEX/Poseidon data.

Summary (3 min read)

1. Introduction

  • An important consideration in large-scale ocean dynamics are the spatial variations in topography and the Coriolis parameter f.
  • In other work (Tyler and Käse 2000a, hereafter TK1) the authors introduced the ‘‘string function’’ for the case of a homogeneous one-layer model and in a companion paper (Tyler and Käse 2000b), validated these theoretical results using results from a primitive equation model.
  • In summary, two points are noteworthy: first, the geostrophic momentum is properly viewed as having a flow part and a propagation part and, second, the velocity of the propagation can be calculated from a scalar string function that encapsulates beta, topographic, and finite amplitude effects simultaneously.
  • It is, however, useful for the discussion in this paper to describe results for a separate baroclinic mode and then include consideration of the barotopic coupling terms.
  • The authors briefly discuss other previous work, restricting ourselves mostly to those works with which they will later make comparisons.

2. Formulation

  • Let us consider the two-layer model shown in Fig.
  • The authors will consider dynamics described by the shallowwater, hydrostatic momentum equations in layer integrated flux form.
  • When r1 ± r2 the starred term remains and the equation above, which the authors will still refer to as describing the barotropic mode, shows coupling with the baroclinic effects.
  • A similar description can be given for (22).

3. Case of geostrophic flow

  • In later sections the authors will add in the nongeostrophic flow effects.
  • Note the similarities in these definitions; m̃ always refers to a mass anomaly due to the alterations in layer thick- nesses, while the string function cm 5 c2/ f with the shallow water wave speed c chosen appropriately for the model.
  • This is very much like that for the one-layer model except that the propagation is less sensitive to topography in the open ocean and is typically much slower.
  • While the ratio c m/ may increase entering shallow water, suggestingcmo an increase in nonlinearity, increasing topographic slope increases the background string velocity and may predominate to make the disturbance less nonlinear.
  • This will make the asymmetries between the propagation of positive and negative m̃ more complicated.

4. Case including nongeostrophic flow

  • Because (25) does not involve R1 explicitly, using (6) it can be cast in a form similar to that for the one-layer case described in detail in TK1.
  • Hence, the nongeostrophic effects for the two-layer baroclinic case are similar to those in the one-layer case and the criteria derived in TK1 can be applied.
  • For scales larger than the Rossby radius (5cm/ f ) the dispersion term (involving ]ts2) and the nonlinear advection term (involving ] j) can be neglected relative to ] tm̃ and the nonlinear string velocity contribution, respectively.
  • That is, the dynamics are controlled by advection (propagation really) of mass by the string ve- locity, and advection of angular momentum by the flow velocity.

5. Barotropic coupling

  • As the authors described in the introduction, in the general two-layer case involving finite amplitudes and topography the barotropic and baroclinic modes remain linearly coupled.
  • The two evolution equations, (20) and (23), would need to be solved simultaneously.
  • U must be prescribed and the constraints given by the equations governing the barotropic dynamics have not yet been imposed.
  • In this case the two string velocity terms must balance.
  • The authors emphasize that several assumptions have been made in this section that limit the validity of the results presented.

6. Summary of theoretical results

  • Hence, it is only through the different definitions (see Table 1) for the mass disturbance m̃ and the shallow water wave speed c that the models differ from one another regarding the generalized beta drift described by the terms on the left of (35).
  • Changes in m̃ following the string velocity are due to the convergences in the various momentum fluxes shown on the right.
  • For many cases these right-side terms vanish and the evolution of m̃ is determined entirely by cm.
  • The momentum terms on the right of (35), in sequence, are SF, which in the case of the one-layer model is the Ekman momentum (5 f 21t 3 z), while in the baroclinic models it is the fraction of the geostrophic barotropic momentum in the lower layer [5(h2/h)Sg];.
  • When the authors assume that s2 is geostrophic to first order, the friction term can be transformed into a Laplacian diffusion term for m̃, and for scales larger than the Rossby radius, SI becomes negligible.

7. Comparisons with previous results and discussion

  • In the small amplitude limit, it is simple to show in a manner similar to that done in TK1 that the predictions given by the string function are consistent with the theory for linear topographic and beta Rossby waves.
  • Interpreting (36) in their context, the authors can say that the term involving H1 in (36) is due to propagation of m̃ by the background strings , while the m̃2 term iscmo due to the propagation by the nonlinear component of the strings m.c̃ Cushman-Roisin et al. also extended the analysis of westward drift to a two-layer flat-bottom case.
  • They found equations describing the drift in each layer that agree with those for their two-layer case in the flat-bottom limit when barotropic effects are discarded.
  • The numerical results of Swaters (1998) showed speeds 20% less than the Nof speed and this difference was attributed to the interaction between the dome and the surrounding topographic Rossby wave field.

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M
ARCH
2001 765TYLER AND KA
¨
SE
q 2001 American Meteorological Society
A String Function for Describing the Propagation of Baroclinic Anomalies in the Ocean
R
OBERT
H. T
YLER
*
AND
R
OLF
K
A
¨
SE
Institut fu¨r Meereskunde, Abteilung Theoretische Ozeanographie, Kiel, Germany
(Manuscript received 14 January 1999, in final form 25 May 2000)
ABSTRACT
The authors derive a string function that describes the propagation of large-scale, potentially large amplitude,
baroclinic energy anomalies in a two-layer ocean with variable topography and rotation parameter. The generality
of the two-layer results allows results for the 1-layer, 1.5-layer, inverted 1.5-layer, lens, and dome models to be
produced as limiting-cases. The string function is a scalar field that acts as a streamfunction for the propagation
velocity. In the linear case the string function is simply / f, where c
o
is the background baroclinic shallow
2
c
o
water wave speed, and typically describes propagation poleward on the eastern boundaries, westward (with some
topographic steering) over the middle ocean, and equatorward on the western boundaries. In the more general
nonlinear case, the string function is locally distorted by the anomaly. In the fully nonlinear examples of a lens
or dome, there is no rest or background string function; the string function is generated entirely by the disturbance
and propagation is due to asymmetric distribution of the anomalous mass over the string function contours. It
is shown that conventional beta/topographic propagation results (e.g., beta drift of eddies, the Nof speed of cold
domes) can be obtained as limiting cases of the string function. The string function provides, however, more
general propagation velocities that are also usually simpler to derive. The first baroclinic mode string function
for the global oceans is calculated from hydrographic data. The westward propagation speeds in the ocean basins
as derived from the meridional gradient of the string function are typically two to five times faster than those
expected from standard theory and agree well with the propagation speeds observed for long baroclinic Rossby
waves in the TOPEX/Poseidon data.
1. Introduction
An important consideration in large-scale ocean dy-
namics are the spatial variations in topography and the
Coriolis parameter f. These cause small convergences
in the flow that lead to or alter the propagation of dis-
turbances; examples of this are contained in beta and
topographic Rossby waves, the beta drift of large eddies,
topographic steering, and the propagation of cold domes
along topographic slopes.
In other work (Tyler and Ka¨se 2000a, hereafter TK1)
we introduced the ‘string function’ for the case of a
homogeneous one-layer model and in a companion pa-
per (Tyler and Ka¨se 2000b), validated these theoretical
results using results from a primitive equation model.
In this paper we extend that work by considering a two-
layer model.
In TK1 the string function for a one-layer model was
* Current affiliation: Applied Physics Laboratory, University of
Washington, Seattle, Washington.
Corresponding author address: Dr. Robert H. Tyler, Applied Phys-
ics Laboratory, Ocean Physics Department, University of Washing-
ton, 1013 NE 40th St., Seattle, WA 98105-6698.
E-mail: tyler@apl.washington.edu
easily described as follows: the depth-integrated geo-
strophic momentum can be decomposed into two
parts—a nondivergent part which can be derived from
a mass transport streamfunction and a remaining part
created by the effects of nonuniform rotation parameter,
nonuniform depth, and finite disturbance amplitude. It
was shown in TK1 that, regardless of which combina-
tion of the latter effects contribute, the remaining di-
vergent momentum component can be described as the
product of the mass of the disturbance (due to the dy-
namic perturbation of the layer thickness) and a prop-
agation velocity. This propagation velocity is nondiv-
ergent and can be written in terms of a streamfunction,
which we have called the string function. In summary,
two points are noteworthy: first, the geostrophic mo-
mentum is properly viewed as having a flow part and
a propagation part and, second, the velocity of the prop-
agation can be calculated from a scalar string function
that encapsulates beta, topographic, and finite amplitude
effects simultaneously.
In this paper we derive the string function for the
propagation of baroclinic anomalies in a two-layer mod-
el. Under the general conditions involving nonuniform
topography and rotation parameter and finite ampli-
tudes, the baroclinic and barotropic modes are linearly
coupled and therefore are not truly modal. It is, however,
useful for the discussion in this paper to describe results

766 V
OLUME
31JOURNAL OF PHYSICAL OCEANOGRAPHY
F
IG
. 1. Schematic of models.
for a separate baroclinic mode and then include con-
sideration of the barotopic coupling terms.
We briefly discuss other previous work, restricting
ourselves mostly to those works with which we will
later make comparisons. Further references describing
related work are given in Tyler and Ka¨se (2000a,b) and
the references cited therein.
With exceptions (e.g., Smith and O’Brien 1983;
Straub 1994), topographic and beta effects have mostly
been examined separately. Earlier work describing the
westward propagation of eddies due to the beta effect
(e.g., Flierl 1977; McWilliams and Flierl 1979; Nof
1981, 1983a; Killworth 1983; Cushman-Roisin 1986;
Shapiro 1986) have been reviewed and a general for-
mulation for the westward drift has been given (Cush-
man-Roisin et al. 1990). In one of the rarer works ded-
icated to super-Rossby scales (Matsuura and Yamagata
1982) it is also seen that nonlinearities due to the thick-
ness anomalies become important and can lead to ten-
dencies that counter those of dispersion. In this case,
governing equations can take a Korteweg–deVries form
for which soliton solutions are possible.
Propagation of eddies and domes along sloped to-
pography have usually been studied while ignoring the
beta effect (e.g., Nof 1983b; Swaters 1998; Swaters and
Flierl 1991). A simple formula given by Nof (1983b)
using a reduced-gravity model has been shown to have
some success in predicting the correct order of mag-
nitude of experimentally observed and numerically
modeled propagation rates.
In the next section, we present the formulation of the
string function for the two-layer model and derive an
evolution equation describing the propagation of the
potential energy anomaly, or more specifically in this
case, the associated mass (thickness) anomaly due to
the displaced sea surface and interface. In section 3, we
analyze the evolution equation for the case assuming
the flow is entirely geostrophic, while in section 4 we
include nongeostrophic effects. In section 5 we examine
the barotropic coupling term appearing in the baroclinic
evolution equation, and in the final three sections we
summarize, compare with previous results, and discuss
motivation for future development of the string function.
2. Formulation
Let us consider the two-layer model shown in Fig. 1.
Conservation of mass in each layer implies

M
ARCH
2001 767TYLER AND KA
¨
SE
] (
rh
2
rj
) 1 = · s 5 0 (1)
t 11 1
] (
rj
) 1 = · s 5 0, (2)
t 22
where
r
1
and
r
2
are the densities in layer one (upper)
and layer two (lower),
h
and
j
are the upward surface
and interface displacements, and s
1
and s
2
are the mo-
mentum per unit area for layer one and two (i.e., the
integral over the layer depth of the product of density
and the horizontal components of velocity).
We will consider dynamics described by the shallow-
water, hydrostatic momentum equations in layer inte-
grated flux form. In anticipation of typically geostrophic
balances we write these momentum equations in the
following form:
gh
1
s 52 =(
rh
) 3 z 1 R , (3)
111
f
gh
2
s 52 =(
rh
1 (
r
2
r
)
j
) 3 z 1 R , (4)
21212
f
where g is gravity, f is the Coriolis parameter, h
1
and
h
2
are the thickness of layer one and two, and z is the
vertical (upward) unit vector. (Familiar forms of the
flux-form momentum equations can be obtained by tak-
ing the cross product with z of the forms above and
using algebra after expanding R
1
, R
2
as described next.)
When R
1
5 R
2
5 0, (3) and (4) describe layer mo-
menta that are in geostrophic balance with the pressure-
gradient terms. More generally, R
1
and R
2
describe ad-
ditional nongeostrophic contributions to the layer mo-
menta. Allowing for nonlinear advection, wind stress
(
t
) momentum flux into layer 1, a Rayleigh friction term
in layer 2 (with Rayleigh coefficient b which may be
nonuniform and has units s
21
), we write
21 21
R 5 f (
t
2]s 2](m ss )) 3 z (5)
1 t 1 j 111j
21 21
R 5 f (2bs 2]s 2](m s s )) 3 z, (6)
22t 2 j 222j
where m
1
5
r
1
h
1
and m
2
5
r
2
h
2
are the columnar mass
in each layer and summation over repeated indices (j
referring to the two horizontal components perpendic-
ular to z) is implicit.
Let us now define several functions that will be used
in later sections. The first set, which pertains to baro-
tropic dynamics, is
S 5 s 1 s (7)
12
˜
M 5
rh
1 (
r
2
r
)
j
(8)
121
1/2
C 5 (gh) (9)
2
C
C5 (10)
m
f
C 5 =C3z, (11)
mm
which in sequence are the total momentum S, the total
mass anomaly M
˜
due to the disturbed layer thicknesses,
the barotropic shallow-water wave speed C, and the bar-
otropic string function C
m
that acts as a streamfunction
for the barotropic generalized beta drift velocity, C
m
(or
simply ‘string velocity’’).
We define a similar set for the baroclinic quantities
using lower case:
m
˜
5
rj
(12)
1
1/2
g9hh
12
c 5 (13)
12
h
r
2
r
21
g95 g (14)
r
1
2
c
c
5 (15)
m
f
c 5 =
c
3 z. (16)
mm
a. Barotropic mode
Let us add Eqs. (3) and (4) to gain a description of
the total momentum S. Using (7)–(12) we write this as
S 52=(M
˜
C
m
) 3 z 1 M
˜
C
m
1 R
1
1 R
2
1 =()3 z 2 m˜ ,m
˜
c
* c*
mm
(17)
where
g9h
1
c
* 5 (18)
m
f
c* 5 =
c
* 3 z. (19)
mm
To interpret Eq. (17) let us first consider the limit of
r
1
5
r
2
. In this case g950 and the starred quantities
vanish. The equation then is identical to the one-layer
homogeneous case described at length in TK1. Accord-
ingly, (17) states that the total momentum S is given by
a nondivergent component of geostrophic momentum
(M
˜
C
m
is a mass transport streamfunction for this non-
divergent flow), a component of momentum due to the
propagation of the total anomalous mass M
˜
along the
strings with velocity C
m
, and nongeostrophic compo-
nents R
1
and R
2
. More generally, when
r
1
±
r
2
the
starred stratification-dependent quantities remain.
An evolution equation for M
˜
is obtained by using the
divergence of (17) together with (8) and the mass con-
servation equations (1), (2) to yield
]
t
M
˜
1 = ·(M
˜
C
m
2 m˜ ) 1 = ·(R
1
1 R
2
) 5 0.c*
m
(20)
Again, in the limit
r
1
5
r
2
the starred quantity van-
ishes and (20) is the same as that studied previously in
TK1 for a homogeneous one-layer fluid. As brief ex-
amples that were described in TK1, when the flow is
geostrophic R
1
5 R
2
5 0 and (20) simply describes
generalized beta drift of M
˜
with the barotropic string
velocity C
m
, while, when the flow is assumed to be
steady and wind stress is retained in the nongeostrophic
term, (20) describes steady propagation of M
˜
away from
the Ekman sources and is a generalized Sverdrup bal-

768 V
OLUME
31JOURNAL OF PHYSICAL OCEANOGRAPHY
T
ABLE
1. Definition of mass anomaly m˜, squared shallow water
wave speed c, and string function
c
m
5 c
2
/f (also separated into
background parts) for models shown in Fig. 1.
c
and nonlinear
c
˜
mm
o
A general form of m˜ that encompasses all of the models (except the
one-layer) is m˜ 5
r
1
(2h
1
1 H
1
) and describes the interfacial mass
anomaly with respect to the rest state. Convenient model-specific
forms shown can be obtained through sign changes and interchange-
ment of
r
1
and
r
2
under the Boussinesq approximation.
Model m˜c
2
c
m
c
m
o
c
˜
m
1-layer
rh
gh
gh
f
gH
f
gm˜
f
r
2-layer
rj
2
g9hh
12
h
g9hh
12
fh
g9HH
12
fh
g9m˜ 2Hm˜
1
2112
12
f
r
h
r
h
o 2
1.5-layer
rj
2
g9h
1
g9h
1
f
g9H
1
f
gm˜
f
r
1
1.5-lay. (inv.)
rj
2
g9h
2
g9h
2
f
g9H
2
f
gm˜
2
f
r
2
Lens
r
h
11
g9hh
12
h
g9hh
12
fh
0
2
g9 m˜m˜
2
2
12
f
rr
h
11
Dome 2
r
h
11
g9hh
12
h
g9hh
12
fh
0
2
g9 m˜m˜
2
2
12
f
rr
h
11
ance allowing for nonuniform topography. In the case
of uniform topography C
m
reduces to the westward long
Rossby wave speed
b
C
2
/ f
2
, while in the case of dom-
inant topographic control, C
m
produces the long topo-
graphic Rossby velocity. More generally, both topo-
graphic steering and beta drift enter into C
m
.
When
r
1
±
r
2
the starred term remains and the equa-
tion above, which we will still refer to as describing the
barotropic mode, shows coupling with the baroclinic
effects.
b. Baroclinic mode
The layer momentum equations (3) and (4) can be
combined into forms that provide more insight into the
baroclinic dynamics:
h
1
s 5 =(m
˜
c
) 3 z 2 m
˜
c 1 R 1 S
1 mm1
h
h
1
2 (R 1 R ), (21)
12
h
h
2
s 52=(m
˜
c
) 3 z 1 m
˜
c 1 R 1 S
2 mm2
h
h
2
2 (R 1 R ), (22)
12
h
where several of the definitions given in (5)–(16) have
been used and it is noted that h 5 h
1
1 h
2
.
Above, (21) describes the momentum in layer one as
being the sum of several terms. Taken in order these
are 1) nondivergent geostrophic mass transport (given
by the streamfunction m˜
c
m
) due to interface displace-
ment; 2) the momentum due to the generalized beta drift
propagating the interface mass anomaly m˜ with velocity
c
m
; 3) the nongeostrophic components R
1
of the mo-
mentum in layer 1; 4) the last two terms which, taken
together, describe the fraction of the geostrophic baro-
tropic momentum that occurs in layer 1. A similar de-
scription can be given for (22).
Similarly, we can produce an evolution equation for
m˜ by combining (2), (21), and (22) while using (7),
(12), and (16) to yield:
] m
˜
1 = ·(m
˜
c )
tm
hhh
212
52= · S 2 = · R 2 R . (23)
21
12 1 2
hhh
Equation (23) is an evolution equation for the dy-
namic disturbance of the interface separating two layers.
The form of the equation anticipates a situation that is
predominantly baroclinic and predominantly geostroph-
ic. In this case the barotropic and nongeostrophic terms
appearing on the right are small and the dominant bal-
ance is given by the two terms on the left. But the terms
on the left simply describe propagation given by the
baroclinic string function. In the following sections we
analyze (23) under varying assumptions.
3. Case of geostrophic flow
In this section we examine (23) under the assumption
that the flow is in geostrophic balance (i.e., R
1
5 R
2
5 0). In later sections we will add in the nongeostrophic
flow effects.
Under this assumption (23) can be written as
h
2
] m
˜
1 = ·(m
˜
c ) 52= · S , (24)
tm
12
h
which, because c
m
is nondivergent, can be interpreted
as stating that the material derivative of m˜ following the
string velocity is zero unless the barotropic term on the
right is nonvanishing. The term on the right of (24)
simply describes the convergence of that fraction of the
total barotropic momentum S that occurs in layer two.
This coupling term is of interest and we will return to
it in section 5. First, however, we wish to discuss the
terms on the left, in particular we compare them with
the one-layer case, and we will assume for expediency
that S 5 0 to start.
In this case (and because c
m
is nondivergent) (24) is
simply ]
t
m˜ 1 c
m
· =m˜ 5 0 describing propagation of
m˜ along the strings just as in the one-layer case studied
in TK1 except that the definition of the quantities ap-
pearing is different in each case. These definitions for
the one-layer and two-layer models, as well as other
limiting-case models, are summarized in Table 1. Note
the similarities in these definitions; m˜ always refers to
a mass anomaly due to the alterations in layer thick-

M
ARCH
2001 769TYLER AND KA
¨
SE
nesses, while the string function
c
m
5 c
2
/ f with the
shallow water wave speed c chosen appropriately for
the model. Also, for a flat-bottomed ocean with van-
ishing m˜ the string function contours (strings) are
aligned with lines of latitude and c
m
5 =
c
m
3 z, cal-
culated using the appropriate string function, correctly
reduces to the westward long Rossby speed (
b
c
2
/ f
2
for
a beta plane).
Despite these similarities there is an important dif-
ference in that the two-layer string function has a dif-
ferent dependence on topography and m˜ than in the one-
layer case. Typically, when h
1
K h
2
, the two-layer bar-
oclinic string function will be less sensitive to topog-
raphy and more sensitive to m˜ than for the one-layer
case.
In Table 1 we break the string function into two com-
ponents: a component due to the background con-
c
m
o
figuration (i.e., corresponding to the part of
c
m
inde-
pendent of m˜ ) and the part
m
that depends on m˜. Let
c
˜
us first discuss the background string function and
c
m
o
associated string velocity 53z. As described,c =
c
mm
oo
with uniform topography (and with H
1
uniform) the
background baroclinic string function 5 g9H
1
H
2
/
c
m
o
( fH), where H 5 H
1
1 H
2
, varies only with f and
produces the westward long Rossby wave velocity. In
regions where topographic changes dominate such
c
m
o
as near the coasts, 52g9(H
1
/H
2
)
2
f
21
z 3 =H. Ac
m
o
zero-value string is located where H
2
vanishes and is
the boundary for small amplitude m˜ propagation. The
general tendency is then for propagation poleward on
the eastern boundaries, across the ocean interior and
equatorward along the western boundary. This is very
much like that for the one-layer model except that the
propagation is less sensitive to topography in the open
ocean and is typically much slower.
In the 1.5-layer model where we assume an infinite
lower layer, c 5 (g9H
1
)
1/2
and the string function is
therefore 5 g9H
1
/ f, giving again just the westward
c
m
o
long Rossby wave velocity. As can be expected, there
is no dependence on topography. Similarly, in the in-
verted 1.5-layer model (infinite layer on top) c 5
(g9H
2
)
1/2
and 5 g9H
2
/ f, which will typically be very
c
m
o
sensitive to topography. In the case where topographic
variations dominate, the inverted 1.5-layer string ve-
locity is proportional to the bottom slope.
Now consider the effects of finite amplitude m˜. As
seen in Table 1,
m
in the one-layer case is linear in m˜,
c
˜
and therefore the string anomalies have the same sign
as m˜.For example, an anticyclonic eddy (m˜ . 0) would
add a perturbation to the background strings appearing
as a bow extending poleward and/or toward shallower
water that propagates along the strings; if m˜ is suffi-
ciently large, the bow can even break off, forming a
locally closed string. In contrast, the two-layer
m
has
c
˜
both a term linear in m˜ (similar to the one-layer case
but multiplied by a factor 2H
1
/H 2 1) and a quadratic
term in m˜, and
m
need not have the same sign as m˜.
c
˜
Indeed, the sign of two-layer
m
can change during the
c
˜
propagation as, for example, when a propagating m˜ fol-
lows the strings into shallower water.
Consideration of the amplitude of m˜ and
m
and the
c
˜
changes in amplitude that can occur during propagation
is important for determining the degree of nonlinearity
in the generalized beta drift; while small amplitude dis-
turbances may follow the background strings in a man-
ner that can be predicted from the outset by examining
the background string function, even once small am-
plitude disturbances can become nonlinear without any
external forces, eventually breaking off to form new
closed strings that interact with the background strings
and perhaps other closed strings. In this case involving
breaking and re-attaching strings, the string function still
gives the proper propagation velocities, but only in an
instantaneous sense because the string configuration is
constantly evolving. As described at the end of the last
paragraph, sign reversals of
m
are even possible when
c
˜
strings cross bathymetric contours and enter a region
where 1 2 2H
1
/H becomes less than zero (i.e., when
H
2
, H
1
). But note to see that this can occur, variations
in f must be included; otherwise the background strings,
at least, would not cross bathymetric contours.
It is also important to note that, in determining the
degree of nonlinearity in the generalized beta drift that
simple ratios of interface displacement to layer thickness
are not satisfactory. Indeed, even the ratio
c
m
/ , while
c
m
o
usually better, is still inadequate. A more reliable in-
dicator is the ratio |=
c
m
|/ because it estimates the
|=
c
|
m
o
ratio of the disturbance in the string velocity and the
background string velocity. For example, while the ratio
c
m
/ may increase entering shallow water, suggesting
c
m
o
an increase in nonlinearity, increasing topographic slope
increases the background string velocity and may pre-
dominate to make the disturbance less nonlinear.
We see from Table 1 that the two-layer
c
m
also has
a term quadratic in m˜,which acts to decrease the local
value of the string function regardless of the sign of m˜.
This will make the asymmetries between the propaga-
tion of positive and negative m˜ more complicated. But
it is important to note that the terms in m˜ (or m˜
2
)do
not complicate the string velocity as much as may ap-
pear because the gradients in m˜ do not play a role. This
is because any part of c
m
5 =
c
m
3 z, which depends
on =m˜ 3 z, is perpendicular to =m˜ and therefore cannot
advect m˜.This is related to the non-Doppler effect (Rhi-
nes 1989).
As can be seen in Table 1, the lens model (1.5-layer
model with H
1
0) and dome model (inverted 1.5-
layer model with H
2
0) have no background string
function and any propagation is entirely nonlinear.
These models will be discussed in a later section where
we make comparisons with previous results. We also
note that, in all the models discussed,
c
m
is defined such
that it is always positive and the
c
m
5 0 contour has
the physical significance of describing the location of
outcropping.

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  • ...Circular eddies of different size (Nof (1984)) as well as the leading edge of a continuous dense bottom current (Tyler & Kæse 2001; Wåhlin 2002) move with the same speed, which is proportional to the slope of the bottom and the reduced gravity, and inversely proportional to the Coriolis parameter....

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TL;DR: In this paper, the authors used data from a cable-based observational system for long-term monitoring of barotropic flow in the Baltic Sea and found that the relative importance of the barotropic forcing tends to weaken during summer.
Abstract: The possibility of using data from a cable-based observational system for long-term monitoring of barotropic flow in the Baltic Sea was investigated. Measurements were made of the induced potential differences between Visby on the island of Gotland and Vastervik on the Swedish mainland and a yearlong period was studied in order to ensure the presence of seasonal fluctuations. The predictions from a 2D electric-potential model, forced by velocity fields from a shallow-water circulation model, proved to be well correlated with the observations. A winter and a summer period were selected for a thorough analysis, the results of which indicated a stronger correlation during winter. This implies that the relative importance of the barotropic forcing tends to weaken during summer. The spatial coverage of the induced potential differences for the cable region was found to encompass a considerable part of the Baltic proper. The correlation study indicated that the winter circulation in the Baltic proper s...

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References
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TL;DR: In this paper, the first baroclinic gravity-wave phase speed c1 and the Rossby radius of deformation l1 are computed from climatological average temperature and salinity profiles.
Abstract: Global 1 83 18 climatologies of the first baroclinic gravity-wave phase speed c1 and the Rossby radius of deformation l1 are computed from climatological average temperature and salinity profiles. These new atlases are compared with previously published 5 83 58 coarse resolution maps of l1 for the Northern Hemisphere and the South Atlantic and with a 1 83 18 fine-resolution map of c1 for the tropical Pacific. It is concluded that the methods used in these earlier estimates yield values that are biased systematically low by 5%‐15% owing to seemingly minor computational errors. Geographical variations in the new high-resolution maps of c1 and l1 are discussed in terms of a WKB approximation that elucidates the effects of earth rotation, stratification, and water depth on these quantities. It is shown that the effects of temporal variations of the stratification can be neglected in the estimation of c1 and l1 at any particular location in the World Ocean. This is rationalized from consideration of the WKB approximation.

1,290 citations

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12 Apr 1996-Science
TL;DR: The TOPEX/POSEIDON satellite altimeter has detected Rossby waves throughout much of the world ocean from sea level signals with ≲10-centimeter amplitude and ≳500-kilometer wavelength.
Abstract: Rossby waves play a critical role in the transient adjustment of ocean circulation to changes in large-scale atmospheric forcing. The TOPEX/POSEIDON satellite altimeter has detected Rossby waves throughout much of the world ocean from sea level signals with ≲10-centimeter amplitude and ≳500-kilometer wavelength. Outside of the tropics, Rossby waves are abruptly amplified by major topographic features. Analysis of 3 years of data reveals discrepancies between observed and theoretical Rossby wave phase speeds that indicate that the standard theory for free, linear Rossby waves is an incomplete description of the observed waves.

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Journal ArticleDOI
TL;DR: In this paper, the evolution of an axially symmetric vortex is calculated with a quasi-geostrophic, adiabatic, hydrostatic β-plane, two vertical mode model.
Abstract: The evolution of an isolated, axially symmetric vortex is calculated with a quasi-geostrophic, adiabatic, hydrostatic. β-plane, two vertical mode model. The circumstances of greatest interest are those of weak friction and large vortex amplitude (strong nonlinearity). Systematic studies are made of the consequences of varying the frictional coefficient, the vortex amplitude, the vortex radius (relative to the deformation radius), the degree of nonlinear coupling between the two vertical modes and the initial vertical structure of the vortex. Results of note include the following. Within the approximation of a single vertical mode model (i.e., in the absence of modal coupling), a baroclinic vortex has an increased westward and a finite meridional propagation speed when its amplitude is greater than infinitesimal. Both of these speeds, however, are limited by the wave speeds (as determined from infinitesimal amplitude theory) of the weak dispersion field outside the vortex. The vortex amplitude dec...

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"A String Function for Describing th..." refers background in this paper

  • ...Earlier work describing the westward propagation of eddies due to the beta effect (e.g., Flierl 1977; McWilliams and Flierl 1979; Nof 1981, 1983a; Killworth 1983; Cushman-Roisin 1986; Shapiro 1986) have been reviewed and a general formulation for the westward drift has been given (Cushman-Roisin et al....

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  • ...We close this section by drawing attention to a case previously discussed by Shi and Nof (1994) in a similar theoretical context....

    [...]

  • ...The Rhines (1989) paper also contains other discussions (e....

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  • ...Earlier work describing the westward propagation of eddies due to the beta effect (e.g., Flierl 1977; McWilliams and Flierl 1979; Nof 1981, 1983a; Killworth 1983; Cushman-Roisin 1986; Shapiro 1986) have been reviewed and a general formulation for the westward drift has been given (Cushman-Roisin et…...

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Journal ArticleDOI
TL;DR: In this paper, it is argued that the major changes to the unperturbed wave speed will be caused by the presence of baroclinic east-west mean flows, which modify the potential vorticity gradient.
Abstract: Planetary or Rossby waves are the predominant way in which the ocean adjusts on long (year to decade) timescales. The motion of long planetary waves is westward, at speeds $ 1c m s 2 1. Until recently, very few experimental investigations of such waves were possible because of scarce data. The advent of satellite altimetry has changed the situation considerably. Curiously, the speeds of planetary waves observed by TOPEX/Poseidon are mainly faster than those given by standard linear theory. This paper examines why this should be. It is argued that the major changes to the unperturbed wave speed will be caused by the presence of baroclinic east‐ west mean flows, which modify the potential vorticity gradient. Long linear perturbations to such flow satisfy a simple eigenvalue problem (related directly to standard quasigeostrophic theory). Solutions are mostly real, though a few are complex. In simple situations approximate solutions can be obtained analytically. Using archive data, the global problem is treated. Phase speeds similar to those observed are found in most areas, although in the Southern Hemisphere an underestimate of speed by the theory remains. Thus, the presence of baroclinic mean flow is sufficient to account for the majority of the observed speeds. It is shown that phase speed changes are produced mainly by (vertical) mode-2 east‐west velocities, with mode-1 having little or no effect. Inclusion of the mean barotropic flow from a global eddy-admitting model makes only a small modification to the fit with observations; whether the fit is improved is equivocal.

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Journal ArticleDOI
TL;DR: In this article, a generalized formula for the determination of the westward drift of mesoscale eddies under the planetary (beta) effect has been derived within the confines of a single-layer model.
Abstract: Since the pioneering work of Nof, the determination of the westward drift of mesoscale eddies under the planetary (beta) effect has been a recurrent theme in mesoscale oceanography, and several different formulae have been proposed in the literature. Here, recpatiulation is sought, and, within the confines of a single-layer model, a generalized formula is derived. Although it is similar to Nof's, the present formula is established from a modified definition and with fewer assumptions. It also recaptiulates all other formulae for the one-layer model and applies to a wide variety of situations, including cases when the vortex develops a wake of Rossby waves or undergoes axismmetrization. Following the derivation of the formula, a physical interpretation clarifies the migration mechanism, which can be divided between a self-induced propulsion and a reaction from the displaced ambient fluid. Numerical simulations with primitive and geostrophic equations validate the formula for a variety of length sc...

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Frequently Asked Questions (2)
Q1. What are the contributions mentioned in the paper "A string function for describing the propagation of baroclinic anomalies in the ocean" ?

The authors derive a string function that describes the propagation of large-scale, potentially large amplitude, baroclinic energy anomalies in a two-layer ocean with variable topography and rotation parameter. The string function provides, however, more general propagation velocities that are also usually simpler to derive. 

Important recent papers ( Dewar 1998, Killworth et al. 1997 ) address this topic ; yet further work to settle this issue is required. Of course the results the authors have presented apply to a two-layer ocean, not one that is realistically stratified ; any comparisons are intended only to motivate further research rather than to explain the observations. Still, it is premature to offer this as an explanation for the observed phase speeds before further fundamental studies of the string function are given. 4. As can be seen, the observed phase speeds are typically a couple to several times greater than that theoretically expected, and the discrepancy shows an increase poleward.