DECEMBER 2013
MARSHALL AND JOHNSON
2699
Propagation of Meridional Circulation Anomalies along Western
and Eastern Boundaries
DAVID P. MARSHALL
Department of Physics, University of Oxford, Oxford, United Kingdom
HELEN L. JOHNSON
Department of Earth Sciences, University of Oxford, Oxford, United Kingdom
(Manuscript received 17 June 2013, in final form 31 August 2013)
ABSTRACT
Motivated by the adjustment of the meridional overturning circulation to localized forcing, solutions are
presented from a reduced-gravity model for the propagation of waves along western and eastern boundaries.
For wave periods exceeding a few months, Kelvin waves play no role. Instead, propagation occurs through
short and long Rossby waves at the western and eastern boundaries, respectively: these Rossby waves
propagate zonally, as predicted by classical theory, and cyclonically along the basin boundaries to satisfy the
no-normal flow boundary condition. The along-boundary propagation speed is cLd/d, where c is the internal
gravity/Kelvin wave speed, Ld is the Rossby deformation radius, and d is the appropriate frictional boundary
layer width. This result holds across a wide range of parameter regimes, with either linear friction or lateral
viscosity and a no-slip boundary condition. For parameters typical of contemporary ocean climate models, the
propagation speed is coincidentally close to the Kelvin wave speed. In the limit of weak dissipation, the
western boundary wave dissipates virtually all of its energy as it propagates toward the equator, independent
of the dissipation coefficient. In contrast, virtually no energy is dissipated in the eastern boundary wave. The
importance of background mean flows is also discussed.
1. Introduction
Wave propagation along western and eastern boundaries is a fundamental process through which the ocean
adjusts to changes in its boundary conditions (e.g.,
Wajsowicz 1986; Kawase 1987; Fig. 1). High-frequency
variability can propagate large distances along meridional boundaries through coastal Kelvin waves, topographic waves, or some combination thereof (Huthnance
1978). However, variability in the meridional overturning
circulation (MOC), heat content, and regional sea level
change also occurs on seasonal-to-decadal time scales, on
which other forms of boundary wave may dominate the
adjustment process.
The adjustment of the ocean to a change in high-latitude
buoyancy forcing varies widely across ocean general circulation models (OGCMs) and coupled climate models.
Corresponding author address: Dr. David P. Marshall, Clarendon
Laboratory, Department of Physics, Parks Road, University of
Oxford, Oxford OX1 3PU, United Kingdom.
E-mail: marshall@atm.ox.ac.uk
DOI: 10.1175/JPO-D-13-0134.1
Ó 2013 American Meteorological Society
The time scale on which anomalies propagate from high
to low latitudes, the signal amplitude, and the inferred
mechanisms all differ significantly. In the subpolar gyre
north of about 408N, several studies have suggested that
advection—rather than boundary wave propagation—
dominates the response of the MOC (e.g., Gerdes and
K€
oberle 1995; Eden and Greatbatch 2003; Getzlaff et al.
2005; Zhang et al. 2011). This is perhaps not surprising
given the strong mean boundary currents in the surface
ocean and at depth, which are both directed equatorward here, and the recently reported interior pathways
for North Atlantic Deep Water (Bower et al. 2009).
Farther south, however, many studies see equatorward
propagation of boundary anomalies at speeds significantly faster than the advective velocities associated
with the deep western boundary current [e.g., D€
oscher
et al. 1994; Capatondi 2000; Eden and Willebrand 2001;
Goodman 2001; Dong and Sutton 2002; Getzlaff et al.
2005; Roussenov et al. 2008; Chiang et al. 2008; Hawkins
and Sutton 2008; K€
ohl and Stammer 2008; Zhang 2010;
Zhang et al. 2011; Hodson and Sutton 2012; for exceptions, see Marotzke and Klinger (2000) and Buckley
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JOURNAL OF PHYSICAL OCEANOGRAPHY
et al. (2012)] and infer that these are Kelvin wave signals, traveling slower than the first baroclinic gravity
wave speed (which is on the order of 1 m s21) owing to
numerical effects. Anomalies in these models take between a month (e.g., Dong and Sutton 2002) and a decade (e.g., Capatondi 2000) to reach the tropics.
The results of Hsieh et al. (1983) are often invoked to
explain these boundary wave propagation differences in
terms of the effect of model grid and resolution. Other
authors cite the different forcing mechanisms (e.g.,
Marotzke and Klinger 2000; Johnson and Marshall 2002a)
as explanation. In this paper we analyze the propagation
characteristics of waves along meridional boundaries in
a reduced-gravity model, at both high and low frequency
and in the presence of both linear and lateral friction.
Motivated by the disparate results of OGCMs in response
to variable buoyancy forcing at high latitudes, and building on the work previously applied to coastal and tropical applications (e.g., Clarke 1983; Grimshaw and Allen
1988; Clarke and Shi 1991), we aim to present a unified
view of first-mode baroclinic wave propagation relevant
to large-scale ocean adjustment on time scales longer
than a few months. We show that the propagation speed
is not the Kelvin wave speed, but in fact varies inversely
with the pertinent viscous boundary layer width; because
the latter differs among OGCMs, this may offer a partial
explanation for the spread in the propagation speed of
Atlantic MOC anomalies observed in the studies above
and others like them.
There are four outstanding questions concerning wave
propagation along meridional boundaries that we plan to
address in this paper (Fig. 1):
d
d
d
d
What is the relevant property that propagates along
the boundary?
At what speed does it propagate?
How much energy is dissipated?
What is the amplitude of western boundary waves
generated by incident long Rossby waves from the
ocean interior, and how much of the incident wave
energy is dissipated?
In the remainder of this section, we take each of these
questions in turn and provide some background context
and motivation. We also discuss the implications and
relevance for the detection, monitoring and prediction
of MOC change.
a. What propagates?
First-mode baroclinic waves propagate an anomaly in
pycnocline depth along the boundary, with an associated
signal in pressure, density, velocity, and often temperature. However, the amplitude of the anomaly need not be
constant along the propagation path. Wajsowicz and Gill
VOLUME 43
FIG. 1. Schematic illustrating the role of boundary waves in the
adjustment of the upper limb of the MOC to localized forcing
through changes in deep water formation at high latitudes (downward arrow), or through westward-propagating Rossby waves and
eddies impinging on the western boundary (rings and solid arrow).
The boundary waves propagate cyclonically around each hemispheric basin (dashed arrows).
(1986) show that Kelvin waves on an f plane are attenuated by friction, and that their amplitude decays as
a result. Johnson and Marshall (2002a) illustrate in a
reduced-gravity model that, even in the absence of friction, when f varies with latitude the amplitude of a western
boundary wave is reduced during equatorward propagation. On the eastern boundary, they see no corresponding
increase in wave amplitude with latitude; instead, long
Rossby waves are radiated into the ocean interior. This
asymmetry between eastern and western boundaries is
a key element of their ‘‘equatorial buffer’’ mechanism.
There is a substantial body of literature on the nature
of eastern boundary wave propagation and the radiation
of long Rossby waves. This has originated largely from
the coastal oceanography community, motivated by
the analysis of coastal sea level observations, and the
propagation of ENSO-related signals out of equatorial
regions. Clarke (1983) showed that motion at the eastern boundary of an ocean basin can be described in
terms of either westward-propagating long Rossby
waves, or coastal Kelvin waves, with a critical latitudedependent frequency below which energy is radiated
into the interior, rather than coastally trapped. He
confirmed the result of Moore (1968) that the amplitude
of a Kelvin wave propagating on a b plane is proportional
pffiffiffi
to y at high frequencies, where y is the meridional
distance north of the equator.
DECEMBER 2013
MARSHALL AND JOHNSON
Wajsowicz (1986), Grimshaw and Allen (1988), and
Clarke and Shi (1991) built on these results. Taking into
account the coastline orientation (Schopf et al. 1981),
Clarke and Shi (1991) calculate critical frequencies
for the World Ocean’s boundaries, which range from
a month or two in the tropics to a year at midlatitudes.
Grimshaw and Allen (1988) note that a consequence of
these two regimes is that, while at high frequencies the
pffiffiffi
amplitude of coastal disturbances varies as y on both
eastern and western boundaries, at lower frequencies
this dependence changes. McCalpin (1995) finds that on
eastern boundaries at low frequency the amplitude of
the wave disturbance is in fact constant along the eastern
boundary, consistent with the earlier findings of Clarke
(1983) and Grimshaw and Allen (1988).
A change in the amplitude of wave signals as they
propagate along meridional boundaries has important
consequences for the diagnosis of waves in ocean models.
Because of the coarse spatial and temporal resolution of
typical OGCM output, together with the presence of
a background mean flow and mesoscale eddy field, the
propagation of anomalies along the boundary can be
hard to detect. Many OGCM studies calculate propagation speed based on the arrival time of a given amplitude of anomaly at a particular latitude (e.g., Marotzke
and Klinger 2000). This may well be a dangerous strategy
because one cannot infer wave speed through tracking
a contour of density if the amplitude is changing as the
wave propagates. An understanding of how we expect
wave amplitudes to change along eastern and western
boundaries, as a function of frequency, may help us to
more clearly identify these signals in models. It will also
have implications for the detection and monitoring of
change in the ocean.
Note that basin-mode theory (e.g., Liu et al. 1999;
Cessi and Louazel 2001; Cessi and Paparella 2001; Cessi
and Primeau 2001; Primeau 2002) assumes the amplitude
of anomalies to be uniform along the eastern boundary at
all times. This will be discussed in more detail in the
following section.
b. At what speed?
Baroclinic coastal Kelvin waves
pffiffiffiffiffiffiffipropagate at the internal gravity wave speed c 5 gr h, where gr is the reduced gravity, and h is the equivalent depth of the
baroclinic mode. Any signal propagation occurring on
the western boundary of ocean models on time scales
from months to a decade is generally assumed to be the
result of Kelvin waves, slowed by low stratification
(Greatbatch and Peterson 1996) or numerical effects,
especially on a B grid (Hsieh et al. 1983). At high
frequencies this may well be the correct interpretation,
but the physical argument outlined in the previous
2701
section suggests that at frequencies significantly lower
than v 5 bLd 5 bc/f (i.e., on time scales longer than
about 2 months) coastally trapped Kelvin wave propagation is no longer expected on either eastern or western
boundaries. This does not mean that propagation along
the boundary is prohibited; simply that it need not occur
at the gravity wave speed and need not be coastally
trapped.
Primeau (2002) suggested that Kelvin waves play no
role in the adjustment of the ocean to a change in forcing
on all but the shortest of time scales. He concluded
that, instead, internal gravity waves act to distribute
pressure anomalies instantaneously along basin boundaries. This is consistent with the results of Clarke (1983)
and Grimshaw and Allen (1988), both of whom point
out that, in the absence of friction, there is no phase
propagation along eastern boundaries, with all points on
the boundary responding to an incident disturbance simultaneously. Cessi and Otheguy (2003) come to similar
conclusions; in fact, this instantaneous redistribution
of anomalies uniformly along the eastern boundary is
central to the theory of basin modes. Clarke (1983) also
considered the inclusion of linear damping in each of the
momentum and continuity equations, obtaining a finite
propagation speed proportional to the damping coefficient with linear friction in the continuity equation,
but still synchronous variation along the boundary with
linear friction in the momentum equation.
Note that, in closed-basin quasigeostrophic (QG)
models, a ‘‘consistency condition’’ (McWilliams 1977) is
used to ensure global mass conservation, determining
the boundary value of the streamfunction, which is constant along the boundary but varies in time. This instantaneous adjustment is the QG model equivalent of
the coastal Kelvin wave and allows for the radiation of
long Rossby waves into the interior (see, e.g., Milliff and
McWilliams 1994; McCalpin 1995).
Other boundary propagation mechanisms besides
numerical Kelvin waves and advection have of course
been proposed. Killworth (1985) and Winton (1996)
both discuss propagation by viscous boundary waves,
associated with the effect of friction at the sidewall. In
Killworth (1985) the divergence associated with the
wave at the coast occurs over one grid box due to the nonormal flow condition, and the wave propagates at a
speed c2/fDx, where Dx is the grid resolution in the zonal
direction. In the presence of a sloping boundary and/or
a continental shelf, topographically trapped wave modes
as well as hybrid waves become possible (Huthnance
1978; Elipot et al. 2013) and can travel significantly faster
than first-mode baroclinic Kelvin waves (O’Rourke 2009).
We restrict our attention to the vertical sidewall case in
this paper, because even this exhibits rich behavior.
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We will show analytically in section 2, using the same
equation set that results in pure Kelvin wave propagation at high frequencies, that at low frequencies the
along-boundary propagation speed depends strongly on
the frictional parameters of the model, and hence varies
among OGCMs. These low-frequency waves may in
some models travel along the boundary at approximately Kelvin wave speeds (e.g., Yang 1999; Johnson
and Marshall 2002a) but this is simply fortuitous. This
model dependence is a worrying state of affairs because
we rely on these OGCMs, coupled to atmospheric circulation models, to predict the rate at which the ocean
will respond to future forcing changes associated with
increased greenhouse gas concentrations. Correctly
representing the ocean adjustment mechanisms to anticipated changes at high latitudes is essential if we are to
make accurate projections of global and particularly
regional climate.
A proper representation of boundary wave propagation speeds is also important for our understanding of
internal climate variability. Several recent studies suggest that the ocean exhibits damped, decadal oscillations
that are stochastically excited by atmospheric forcing
(e.g., Greatbatch and Peterson 1996; Cessi and Louazel
2001; Cessi and Paparella 2001; Eden and Greatbatch
2003; Dong and Sutton 2005; Danabasoglu 2008;
Frankcombe et al. 2009; Czeschel et al. 2010; Danabasoglu
et al. 2012; Kwon and Frankignoul 2012; Delworth and
Zeng 2013). These modes of variability can have significant climate fingerprints (e.g., Knight et al. 2005;
Hurrell et al. 2006; Zhang and Delworth 2006; Mahajan
et al. 2011). The mechanisms and time scales underlying
the oscillations differ from one model to another, with
boundary adjustment processes often implicated (e.g.,
Greatbatch and Peterson 1996; Hawkins and Sutton
2007). As a result of the diversity of model behavior,
there is little consensus on the causes of decadal-tomultidecadal variability in the real climate system.
Understanding the mechanisms and time scales involved
would facilitate attempts to develop the capacity for
decadal climate prediction in the Atlantic sector (e.g.,
Latif et al. 2006; Msadek et al. 2010; Robson et al.
2012).
Note that the speed of boundary wave adjustment
has implications beyond the ocean basin in which a
change in forcing occurs. In response to high-latitude
forcing in the North Atlantic, for example, fast boundary
waves, together with equatorial Kelvin waves, carry
the signal of a thermocline displacement to all the
major ocean basins on relatively short time scales (e.g.,
Huang et al. 2000; Goodman 2001; Cessi and Otheguy
2003; Johnson and Marshall 2004). While the amplitude
of variability on anything shorter than multidecadal
VOLUME 43
time scales may be small outside the hemispheric
basin of forcing (Johnson and Marshall 2002b), the
implications for sea level are significant (Hsieh and
Bryan 1996; Landerer et al. 2007). Fast atmospheric
teleconnections via tropical air–sea interactions may
also result in a global response (e.g., Dong and Sutton
2002).
c. Energetics and western boundary interactions
The mechanical energy budget of the global ocean
circulation has received a lot of attention in recent years
(e.g., Munk and Wunsch 1998; Wunsch and Ferrari 2004;
Tailleux 2009; Hughes et al. 2009), with mechanisms of
energy loss from the geostrophic modes including bottom drag (Sen et al. 2008), loss of balance (Molemaker
et al. 2005), exchange of energy with pre-existing internal waves (Polzin 2008), and generation of internal waves
by geostrophic flow over rough topography (Marshall and
Naveira Garabato 2008; Nikurashin and Ferrari 2010;
Nikashurin et al. 1998). Recently, Zhai et al. (2010)
demonstrated in idealized model calculations and altimetric data that there is a significant sink of geostrophic
eddy energy when westward-propagating eddies (e.g.,
Chelton et al. 2007) reach the western boundary—the
so-called Rossby graveyard. In the model calculations,
the amount of energy dissipation appears to be independent of model resolution and of the magnitude
and nature of momentum dissipation. In the ocean, it
is unclear how much of the energy is merely backscattered to the mean flow (cf. Starr 1968) or dissipated
through vertically breaking boundary waves (Dewar
and Hogg 2010).
Irrespective of how energy is dissipated in the ocean,
in coarse-resolution models, it is likely that a significant
amount of energy is dissipated in western boundary
waves. In section 4, we show that this rate of energy
dissipation is independent of the magnitude of the local
friction and, in section 5, we show that the vast majority
of the energy incident on the western boundary in the
form of long Rossby waves is dissipated locally on
reaching the western boundary.
The energy dissipation implicit in long waves reaching
a western boundary is intimately related to the strong
reduction in dynamic height variability at the western
margin of an ocean basin. This reduction is crucial in
preventing MOC variations inferred from boundary
hydrographic and other data (Cunningham et al. 2007;
Kanzow et al. 2007) from being swamped by eddy variability (Wunsch 2008). This issue has been previously
discussed using a stripped-down version of the linear
wave solutions presented in this manuscript (Kanzow
et al. 2009) but nevertheless serves as an important
motivation for the present study.
DECEMBER 2013
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MARSHALL AND JOHNSON
d. Structure of the paper
The paper is structured as follows. In section 2, we
solve for the linear boundary waves in the presence of
a meridional boundary and linear friction. In section 3,
we discuss the behavior of meridional transport anomalies along western and eastern boundaries. In section 4,
we analyze the energetics of western and eastern boundary waves. In section 5, we study the western boundary
waves generated through incident long waves and the
location and magnitude of energy dissipation associated
with these interactions. In section 6, we solve for linear
boundary waves in the presence of lateral friction and
a no-slip boundary condition. In section 7, we investigate
the impact of background mean flow on the boundary
wave solutions. Finally, in section 8 we present a brief
concluding discussion.
We consider solutions to the reduced-gravity equations, linearized about a state of rest. We adopt a semigeostrophic approximation (Hoskins 1975) in which we
assume that geostrophic balance holds in the zonal
momentum equation, but not in the meridional momentum equation where we know that the Coriolis acceleration vanishes at the boundary.
Thus,
›h0
5 0,
›x
›y 0
›h0
1 byu0 1 gr
5 2ry 0 , and
›t
›y
0
›h0
›u ›y0
1 h0
1
5 0.
›t
›x ›y
!
› ›2 h0 h0
›h0
›2 h0
1
r
1
b
2
5 0,
›t ›x2 L2d
›x
›x2
(2.5)
Ld ( y) 5
pffiffiffiffiffiffiffiffiffi
gr h0
c
5 .
by
by
(2.6)
The no-normal flow boundary condition (2.4), when
substituted in (2.2) and using (2.1), takes the form:
›2 h0
›h0
›h0
1 by
1r
5 0 (x 5 0).
›t›x
›y
›x
(2.7)
(2.1)
Following Clarke and Shi (1991), we now seek solutions of the form
(2.2)
h0 5 A( y)eik(y)x e2ivt ,
(2.3)
The velocities in the zonal x and meridional y directions
are u and y, respectively, h is the layer thickness, and r is
the coefficient of linear friction. We adopt a b-plane
approximation such that the Coriolis parameter f 5 by.
Primes indicate linear perturbations and zero subscripts the
background mean state.
For simplicity, we restrict our attention to north–
south coastlines located at x 5 0 (Fig. 2), along which
a no-normal flow boundary condition is applied
u0 5 0 (x 5 0).
From (2.1) to (2.3), we can derive a vorticity equation:
where
2. Wave solutions with linear friction
2byy 0 1 gr
FIG. 2. Schematic showing the domain considered for the western
and eastern boundary wave solutions.
where real parts is understood, and we anticipate that
the zonal wavenumber k( y) varies with latitude because
of the variation of the Coriolis parameter and deformation radius with latitude; for example, to allow the
zonal decay scale of coastal Kelvin waves to vary with
latitude.
Substituting this trial solution into (2.5) gives
(v 1 ir)k2 1 bk 1
v
5 0,
L2d
(2.9)
to which the general solution is
(2.4)
Thus, western boundary solutions correspond to the
half-plane x $ 0 and eastern boundary solutions correspond to the half-plane x # 0 (Fig. 2). The orientation of
the boundaries has some influence on the solution (e.g.,
Grimshaw and Allen 1988).
(2.8)
k5
pffiffiffiffiffiffiffiffiffiffiffi
b
21 7 1 2 l ,
2(v 1 ir)
(2.10)
l5
4v(v 1 ir) 4v(v 1 ir) 2
5
y .
c2
b2 L2d
(2.11)
where
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JOURNAL OF PHYSICAL OCEANOGRAPHY
The convention we adopt in this and subsequent equations is that the first root corresponds to the western
boundary solution and the second root to the eastern
boundary solution.
Substituting the trial solution into the boundary condition (2.7) gives an equation for the amplitude variation
along the boundary:
dA
(v 1 ir)
52
kA.
dy
by
(2.12)
First we consider the weakly damped Kelvin wave
limit (l 1, r v) equivalent to v bLd/2. Typical
values of b around 2 3 10211 m21 s21 and Ld around 5 3
104 m gives v 5 3 1027 s or periods much shorter than
3–4 months (on which time-scale damping can safely be
assumed to be weak).
The leading-order solution for the zonal wavenumber
is
ib
l1/2 (61 1 il21/2 1 . . . )
2(v 1 ir)
i
ir 21/2
11
(61 1 il21/2 1 . . . )
5
Ld
v
5
(2.13)
where we have neglected O(l21) and O(r2/v2) terms in
the expansion.
Thus, to leading order
h0 ’ A(y)e7x/Ld e2ivt ;
(2.14)
the Kelvin wave decays away from the boundary over Ld
as expected, but note that the deformation radius decreases with increasing latitude.
2) MERIDIONAL STRUCTURE
The real component of k is of little consequence for
the zonal structure of the wave, but has an important
impact on its meridional structure and propagation.
Substituting (2.13) into the boundary condition (2.12)
gives
dA
iv r
1
5 7 6 1 1 . . . A.
dy
c 2c 2y
rffiffiffiffiffi
y 7ivy/c 6ry/2c
e
e
.
y0
(2.15)
In contrast to the f-plane limit, the wave amplitude (in h)
varies as the square root of latitude y; this result was first
derived by Moore (1968). The wave propagates cyclonically around the coastline at c and decays because
of friction over a length scale 2c/r.
Now consider the low-frequency limit l 1. The
analysis presented below applies irrespective of the
relative magnitudes of v and r. We first discuss
the Rossby wave limit v r and defer discussion of
the Stommel boundary layer limit v r until section 2c.
1) ZONAL STRUCTURE
b
l
21 7 1 2 1 . . .
2(v 1 ir)
2
b
l
v
’2
12
and 2 2 ,
(v 1 ir)
4
bLd
k5
pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
ib
l1/2 (6 1 2 l21 1 il21/2 )
2(v 1 ir)
i
r
b
1 ... ,
6
2
Ld 2vLd 2v
A ’ A0
The leading-order expression for the zonal wavenumber (2.10) is now
1) ZONAL STRUCTURE
56
The leading-order solution for A is
b. Rossby wave limit (l 1, r v)
a. Kelvin wave limit (l 1, r v)
k5
VOLUME 43
(2.16)
where (2.11) has been used to substitute for l in the final
expression. The first, western boundary root corresponds to a short Rossby wave solution, damped over
a zonal scale v2/rb. The second, eastern boundary root
corresponds to a long Rossby wave solution, independent of the size of r at leading order. Physically, as the
angular frequency is reduced, short Rossby waves become shorter and eventually find the frictional boundary
layer scale r/b (Stommel 1948), whereas long Rossby
waves become longer and friction remains unimportant.
2) MERIDIONAL STRUCTURE
The meridional structure of the waves is determined
by substituting (2.16) into the boundary condition (2.12).
First for the western boundary short-wave solution,
we find
dA
1 v(v 1 ir)y . . .
A.
5 2
1
dy
y
c2
The solution is
;
A 5A
y
0y
e2v
y /2c2 2ivry2 /2c2
2 2
e
1 ...
0
2
y
’ A0 e2ivry0 (y2y0 )/c .
y0
(2.17)
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MARSHALL AND JOHNSON
Here we have set e2v y /2c ’ 1 because v2y2/c2 1, and
Taylor expanded in y about a mean latitude y0 absorbing
the leading term in the Taylor expansion into the complex phase A0.
Thus the amplitude of the western boundary shortwave solution decays linearly with decreasing latitude
and propagates equatorward along the boundary at a
speed
2 2
cw 5
2
c2
by0 dS
5c
Ld
,
dS
(2.18)
where dS 5 r/b is the Stommel frictional boundary layer
width (Stommel 1948). Thus, the boundary wave propagates at the classical Kelvin wave speed, scaled by the
parameter Ld/dS. We write (2.18) in this form because, as
we shall see in due course, the same result emerges
across many regimes, including with lateral friction and
a no-slip boundary condition, provided that dS is replaced
by the pertinent frictional boundary layer width.
For a typical ocean climate model the frictional
boundary layer width is chosen to be comparable to the
grid spacing, which in turn is coincidentally close to the
Rossby deformation radius. Thus the boundary anomalies can be expected to propagate at roughly the classical Kelvin wave speed, but the boundary propagation
speed is likely to vary between different models and
with latitude.
For the eastern boundary wave, we instead have
dA v(v 1 ir)y
A1 ... ,
5
dy
c2
to which the leading-order solution is
A 5 A0 eivry
2
/2c2
1 ...
2
’ A0 eivry0 (y2y0 )/c .
(2.19)
(2.20)
We have again assumed v2y2/c2 1 and Taylor expanded the solution about a reference latitude. In contrast to the western boundary solution, the amplitude A
does not vary, at leading order, along the boundary. The
wave propagates poleward at the same speed as the
western boundary solution,
ce 5 c
Ld
.
dS
(2.21)
c. Stommel boundary layer limit (l 1, r v)
The preceding analysis in section 2b also applies in the
limit v r, which we briefly comment on here. As the
frequency decreases, short Rossby waves become
increasingly shorter, such that friction becomes increasingly important, whereas long Rossby waves
become increasingly longer and friction remains unimportant. This is clear when one considers the solution for the western boundary wave in section 2b. If
r v, then
k’
ib
l
ib
12 1 ... ’ ,
r 2 iv
4
r
(2.22)
and
h0 ’ A(y)e2x/dS e2ivt .
(2.23)
The solution is a frictional western boundary layer of
width dS (Stommel 1948), with a flow direction that oscillates in time [also see Pedlosky (1965)]. As in section
2b, the solution propagates along the boundary at the
speed given in (2.18).
3. Meridional transport anomalies
Meridional volume transport anomalies are related to
the layer thicknesses anomalies on the western h0w and
eastern h0e boundaries by integrating the geostrophic
velocity anomaly multiplied by the mean layer thickness
across the basin:
T0 5
gr h0 0
(h 2 h0w ) .
by e
(3.1)
These can be interpreted as meridional overturning
transport anomalies associated with each baroclinic
mode in a basin of uniform depth and stratification.
Western boundary wave solutions all decay rapidly in
the zonal direction. Hence, meridional transport anomalies associated with these western boundary waves
c2 h0w
Tw0 5 2
b y
(3.2)
are confined to the western boundary region.
Conversely, eastern boundary waves do not decay (to
leading order) as they cross the basin. Nevertheless, it
makes sense mathematically to define the meridional
transport anomalies associated with the eastern boundary waves as
Te0 5
c2 h0e
;
b y
(3.3)
this decomposition also makes physical sense if we anticipate a result from section 5 that the layer thickness
(or pressure) anomalies associated with long waves are
2706
JOURNAL OF PHYSICAL OCEANOGRAPHY
vastly reduced in the vicinity of the western boundary
[also see Kanzow et al. (2009)].
VOLUME 43
0
1
› @h0 y 02 gr h02 A
1
1 $ (c2 h0 u0 ) 5 2h0 ry 02 ,
›t
2
2
(4.1)
a. Kelvin wave limit
In the Kelvin wave limit, layer thickness anomalies on
both western and eastern boundaries are proportional to
pffiffiffi
y. Thus the meridional transport anomalies vary as
1
jTw0 j, jTe0 j } pffiffiffi ,
y
(3.4)
that is, the meridional transport anomalies are largest at
low latitudes.
b. Rossby wave and Stommel boundary layer limits
In both the Rossby wave and Stommel boundary layer
limits, there is an asymmetry in the variation of the layer
thickness anomalies with latitude.
On the western boundary, the layer thickness anomalies are proportional to y, and hence
jTw0 j ’ constant .
y0 5
(3.5)
In contrast, on the eastern boundary, the layer thickness
anomalies are independent of y, and hence
1
jTe0 j } ,
y
where the overbar represents a time average over a wave
period. Thus, the wave energy flux in (4.1) is c2 h0 u0 . This
energy flux is degenerate in the sense that any rotational
gauge can be added to the flux without changing the
energy equation (e.g., Longuet-Higgins 1964; Pedlosky
1987; Orlanski and Sheldon 1993; Chang and Orlanski
1994). For statistically steady wave solutions, the time
derivative vanishes and the divergence of the energy flux
is balanced by frictional energy dissipation.
We now consider the dominant terms in this energy
balance for each of the wave solutions derived in section 2.
In the derivation of these wave energy fluxes, recall that
real parts of h0 , u0 , and y0 are understood before taking
quadratic products. From the momentum Eqs. (2.1) and
(2.2), the form of the trial solution (2.8), and the boundary
condition (2.12), it follows that
(3.6)
that is, they decay with increasing latitude as the wave
propagates poleward.
This asymmetry is consistent with the theoretical
model of Johnson and Marshall (2002a) for the adjustment of the MOC to forcing anomalies and, in particular, to the ‘‘equatorial buffer’’ mechanism introduced
to explain the asymmetry between the response of the
MOC to high-latitude localized forcing on each side of
the equator, and between the western boundary layers
and basin interior.
4. Energetics
We now turn to the energy budget of the wave solutions. The purpose of this short section is to demonstrate
a fundamental asymmetry between the western and
eastern boundary waves, with the majority of the energy
being dissipated in the former but conserved in the latter. Subsequently in section 5, we consider the generation of short waves at the western boundary by long
waves incident from the ocean interior.
The energy equation is derived from the momentum
Eqs. (2.1)–(2.3):
ikgr 0
h,
by
(4.2)
and
g 1 dA
dk (v 1 ir)
u0 5 2 r
1 ix 1
k h
dy
by
by A dy
ig x dk 0
h.
52 r
by dy
(4.3)
In the case of the meridional energy flux, integrating
h0 3 (2.1) further gives:
ð
h0 y 0 dx 5
wave
gr
[h0 2 ],
2by
(4.4)
where the integral (with square brackets indicating the
limits of integration) is across the relevant western or
eastern boundary wave. These relations are valuable in
identifying the components of u0 and y 0 that are in phase (or
antiphase) with h0 and hence contribute to the energy flux.
a. Kelvin wave
For Kelvin waves in the limit of vanishing friction
(r 5 0), the net meridional energy flux in (4.1) is
ð
c2 g jA j2
c2 h0 y0 dx 5 7 r 0 ,
4by0
wave
(4.5)
independent of latitude. Hence, in the Kelvin wave
solution, the variation of the wave amplitude on the
boundary as the square root of latitude can be interpreted
DECEMBER 2013
as being precisely what is needed to conserve wave energy (Moore 1968). Finite friction merely modifies this
flux to account for the energy dissipated.
The zonal energy flux, obtained by substituting (2.13)
in (4.3), is
cg jA j2
c2 h0 u0 5 6 r 0 xe72x/Ld .
2y
(4.6)
Note that this vanishes both on the boundary at x 5 0
and as x /6‘. Thus, the zonal energy flux is precisely
that required to broaden/narrow the Kelvin wave as it
propagates equator-/poleward but vanishes in the zonally integrated energy budget.
b. Short Rossby wave/Stommel boundary layer
The zonal energy flux, obtained by substituting the
western boundary solution in (2.16) into (4.3), is
c2 h0 u0 5 0 ,
(4.7)
because h0 and u0 are out of phase. In contrast to the
Kelvin wave, the wavelength of the short Rossby wave/
width of the Stommel boundary layer is independent of
latitude and hence there is no need for a zonal energy
flux.
The net meridional energy flux, using (4.4) and noting
h0 / 0 as x / ‘, is
ð‘
0
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MARSHALL AND JOHNSON
c2 g jA j2
c2 g jA j2
c2 h0 y 0 dx 5 2 r s 5 2 r 20 y.
4by
4by0
(4.8)
The meridional energy flux is equatorward and decreases linearly with decreasing latitude. This energy is
lost to dissipation at a rate
ð‘
0
h0 ry0 2 dx 5
c2 gr jA0 j2
,
4by20
(4.9)
independent of latitude. Note that the dissipation is also
independent of the drag coefficient r. Physically, as r is
reduced, the pointwise energy dissipation reduces, but
occurs over a proportionately larger area as the short
Rossby waves propagate further away from the boundary; these effects compensate at leading order.
In reality, this expression must break down close to
the equator, when y approaches the equatorial deformation radius. Thus, a small amount of energy can be
expected to leak into an equatorial Kelvin wave, but this
is a small residual of the meridional energy flux within
the short Rossby wave at higher latitudes.
c. Long Rossby wave
In the long Rossby wave solution, both u0 and y 0 are
out of phase with h0 and hence both the zonal and meridional energy fluxes vanish, as does the dissipation at
leading order. As mentioned above, the energy flux is
degenerate to an arbitrary rotational flux; thus, the result that the energy fluxes vanish in the long-wave solution should be interpreted as indicating only that there
is no divergence of the energy flux.
5. Long waves incident on a western boundary
a. Statement of problem
We now consider a problem of particular relevance to
the ocean energy budget and to monitoring the meridional overturning circulation: the fate of long Rossby
waves incident on a western boundary and the amplitude of the boundary wave anomaly excited on the
western boundary. This problem is relevant to the ocean
energy budget because analysis of satellite altimeter
data and numerical model calculations suggests that the
western boundary current acts an eddy graveyard and
eddy energy sink (Zhai et al. 2010). Moreover, associated with this eddy energy sink is a reduction in the
amplitude of dynamic height variability at the western
boundary, also consistent with altimetric and hydrographic observations. The latter turns out to be crucial
for end-point monitoring of the MOC (Cunningham
et al. 2007), as discussed by [Kanzow et al. 2009; also see
Wunsch (2008)].
We consider a long Rossby wave incident on a western
boundary over a confined latitude band ys # y # yn.
Because the incident wave is long relative to the short
wave it will excite, it suffices to model the incident wave
through a prescribed (complex) wave amplitude that
varies with latitude and time
h0l 5 Al (y)e2ivt ,
where h0l vanishes for y # ys and y $ yn by assumption.
On reaching the western boundary, a short-wave or
frictional boundary layer solution
h0s 5 As (y)e(iks x2vt)
is excited to satisfy the no-normal flow western boundary condition. Note that while h0s vanishes for y $ yn
(because the short-wave meridional energy flux is
equatorward), h0s need not vanish for y # ys.
A preliminary version of this analysis, valid in the
limit of vanishing friction, was presented in Kanzow
et al. (2009).
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JOURNAL OF PHYSICAL OCEANOGRAPHY
VOLUME 43
b. Boundary amplitude condition
The boundary condition (2.7) now applies to the sum
of the long- and short-wave solutions and can be written
d
(v 1 ir)
(A 1 Al ) 5 2
ks As .
dy s
by
(5.1)
Note that the term involving Al on the right-hand side of
(5.1) is neglected because we have assumed that the
wavenumber of the incident long wave is vanishingly
small, jklj jksj (and we anticipate that As and Al are of
similar magnitude).
Substituting for ks gives
A
d
(A 1 Al ) 5 s 1 . . . .
dy s
y
(5.2)
As in Kanzow et al. (2009), we can use the identity
(A 1 A )
d As 1 Al
1 d
(As 1 Al ) 2 s 2 l
5
dy
y dy
y
y
to rewrite this result as
A
d As 1 Al
5 2 2l 1 . . . .
dy
y
y
(5.3)
Finally, integrating (5.3) over the latitude of the incident
long wave gives
As 1 Al 5 y
;
ðy
n
ys
Al
dy 1 . . .
y2
Dy
A,
y l
(x )
(5.4)
c. Energetics
The foregoing analysis raises a number of questions
about the energetics of the interaction between the incident long wave and the short wave generated at the
western boundary:
d
d
The natural way to pose the problem is to evaluate the
zonal energy flux of the incident long wave grh0h0l u0l , the
(zonal and meridional) energy flux of the reflected short
wave grh0h0s u0s , and the energy dissipation in the short
wave 2h0 ry 0s 2 . However, for two waves of the same
frequency, as considered here, both the wave energy and
energy flux are modified by the cross terms representing
the interference of the two waves.1 Thus, even the concept of short- and long-wave energies, and short- and
long-wave fluxes, is problematic. However, these difficulties can be avoided if one integrates over a much
larger area than the zonal decay scale of the short waves,
as sketched in Fig. 3, such that the cross-interaction
energy fluxes can be neglected.
Firstly, the incident long-wave energy flux is
What fraction of the incident energy reflects zonally as
a short wave?
What fraction of the energy is fluxed equatorward
along the boundary?
What fraction of the energy is dissipated by friction?
ðy
c2 gr d
jAl j2 dy
ys 4by dy
ð
c2 gr yn jAl j2
dy .
5
4b ys y2
Fl 5
where Dy 5 yn 2 ys. Thus, the net wave amplitude on the
boundary As 1 Al is a factor Dy/y 5 bDy/f smaller than
the amplitude of the incident long wave Al.
Note that equatorward of the incident long wave, (5.3)
reduces to the result that As/y is constant as found for
a solitary short wave in (2.17).
d
FIG. 3. Schematic showing the energy fluxes involved when
westward-propagating long Rossby waves excite short Rossby
waves at a western boundary. The short wave is greatly reduced in
amplitude equatorward of the incident long wave. The mathematical variables are defined in the text.
n
(5.5)
Here, we have used the fact that (2.2) reduces to geostrophy in the limit y 0l / 0 (equivalently kl / 0) and
integrated the resultant expression by parts with Al 5 0
at the limits of integration.
As in section 4b, the reflected short-wave zonal energy
flux vanishes equatorward of the incident long wave:
F(sx) 5 0.
(5.6)
1
The cross-interaction terms in the wave energy budget only
vanish when the wave frequencies are different. Consider, for example, two equal and opposite waves that destructively interfere.
Each wave has finite energy, yet the sum of the two waves has zero
energy. This paradox is resolved when one realizes that the crossinteraction terms provide a negative energy that exactly cancels the
energies of each of the constituent waves.
DECEMBER 2013
2709
MARSHALL AND JOHNSON
Over the latitude band of the incident long wave, the
short-wave energy flux is complicated by the modified
boundary condition (5.3) and by the need to account for
the interference between the short and long waves.
Nevertheless, it is clear that these terms decay as one
moves away from the boundary and hence do not affect
the integral energy budget.
So where does the incident long-wave energy go?
The first option is that the energy is fluxed equatorward by the short waves:
c2 g
F(sy) 5 2 r jAs j2 ,
4by
(5.7)
as in (4.8).
The second option is that the energy is dissipated.
Because the meridional velocity associated with the
long waves is vanishingly small by assumption, the net
energy dissipation integrated across the boundary layer
is simply:
ð
D 5 2 h0 ry 0 2s dx
5
c2 gr
4b
analogous to the rip currents generated at a beach (see
Marshall et al. 2013).
6. Wave solutions with lateral friction
In section 2, we derived the properties of Kelvin and
boundary Rossby waves in the presence of linear friction, the main advantage being that the mathematics is
relatively straightforward and solutions can be categorized in terms of a single nondimensional parameter. In
this section, we summarize the results of incorporating
lateral friction and a no-slip boundary condition. After a
brief statement of the mathematical problem and derivation of the governing equations, we state the results
for the boundary propagation speed and variation of
wave amplitude along the western and eastern boundaries in three limiting cases corresponding to those
studied in section 2. Details of the derivations, which
involve the application of singular perturbation theory,
are sketched in an appendix.
a. Governing equations
The meridional momentum equation is now
2
jAs j
.
y2
›y0
›h0
›2 y 0
1 byu0 1 gr
5n 2 ,
›t
›y
›x
(5.8)
Note, again, that the energy dissipation is independent
of the linear drag coefficient.
Now suppose that the incident long-wave energy flux
is mostly confined to a narrow latitude band, as suggested by satellite altimeter measurements of sea surface height variance (e.g., Zhai et al. 2010). Then (5.4)
implies that, to leading order,
As ’ 2Al
over the latitude band of the incident long waves. In
contrast, equatorward of the incident long waves, As is
greatly reduced following (5.4). Thus, it follows that
most of the dissipation occurs within the latitude band of
the incident long waves. Only a small fraction of the
incident energy is fluxed equatorward, and most of that
is dissipated within the western boundary layer, at a
rate that is independent of latitude. Only a very small
amount of energy ultimately escapes into the equatorial
waveguide where the preceding analysis breaks down.
These results are consistent with the Rossby graveyard mechanism proposed and diagnosed by Zhai et al.
(2010) using altimetric data and numerical calculations.
Because the wave energy flux in (4.1) is proportional to
the bolus transport, an interesting corollary is that there
is a convergence of eddy bolus transport at the western
boundary, requiring offshore Eulerian-mean currents,
(6.1)
and the associated vorticity equation is
!
› ›2 h0 h0
›h0
›4 h0
2 n 4 5 0,
2 2 1b
2
›t ›x
›x
›x
Ld
(6.2)
where v is the viscosity.
For boundary conditions, we impose no slip that,
exploiting geostrophy for the meridional flow, can be
written
›h0
5 0 (x 5 0),
›x
(6.3)
and the no-normal flow boundary condition, setting
u 5 0 in (6.1), takes the form
by
›h0
›3 h0
2 n 3 5 0 (x 5 0) .
›y
›x
(6.4)
We now seek trial solutions of the form
#
eika (y)x 1 geikb (y)x 2ivt
e
h 5 A(y)
.
11g
"
0
The form of this solution, with the sum of two exponentials in x, is required to satisfy the two boundary
conditions and may be anticipated from classical boundary
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JOURNAL OF PHYSICAL OCEANOGRAPHY
current theory (e.g., Munk 1950). Division by the coefficient (1 1 g) is to ensure that A(y) is equal to the
wave amplitude on the boundary (at x 5 0).
Substituting this trial solution into the vorticity Eq.
(6.2) gives the depressed quartic equation:
2nk4c 1 ivk2c 1 ibkc 1
iv
5 0 (c 5 a, b) .
L2d
(6.5)
There are four roots, two each for the western and
eastern boundary solutions. This can be solved analytically following the method described by Cardano
(1968). However, the full analytical solution is complicated and provides little physical intuition. Instead we
resort to approximate solutions obtained using singular
perturbation theory (see the appendix).
Once we have the solutions for ka and kb, substituting
the trial solution into the boundary conditions (6.3) and
(6.4) yields
and
(6.6)
The latter condition allows us to determine both the propagation and amplitude variations along the boundary.
b. Summary of solutions
In the appendix, we derive the solutions in three
limiting cases:
(i) a Kelvin wave limit in which propagation is cyclonic around each hemispheric ocean basin and at
the classical Kelvin wave speed c;
(ii) a Rossby wave limit in which propagation is cyclonic
and at the speed
c
Ld
,
dv
pffiffiffiffiffiffiffiffiffiffi
where dv 5 n/2v is the width of an oscillatory
Stokes viscous boundary layer; and
(iii) a Munk boundary layer limit in which propagation
is cyclonic and at the speed
c
In all other respects, the solutions follow those obtained with linear friction in section 2, with the same
variations of the wave amplitude along the western and
eastern boundaries.
Thus, in the low-frequency limit, the waves are slowed
down from the Kelvin wave speed by the ratio of the
deformation radius and the pertinent frictional boundary layer width.
c. Energetics and western boundary interactions
We do not present any details here, but note that all of
the key results from sections 4 and 5 for the energy
fluxes, energy dissipation, and interactions between long
and short waves carry over to the case with lateral friction and no-slip boundary conditions. This is not surprising because the amplitudes and the phases of the
waves, and hence the energy fluxes, are the same outside
the viscous boundary layers.
7. Effect of time-mean flow
k
g52 a,
kb
dA in
5 k k (k 1 kb )A.
dy by a b a
VOLUME 43
Ld
,
dM
where dM 5 (n/b)1/3 is the width of the Munk boundary
layer (Munk 1950).
To this point, we have analyzed wave solutions to the
equations linearized about a state of rest. While it is
tempting to suppose that mean flows simply Doppler
shift the solutions described above, this need not be the
case. For example, the non-Doppler shifting of long
Rossby waves by background mean flows in the reducedgravity model is well established (Liu 1999): physically,
a mean flow both Doppler shifts the waves, but also
modifies the mean potential vorticity gradient, thereby
modifying the intrinsic Rossby wave speed; in the case of
long Rossby waves in the reduced-gravity model, these
two effects are equal and opposite.
A detailed treatment of mean flows lies beyond the
scope of the present manuscript, both because it would
require a large number of additional calculations, and
equally because the technical challenge of including
general mean flows is nontrivial. Nevertheless, here we
include one simple, if somewhat extreme, scenario, mainly
to illustrate the potential importance of mean flows.
Specifically, we consider the case of a mean, inertial
western boundary current within which the potential
vorticity is uniform. For this purpose, we employ a geostrophic vorticity model (Sch€
ar and Davies 1988; Tansley
and Marshall 2000). The equations of motion, including
nonlinear accelerations but neglecting dissipation, are
as follows:
2byy 1 gr
›h
5 0,
›x
(7.1)
›y
›y
›
y2
1 by 1
u1
gr h 1
5 0, and (7.2)
›t
›x
›y
2
DECEMBER 2013
2711
MARSHALL AND JOHNSON
›h ›
›
1 (hu) 1 (hy) 5 0 .
›t ›x
›y
(7.3)
These can be combined to obtain a potential vorticity
conservation equation
›
›
›
1u 1y
Q 5 0,
›t
›x
›y
(7.4)
where
Q5
1
›y
by 1
h
›x
is the potential vorticity.
Setting the potential vorticity to a uniform value Q0
gives an elliptic equation for the time-mean layer
thickness
by 1
gr ›2 h
5 Q0 h ,
by ›x2
(7.5)
the solution to which is
by
^
^
h5
(1 2 e2x/Ld ) 1 hw ( y)e2x/Ld ,
Q0
(7.6)
where
L^d 5
gr
byQ0
1/2
(7.7)
is closely related to the conventional Rossby deformation radius. The time-mean layer thickness on the
boundary hw is a weak function of latitude because the
no-normal flow boundary condition applies to the total,
rather than geostrophic, velocity. The solution, sketched
in Fig. 4, is the classical Fofonoff gyre (Fofonoff 1954),
slightly modified by the nonuniform value of hw on the
boundary.
Because the potential vorticity is uniform, layer thickness anomalies satisfy
›2 h0
gr
5 Q0 h0 ,
by ›x2
(7.8)
to which the solution is
FIG. 4. Schematic showing the solution for the Fofonoff gyre with
uniform potential vorticity, about which linearized wave solutions
are obtained in section 7. Sketched are the layer thickness contours,
which serve as approximate streamlines for the flow. The layer
thickness varies slightly along the boundary because the no-normal
flow boundary condition applies to the total, rather than geostrophic, velocity.
deformation radius, as in the classical Kelvin wave. This
is not surprising for two reasons: (i) Rossby waves rely
on the presence of a background potential vorticity gradient that is absent here; (ii) both the present solution and
the classical Kelvin wave are associated with vanishing
potential vorticity anomalies. Time dependence is left
arbitrary in this solution.
To solve for the meridional propagation of the wave,
we need to apply the no-normal flow boundary condition linearized about the mean state:
›y0 ›
1 (y y 0 1 gr h0 ) 5 0 (x 5 0)
›t ›y w
(7.10)
where y w ( y) is the mean boundary velocity and
y0 5
g ›h0
by ›x
is the perturbation geostrophic velocity.
Substituting for h0 gives:
› A(y, t)
›
A(y, t)
^
[y w (y) 2 byLd (y)] pffiffiffi
5 0.
pffiffiffi 1
›t
›y
y
y
(7.11)
^
h0 5 A(y, t)e2x/Ld .
(7.9)
Note that we have lost the zonal propagation of the short
Rossby wave solution and instead have a solution that
decays over a scale that is closely related to the classical
The general solution is nontrivial. Nevertheless, (7.11) is
in flux form where we can identify
cw (y) 5 y w 2 byL^d ’ yw 2 c
(7.12)
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JOURNAL OF PHYSICAL OCEANOGRAPHY
as the relevant, nondispersive western boundary wave
speed. To leading order, this is the classical Kelvin
wave speed, Doppler shifted by the mean velocity on the
boundary.
While the results of this section apply in just one extreme limit in which the potential vorticity is completely
homogenized, the fact that the Kelvin wave solution is
recovered should provide a further cautionary note
about the propagation speeds of meridional circulation
anomalies in OGCMs. In particular, models that fail to
resolve eddies and hence the Rossby deformation radius, will also fail to resolve the inertial boundary layers
and the mean flows described in this section. It remains
to be determined whether this solution is of any relevance to more realistic cases in which there are finite
potential vorticity gradients and Rossby waves and also
when a no-slip boundary condition is incorporated.
8. Concluding remarks
Boundary waves play a fundamental role in the adjustment of the MOC to changes in surface wind and
buoyancy forcing. Yet, this adjustment varies widely
across different OGCMs, as reviewed in section 1. In this
manuscript, we have analyzed boundary wave solutions
along western and eastern boundaries in a reducedgravity model. Contrary to what is often assumed in
studies of MOC adjustment, but consistent with results
from the coastal oceanography literature (e.g., Clarke
and Shi 1991), boundary propagation occurs not through
Kelvin waves, but through short and long Rossby waves
at the western and eastern boundaries respectively:
these Rossby waves propagate zonally, as predicted
by classical theory, and cyclonically along the basin
boundaries to satisfy the no-normal flow boundary condition. The along-boundary propagation speed is cLd/d,
where c is the internal gravity/Kelvin wave speed, Ld is
the Rossby deformation radius, and d is the appropriate
frictional boundary layer width. This result holds across
a wide range of parameter regimes, with either linear
friction or lateral viscosity and a no-slip boundary condition. However, we note that this result may not carry
over to numerical models in which the lateral resolution
is insufficient to resolve the boundary layers; for example, Hsieh et al. (1983) has found sensitivity of the alongboundary propagation speed to the choice of lateral
boundary condition in coarsely resolved models.
An important corollary is that the boundary propagation speed in OGCMs is likely to be sensitive to model
parameters, offering a plausible explanation for the wide
range of model behavior, as well as some of the discrepancies between models run at eddy-permitting and coarser
resolution. These results are likely to be modified by the
VOLUME 43
inclusion of realistic bottom topography, higher baroclinic modes, background mean flows (as shown in section 7), and will also differ for the adjustment of the
deep limb of the MOC (e.g., Elipot et al. 2013). Nevertheless, we suggest that a cautious approach is apposite when discussing the boundary propagation of MOC
anomalies in any one particular model. A proper representation and dynamical understanding of boundary
wave propagation speeds is important for understanding
internal climate variability, as reviewed in section 1, and
may facilitate attempts to develop the capacity for decadal climate prediction in the Atlantic sector (e.g., Latif
et al. 2006; Msadek et al. 2010; Robson et al. 2012).
One obvious extension of this work is to explain the
lack of coherence of meridional circulation anomalies
observed in models across the Gulf Stream over the past
several decades (e.g., Bingham et al. 2007; Biastoch et al.
2008; Lozier et al. 2010). While this lack of coherence
may be a consequence of the spatial pattern of wind
forcing anomalies (e.g., Zhai et al. 2013, manuscript
submitted to J. Climate), rather than boundary waves
per se, a plausible explanation is that the change in potential vorticity encountered by a western boundary
wave as it propagates across the Gulf Stream excites an
eastward-propagating frontal wave (e.g., Cushman-Roisin
et al. 1993; Cushman-Roisin 1993), to which it loses energy. Results from such a study will be reported in a future manuscript.
Acknowledgments. We are grateful to Sergey Danilov
and two anonymous reviewers for helpful comments on
a preliminary draft. Financial supported was provided
by the U.K. Natural Environment Research Council and
a Royal Society University Research Fellowship (HLJ).
APPENDIX
Derivation of Solutions with Lateral Friction
and No-Slip Boundary Condition
a. Kelvin wave limit
Nondimensionalize the wavenumber by the deformation radius:
;
kc 5 Ld kc ,
(A1)
where henceforth tildes indicate nondimensional variables. The depressed quartic (6.5) becomes
; ;
;
;
i21 d3 kc4 1 ( kc2 1 1) 1 21 kc 5 0,
where
DECEMBER 2013
2713
MARSHALL AND JOHNSON
;
d5
or, in dimensional form,
dM
Ld
(1 1 i)
kb 5 6 pffiffiffi
2
is the nondimensional Munk viscous boundary layer
width with dM 5 (n/b)1/3. The nondimensional parameter
5
v
bLd
is analogous to l1/2 in section 2. For Kelvin waves, we
require 1 analogous to l 1 in section 2a.
To ensure that the Kelvin waves are not swamped by
;
lateral friction, we also require d 1. The appropriate
;
choice turns out to be d 5 O(21/3 ), giving
;
;
;
;
i22 ( d 3 ) k4c 1 ( k2c 1 1) 1 21 kc 5 0.
(A2)
... ,
#
"
dA
iv 1 (1 2 i) n1/2 v1/2 by . . .
1
A.
5 7 1 6 pffiffiffi
dy
c 2y
c2
2
The solution, retaining just the leading terms for the
boundary wave propagation and amplitude variation, is
and
The solutions are
;
k0 5
6i
and
;
k1 5
1
2 ,
2
i
b
2 1 ... .
Ld 2v
(A3)
As in section 1, the first root corresponds to the western
boundary and the second root to the eastern boundary.
Second, to obtain the viscous boundary layer roots, we
expand (A2) in the distinguished limit,
;
;
;
kc 5 k0 1 k1 1
An alternative expansion that admits a short Rossby
wave solution is obtained by nondimensionalizing the
wavenumber by the short Rossby wavenumber:
v
;
kc 5 k .
b
2
giving
N ;4 ;2 ;
k 2 i kc 2 i kc 2 i2 5 0,
2 c
where now 1,
;
;
i( d ) k40 1 k02 5 0
;
; ;
; ;
4i( d3 ) k30 k1 1 2 k0 k1 5 0 .
and
;
k0 5
(1 1 i)
6 pffiffiffi d23/2 21/2
2
N5
pffiffiffi d 6
b2 n
v
5
2
,
dM
v3
and
The solutions are
and
;
k1 5 0,
(A6)
Then
... ,
;3
(A5)
b. Rossby wave limit
or, in dimensional form,
ka 5 6
A0 7ivy/c 6vd by2 /4c2
A
v
e
,
pffiffiffi ’ pffiffiffiffi
ffie
y0
y
that is, the wave propagates cyclonically along the
boundaries at the classical Kelvin wave speed, c, and the
pffiffiffi
wave amplitude varies as y. The latter term represents
the frictional decay of the Kelvin wave and is latitude
dependent because the importance of friction depends
on the relative magnitude of the deformation radius and
the viscous boundary layer width, with the deformation
radius being larger at low latitudes and thus frictional
damping being less important.
giving at the leading two orders:
;2
k0 1 1 5 0
; ;
;
2 k0 k1 1 k0 5 0.
(A4)
This is the classical solution for a viscous Stokes
boundary layer in an oscillatory flow.
Finally, we determine the variation of the wave amplitude along the western and eastern boundaries by
substituting for ka and kb in (6.6):
First, to obtain the Kelvin wave roots, we expand (A2)
in powers of 21,
;
;
;
kc 5 k0 1 21 k1 1
rffiffiffiffi
v ...
1
.
n
dv 5
rffiffiffiffiffiffi
n
2v
(A7)
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JOURNAL OF PHYSICAL OCEANOGRAPHY
is the width of an oscillatory Stokes viscous boundary
layer. We also set N 5 O(2); clearly the results are only
valid when dv dM.
First, to obtain the Rossby wave roots, we expand
in 2,
;
;
;
kc 5 k0 1 2 k1 1
that is, the propagation speed along the boundary is
cw 5 c
... ,
; ;
;
N ;4
k0 2 2i k0 k1 2 i k1 2 i 5 0.
2
The solution, exploiting 2 1, is
and
22 1 . . . ,
2
(A8)
... ,
Finally, we consider a regime in which a viscous
boundary layer solution is obtained at the western boundary. Here, we nondimensionalize the wavenumber by the
deformation radius:
;
kc 5 Ld kc .
and
(A14)
;3 ;4
;
;
d kc 2 i( kc2 2 1) 2 i kc 5 0,
;
N ; ;
4 2 k02 k1 2 2i k1 2 i 5 0.
(1 1 i) 1
6 pffiffiffiffiffiffiffi 1 1 . . . ,
2N 2
(A9)
corresponding to an oscillatory viscous boundary of
width dv. The leading-order real and imaginary terms in
the no-normal flow boundary condition for the western
solution are
;
d5
A.
dM
Ld
is the nondimensional Munk viscous boundary layer
width with dM 5 (n/b)1/3. We set d 5 ;1, which ensures
that viscosity has a leading-order impact at the deformation scale.
We now expand in the small parameter :
;
;
;
kc 5 k0 1 k1 1
... ,
giving
;3 ;4
;
d k0 2 i k0 5 0
; ; ;
;
;
4 d3 k03 k1 2 i k1 2 i( k02 1 1) 5 0.
The solution is
A A0 2i(bd /f )(v/bL2 )y0
d
v 0
’ e
,
y y0
(A15)
where
The leading terms in the solutions for the western and
eastern boundaries are
vbdv y . . .
dA
1
5
2i
1
dy
y
c2
(A13)
The depressed quartic (6.5) becomes
N ;4 ;2
k 2 i k0 5 0
2 0
Ld
.
dv
c. Munk boundary layer limit
giving
;
kb 5
(A12)
that is, the propagation speed along the boundary is
again
ce 5 c
that is, the short and long Rossby wave solutions,
respectively.
To obtain the remaining roots, we expand in the distinguished limit:
;
kc 5 21 k0 1 k1 1
0
A ’ A0 e2i( bdv /f0 )(v/bLd )y ,
The leading terms in the solutions for the western and
eastern boundaries are
and
(A11)
2
vbdv y . . .
dA
v y
A.
5
1i
1
dy
c2
c2
;2
k0 1 k0 5 0
21 1 2 1 iN 1 . . .
Ld
.
dv
At the eastern boundary, we find
giving
;
ka 5
VOLUME 43
and
(A10)
The solutions for the western boundary are
DECEMBER 2013
;
ie2pi/3 ka 5 2 ; 1
3
d
;
ie22pi/3 kb 5 2 ; 1
3
d
2715
MARSHALL AND JOHNSON
12
e22pi/3
!
1 ...
;
d2
12
e2pi/3
and
(A16)
1 ... ,
;
d2
...
;
i kb 5 2 ; 1
3
d
(A17)
and
1
(A18)
!
12 ; 1 ... .
d
(A19)
The western boundary solutions represent the classical
viscous boundary layer solution of Munk (1950), extended to include the leading-order correction due to
the oscillatory time dependence. The eastern boundary
solutions represent the long Rossby wave and no-slip
viscous boundary layer solutions, respectively.
The boundary condition gives for the western
boundary:
dA
1 dM vby
. . . A,
5
2
i
1
dy
y
c2
which takes the same form as the equivalent equation
with linear friction. The solution, expanded about a reference latitude, is again
A A0 2i(bd /f )(v/bL2 )y0
d
M 0
5 e
,
y y0
(A20)
that is, the propagation speed along the boundary is
again
cw 5 c
Ld
.
dM
(A23)
!
and for the eastern boundary,
;
ka 5 2 1
ce 5 c
Ld
.
dM
(A21)
The boundary condition gives for the eastern
boundary:
dA idM vby
A1 ... ,
5
dy
c2
to which the solution is
0
A 5 A0 e2i(bdM /f0 )(v/bLd )y ,
2
(A22)
that is, poleward propagation along the boundary,
without any amplitude change, at a speed
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