ABSTRACT
We examine nonlinear topographic waves in barotropic, rigid lid models with a focus on how depth shapes can or cannot support solitary waves. Comparisons between the full equations and the quasigeostrophic system show that the conditions for vanishing nonlinearity are different and that the transition from cyclonic to anticyclonic solitary waves does not happen for the same topographic shape. For these dynamics, however, such waves only exist for a channel geometry when the long waves can become nondispersive. We therefore examine wave groups since short waves can have an isolated crosstopographic structure. Group solitary waves are found and analyzed with a hyperbolic tangent topography representing the transition from the shelf to deep water.
Keywords:
Topographic Rossby waves; Solitary waves; Solitary groups
INTRODUCTION
Topographic Rossby waves, for which the background potential vorticity (PV) gradient is provided by the bottom slope rather than the gradient in f, have been observed in a variety of oceanic regions. On the New England continental slope, near the Gulf Stream, Thompson (1971)THOMPSON, R. 1971. Topographic Rossby waves at a site north of the Gulf Stream, Deep Sea Res. and Oceanogr. Abstracts 18, 119. and Thompson & Luyten (1976)THOMPSON, R. & LUYTEN, J. 1976. Evidence for bottomtrapped topographic Rossby waves from single moorings, Deep Sea Res. and Oceanogr. Abstracts 23, 629635. identified motions off Cape Hatteras with appropriate space and time scales, and Pickart (1995)PICKART, R. 1995. Gulf Streamgenerated topographic Rossby waves, J. Phys. Oceanogr. 25, 57486. traced these to forcing by eastward propagating meanders of the jet. Louis et al. (1982)LOUIS, J., PETRIE, B. & SMITH, P. 1982. Observations of topographic Rossby waves on the continental margin off Nova Scotia, 12, 4755. observed them off Nova Scotia. Other locations include the Beaufort Gyre (Zhao & Timmermans 2018ZHAO, B. & TIMMERMANS, M. 2018. Topographic Rossby waves in the Arctic Ocean's Beaufort Gyre, Journal of Geophysical Research: Oceans 123(9), 65216530.), the South China Sea (Shu et al. 2016SHU, Y., XUE, H., WANG, D., CHAI, F., XIE, Q., CAI, S., CHEN, R., CHEN, J., LI, J. & HE, Y. 2016. Persistent and energetic bottomtrapped topographic Rossby waves observed in the southern South China Sea, Scientific reports 6(1), 18., Wang et al. 2019WANG, Q., ZENG, L., SHU, Y., LI, J., CHEN, J., HE, Y., YAO, J.AND WANG, D. & ZHOU, W. 2019. Energetic topographic Rossby waves in the northern South China sea, Jour. Phys. Oceanog. 49(10), 26972714.), and east Australia (Garrett 1979GARRETT, C. 1979. Topographic Rossby waves off east Australia: identification and role in shelf circulation, J. Phys. Oceanogr. 9, 244253.). Finally, as appropriate for this collection, Oey & Lee (2002)OEY, L. & LEE, H. 2002. Deep eddy energy and topographic Rossby waves in the Gulf of Mexico, J. Phys. Oceanogr. 32, 34993527. and Hamilton (2009)HAMILTON, P. 2009. Topographic Rossby waves in the Gulf of Mexico, Prog. in Oceanogr. 82, 131. found them in the Gulf of Mexico. Clarke (1971)CLARKE, R. 1971. Solitary and cnoidal planetary waves, Geophys. Fluid Dyn. 2, 343354. demonstrated that the shallow water equations could have solitary and cnoidal wave solutions with a sloping bottom in a channel. A zonal shear flow could also provide the necessary potential vorticity gradient (see also (Long 1964LONG, R. 1964. Solitary waves in the westerlies, J. Atmos. Sci. 21, 197200.) and (Larsen 1965LARSEN, L. 1965. Comments of "Solitary Waves in the Westerlies", J. Atmos. Sci. 22, 222224.)). MalanotteRizzoli & Hendershott (1980)MALANOTTERIZZOLI, P. & HENDERSHOTT, M. 1980. Solitary Rossby waves over variable relief and their stability. Part I. The analytical theory., Dyn. Atmos. Oceans 4, 247260. and MalanotteRizzoli (1980)MALANOTTERIZZOLI, P. 1980. Solitary Rossby waves over variable relief and their stability. Part II. Numerical experiments., Dyn. Atmos. Oceans 4, 261294., MalanotteRizzoli (1982)MALANOTTERIZZOLI, P. 1982. Planetary solitary waves in geophysical flows, Advances in Geophysics 24, 147224. followed up on the topographic case using the quasigeostrophic (QG) approximation.
Solitary wave packets for betaplane waves in a channel, were studied by Yamagata (1980)YAMAGATA, T. 1980. On the nonlinear modulation of planetary topographic waves in a rotating stratified ocean, Geophysical & Astrophysical Fluid Dynamics 16(1), 3554. again in a channel with the QG simplification. These can arise from whatever forcing creates waves in a limited region or from a modulational instability of a sinusoidal wave. Since the group velocity depends on the wavenumber and the packet has, of course, a range of wavenumbers, the envelope will deform under linear dynamics. As in the case of the solitary wave, the nonlinearity can counteract this and result in a packet travelling without change of form.
In this paper, we explore both kinds of nonlinear topographic Rossby waves. For the solitary waves, we examine in some detail which topographies support cyclonic or anticyclonic waves. For the wave packet, we pick a topography with a shelf, a slope, and then a flat bottom further offshore. This can support waves confined over the slope, especially when the wavelength is small.
BAROTROPIC MODEL EQUATIONS
We will use the barotropic, rigid lid, shallow water equations as the simplest model system for representing topographic Rossby waves; the rigid lid filters out the surface gravity waves. We are left with conservation of potential vorticity and a slightly more complex inversion relationship between the PV and the masstransport streamfunction.
We write the potential vorticity as
with $${H}_{0}=H\left(0,0\right)$$. The f_{0} just removes a potentialy large constant.
Defining $$\lambda ={H}_{0}/H$$ and also $$h=H/{H}_{0}$$ (so we can use whichever is convenient) allows us to write the potential vorticity (PV) equation as
with
It is more convenient to introduce the operator $$\mathcal{L}=\nabla \cdot \lambda \nabla $$ and use the vorticity equation
or, written just in terms of the streamfunction,
The quasigeostrophic (QG) version replaces λ by one in the definition of vorticity and in its advection; the second term gives the planetary and topographic beta
The QGPV is then
and satisfies
If $$\lambda 1$$ is, as required, small (order $$\delta H/{H}_{0}$$), then we expand and drop all higher order terms. If we write the QG system in terms of the topographic elevation $$B\left(x\right)={H}_{0}H\left(x\right),\lambda $$ becomes $${\left(1B/{H}_{0}\right)}^{1}$$ and the QG topographic beta is just $${f}_{0}B\left(x\right)/{H}_{0}$$ (dropping terms higher order in $$\left.B/{H}_{0}\right)$$. The QGPV takes the familiar form
We will use $$H=H\left(y\right)$$ for the rest of the paper.
LINEAR BUT ALSO NONLINEAR SOLUTIONS
Just as a single Rossby wave in the betaplane barotropic vorticity equation is a nonlinear solution, we can ask when that also holds in the full equation (1). A steadily propagating solution will have
with $$R={\int}^{y}h$$. We let
so that
The x dependent and x independent parts must balance separately
so that K^{2} is a total wavenumber, while
relates the propagation speed to the topography and the wavenumber. Taking a y derivative gives
so that the linear solution will satisfy the nonlinear equations when the topography has the form
with α a constant. The dispersion relation is simply
Solving the equation for the topography gives
(with the condition $$\lambda \left(0\right)=1$$). In terms of the depth, the topography must satisfy
On the f plane, this says our topography is
Thus we have related the wave speed to the shape and slope of the topography. In particular, if the bottom elevation is
we find
This is exactly the QG result, but here the slope is not constant, and the structure of the wave is not sinusoidal in y . Note that this can only work in a bounded region, since H will blow up for $$y=1/\left(2\alpha \right)$$. Therefore, we need at least one wall at some y north of this point.
SOLITARY WAVES
For the rest of the paper, we will ignore the planetary β; the y direction is now upslope and the x is along the topography. The equations will be nondi mensionalized with the time scale $$1/{f}_{0}$$ and space scale L characterizing the slope $$\left(\frac{\partial H}{\partial y}\sim {H}_{0}/L\right)$$. The equations become
with the QG form having $$\lambda \to 1$$ and retaining only the lowest order term in the expansion of the topographic β .
Clarke (1971)CLARKE, R. 1971. Solitary and cnoidal planetary waves, Geophys. Fluid Dyn. 2, 343354. derived the solitary wave solutions like those here; we are adding the KdV equation and more discussion on the role of the topographic form on the waves. MalanotteRizzoli & Hendershott (1980)MALANOTTERIZZOLI, P. & HENDERSHOTT, M. 1980. Solitary Rossby waves over variable relief and their stability. Part I. The analytical theory., Dyn. Atmos. Oceans 4, 247260. also studied the dynamics, but using the quasigeostrophic equations; qualitatively, these are similar to the cases here with order one variation in amplitude, but there are noticeable quantitative differences.
If there are no closed streamlines in the comoving frame, then we just have
For velocities small compared with the phase speed, we can Taylorexpand the r.h.s.
and evaluate the $${\mathcal{Q}}^{\prime}$$ by y derivatives of
giving
If h decreases monotonically towards positive y then λ increases with y . (In the limit of small depth variations λ is nearly equal to a constant plus the topography height divided by $${H}_{0}$$.)
With these definitions, our Taylorexpanded equation ^{1} 1 The QG form will be the same except ζ=∇2ψ and γ=∂2λ/∂y2; for the sample topographies in table 1 , the values can be quite different from our finitetopography case D0β/c0 is selfadjoint. The result is , is
Although the steps below can be taken in general, we note that there is a particular topography for which the Taylorseries truncates exactly; i.e., $$\mathcal{Q}$$ is a quadratic function of its argument:
In that case, which we will call "quadratic topography", the wave satisfies (3) exactly with
There is no restriction on the amplitude; when, however, there are closed streamlines, $$\mathcal{Q}$$ need not be the same on interior streamlines as it is on the ones extending to infinity. This does not mean that the solution is invalid, but rather that there are many other solutions with different $$\mathcal{Q}\left(R+\psi /c\right)$$ interior functions. Examples of such multivalued $$\mathcal{Q}$$ functions are "modons" (Stern 1975STERN, M. E. 1975. Minimal properties of planetary eddies, J. Mar. Res. 40, 5774., Flierl et al. 1980FLIERL, G., LARICHEV, V., MCWILLIAMS, J. & REZNIK, G. 1980. The dynamics of baroclinic and barotropic solitary eddies, Dyn. of Atmos. and Oceans 5, 141.) and potential vorticity patches (Moore & Saffman 1971MOORE, D. & SAFFMAN, P. 1971. Structure of a line vortex in an imposed strain, in R. M. Olsen J.H., Goldburg A. (ed.), Aircraft Wake Turbulence, Plenum, New York, 339353.); cases with topography include Wang (1992)WANG, X. 1992. Interaction of an Eddy with a Continental Slope, PhD Thesis, MITWHOI., BakerYeboah et al. (2010)BAKERYEBOAH, S., FLIERL, G., SUTYRIN, G. & ZHANG, Y. 2010. Transformation of an Agulhas eddy near the continental slope, Ocean Science 6, 143159. Zhang et al. (2011)ZHANG, Y., PEDLOSKY, J. & FLIERL, G. 2011. Shelf circulation and crossshelf transport out of a bay driven by eddies from an openocean current. Part I: Interaction between a barotropic vortex and a steplike topography, J. Phys. Oceanogr. 41, 889910..
We just summarize Clarke's (1971) procedure: look for an approximate solution of the form
At the lowest order
This determines the meridional structure and, with appropriate boundary conditions, the lowest order phase speed $${c}_{0}$$.
The second order equation is
We multiply by g and integrate (with the integral notated as ‹ • ›), using the fact that the operator
For an isolated disturbance A_{xx} will have the same sign as A so that we can have decaying exponentials when A is small. Taking into account $${\mathcal{Q}}^{\prime}>0\Rightarrow \beta >0$$, this condition requires $${c}_{1}<0$$. As is characteristic of solitary waves, it will travel faster than the fastest long wave. (This is implicit in the QG dispersion relation $$c={c}_{0}/\left(1+{k}^{2}/{l}^{2}\right)$$ but with $${k}^{2}$$ negative.) To match between the decaying solutions to the right and left, there must be a region where the amplitude is large enough so that the second term dominates and makes the second derivative of A be opposite in sign from A . This requires $$A{g}^{2}\gamma <0$$ where g is the gravest mode across the channel and is chosen to be positive. If $${h}^{2}$$ has positive [negative] curvature, the eddy will be cyclonic [anticyclonic].
The usual solution for the x structure of the wave
indeed shows that $${c}_{1}<0$$ and, for positive γ , that the wave is cyclonic with a low in the streamfunction, $${A}_{0}<0$$
KoV
To demonstrate that these are "solitons", we shall show that the timedependent equation for the amplitude is indeed the KortwegdeVries equation, so that faster eddies will pass through slower ones with only a phase shift, and arbitrary, localized disturbance of the right sign will break up into a set of solitons and a weak dispersive field. See the literature on "inverse scattering", e.g., Ablowitz & Segur (1981)ABLOWITZ, M. & SEGUR, H. 1981. Solitons and the inverse scattering transform, Society for Industrial and Applied Mathematics, Philadelphia, 425p..
In the vorticity equation
with
we introduce a long scale for x , a slow time, and small amplitude waves
The order ϵ equation is just
which has the solution
The translation rate $${c}_{0}$$ is the long wave speed  the gravest eigenvalue of
The order ϵ^{3} equation includes the order ϵ^{2} correction to ψ_{0} and terms with the largescale derivatives in x and t
Multiplying by g and integrating over y gives
Substituting $${\mathcal{D}}_{0}g=g\beta /{c}_{0}$$ leads to
This is a form of the KdV equation; therefore the long waves will indeed behave like solitons as long as they all have the same y structure.
TOPOGRAPHIC SHAPE
The shape used for the topography determines the sign of the solitary waves; however, some choices can also be convenient in that the lowest order equa tion for the y structure has a simple form. Fig. 1 shows sample plots of the depth profiles, with characteristic properties listed in table 1 . The case
Depth profiles $$h\left(y\right)$$ for $$\lambda ={e}^{ay}$$ and $${(1+ay)}^{n}$$ with $$n=1,2/3$$ (cyc.), $$1/2$$ (no solitary waves), $$1/4$$ (anticyc.). The corresponding values of $$\gamma \left(0\right)/{a}^{2}$$ are $$2,1,2/9,0,1/8$$. is chosen so that the depth range is $$10{H}_{0}$$ in the interval $$1/2<y<1/2$$. The $$n=1$$ case corresponds to a constant slope and has $$a=18/11$$ with $$\gamma /{a}^{2}=3$$ (cyc.). like the continental slope, has a limited extent connecting two flat regions. But using a representative topographic model $$\lambda =1+d\text{tanh}\left(y\right)$$ indicates the problem. The solutions in the flat regions just have $${\partial}^{2}\psi /\partial {y}^{2}=0$$ so that solutions will not decay. In deed, if we look at the QG form, the y structure problem
gives a simple expression for $$\gamma ={a}^{2}n\left(2n1\right)(1+ay{)}^{2n2}$$ showing that it will be negative for $$0<n<0.5$$ (anticyclones) and positive for $$n>0.5$$ (cyclones). The latter includes our quadratic topography with $$n=2/3$$. The $$n=1/2$$ case, not surprisingly, has $$\gamma =0$$ so that nonlinear steepening cannot balance dispersion. The $$n=1/4$$ topography, which supports anticyclones has a long, fairly flat shelf with a rapid dropoff (Fig. 1). The linear slope, $$n=1$$, used in QG models as an equivalent β no longer has $$\gamma =0$$ so that nonlinear effects will enter at long times.
For $$\psi ={\lambda}^{1/2}\varphi $$, the vorticity operator becomes
and our steadilypropagating equation becomes
The northsouth structure equation with $$g={\lambda}^{1/2}G\left(y\right)$$ is
In table 2, we use this to present forms of the topography which will have analytical solutions for G in the domain $$1/2\le y\le 1/2$$.
The values of B and k will be set by the positions of the walls such that G vanishes. We will have $$c<0$$ so that k is real. Special cases of the Bessel functions are included in the table. Higher order Bessel function or spherical Bessel function solutions will exist for integer or half integer n values, respectively. The sign of k also has to be chosen so that the solutions are oscillatory in the domain; for the Airy functions we need $$k{z}^{1/2}<0$$ so that the minus sign is required.
TWOD SOLITARY WAVES
The solitary wave solutions are not entirely satisfactory; one would like to consider a topography which,
has one solution $$g=\text{tanh}\left(y\right)$$ with $$c=d/2$$. The second solution is linear in y away from the topography
While this suggests we need a twoscale approach with y variation on scales comparable to the length of the solitary wave, we suspect this will still not lead to solutions. The difficulty is that the linear wave solutions which are bounded in y are not nondispersive in the long wave limit, but, as shown in the next section, have $$c\sim d/k$$. So the standard approach of starting with a large scale structure with weak dispersion and weak nonlinearity will not work.
Large amplitude solitons can have closed streamlines, in which case $$\mathcal{Q}\left(R+\psi /c\right)$$ is not determined by the value on the open streamlines. Adding PV anomalies in the interior may give solutions which decay in the flat areas (as would a point vortex), but would not prevent loss of energy by radiation since the speed would still resonate with some waves; c.f, Flierl & Haines (1994)FLIERL, G. & HAINES, K. 1994. The decay of modons due to Rossby wave radiation, Phys. Fluids 6, 34873497..
However, we can expect there will be isolated dipole solutions ("modons") moving eastward. This is apparent if we take d small so that we can use QG and an examine a dipole centered at $$y=0$$ with small enough scale that $$\lambda 1\simeq dy$$. The modons in the barotropic vorticity equation (Larichev & Reznik 1976LARICHEV, V. & REZNIK, G. 1976. Twodimensional Rossby soliton: an exact solution, Rep. U.S.S.R. Acad. Sci. 231, 10771079., Flierl et al. 1980FLIERL, G., LARICHEV, V., MCWILLIAMS, J. & REZNIK, G. 1980. The dynamics of baroclinic and barotropic solitary eddies, Dyn. of Atmos. and Oceans 5, 141.) will be good approximations to solutions.
ENVELOPE SOLITARY WAVES
We will use the hyperbolic tangent inverse topography which has a sloping bottom joining a shallow shelf to the deep ocean
as shown in Fig. 2. We will be using a multiscale expansion of the basic equation
Characteristics of the shelfslope topography for $$\lambda =H\left(0\right)/H=1+d\text{tanh}\left(y\right)$$. The bathymetry is shown as $$H\left(y\right)/H\left(\infty \right)$$.
assuming the packet is long in the x direction compared to the carrier wave and that the amplitude is small.
LINEAR SOLUTION
The linear problem has solutions
The y structure is defined by the eigenvectors of
The operator $${\mathcal{D}}_{1}$$ and its relatives
will appear in solving for the harmonics of the carrier wave. (We have already seen $${\mathcal{D}}_{0}$$ in section 3.)
The QG form simply replaces λ with 1 ; the solutions are
[dimensionally, $$c=\left(df/L\right)/\left({k}^{2}+k/L\right)$$ with L the topographic width and $$df/L$$ acting as β . The eigenfunctions $$g\left(y\right)$$ of the full (nonQG) equation (4), found numerically, are not symmetrical, being peaked in slightly deeper water and showing a more rapid decay offshore than onshore (Fig. 3). This is most noticeable when the ratio of the shelf and offshore depths $$\left(1d\right)/\left(1+d\right)$$ is small. This Wave structure $$g\left(y\right)$$ suggests that QG may be acceptable, given the simplifications aleady in the setup: the neglect of stratification and variations in slope and orientation of the topography. However, we will continue to use numerics.
$$g\left(y\right)$$ for $$k=0.5$$ and $$d=0.2$$ or 0.8 for the full equations and the QG approximation.
The phase speed, c , is also quite simlar to the QG result (Fig. 4); like β plane Rossby waves, these move westward, and the speed decays as $$1/{k}^{2}$$ for short waves. Long waves, however, have a crosstopography structure which widens with the wavelength, so that the average topographic β decreases and $$c\sim 1/k$$.
$$c\left(k\right)d=0.2$$ and $$0.8$$ for the full equations and the QG approximation (dashed).
Wave packets
A packet will travel at the group velocity; this is eastward and varies with the primary wavenumber (Fig. 5).
$${c}_{g}\left(k\right)d=0.2$$ and 0.8 for the full equations and the QG approximation (dashed).
But the envelope itself will change with time under linear dynamics since the packet has a peaked spectrum, and the different waves have slightly different $${c}_{g}$$ values. This evolution occurs on a timescale on the order of the time scale for the waves $$\left(1/\omega \right)$$ times the square of the ratio of packet to carrier wave scale.
The nonlinearity can enter in several ways: the linear solution does not exactly satisfy the nonlinear equations:
and this term will generate both harmonics and a packetscale flow. These interact with the primary wave to alter the group propagation speed and its shape.
In the multiscale approach, we define ϵ to be the ratio of the carrier wave scale to the packet width; then the streamfunction can be written as
and the various derivatives become
with the → indicating that these replacements are used in our dynamical equation (1) (divided by ϵ ). Then ψ is expanded in powers of ϵ , and we end up with a sequence of equations
with the definitions
ZEROTH ORDER: Equation (6) has the linear solution but with a spatially and temporarily varying amplitude
resulting in the eigenvalue/eigenfunction equation above (5) and the characteristics portrayed in Fig. 6.
y structures $$g,{g}_{1},{g}_{2},\partial {g}_{0}/\partial y$$ for $$k=1$$. The function $${g}_{0}$$ limits to linear slopes, positive (negative) for $$y\u2329\u23290,y\u232a\u232a$$: 0 . We plot the $$\partial {g}_{0}/\partial y$$ because that's all that enters into the coefficients below.
FIRST ORDER: The solvability equation, formed by multiplying (7) by the adjoint solution $$\text{exp}\left(\text{\u0131}kx\right)g\left(y\right)$$, averaging over x , and integrating over y , eliminates the left side.^{2} 2 Which is a solution to the zeroth order problem since cL0+β is selfadjoint. From the right side, we have
with $$\u27e8F\u27e9={\int}_{\mathrm{\infty}}^{\mathrm{\infty}}F(y)dy$$. We can verify that the group velocity matches the usual definition
by differentiating the equation (5) for the dispersion relation with respect to k
Again, projecting with g leads to
which is consistent with the expression above. Note that, as expected from the dispersion relation figure, the group velocity is positive so that the packet propagates in the opposite direction as the peaks and troughs.
The first order stream function takes the form
The Jacobian, $$J\left({\psi}_{0},\lambda {\mathcal{L}}_{0}{\psi}_{0}\right)$$ gives only second harmonic terms; these lead to
The $${A}_{1}$$ term arises because there is a nonresonant residual after substituting the expression for $${c}_{g}$$ in the linear terms; this gives
Finally the $${A}_{0}$$ , which is the packetscale flow, represents an x independent solution to the left side. Its amplitude and structure comes from the next order.
SECOND ORDER: The right side of (8) will have terms which do not depend on x ; these produce an equation for $${A}_{0}{g}_{0}$$:
We again project (8) by the adjoint to end up with the equation satisfied by the envelope
with
The calculations are tedious, and details are relegated to ^{Appendix A} APPENDIX A: ALGEBRAIC DETAILS Here, we give the termbyterm decomposition of the righthand side of the second order equation (8) c L 0  β ψ 2 , x =  c L 1 ψ 1 , x  c L 2 ψ 0 , x  c g L 0 ψ 1 , x  c g L 1 ψ 0 , x + β ψ 1 , x + 1 c J x ψ 0 λ β ψ 0 + 1 c J ψ 1 , λ β ψ 0 + J ψ 0 , λ L 0 ψ 1 + J ψ 0 , λ L 1 ψ 0 + β c ψ 0 , T (with subscripts indication partial derivatives). We can split each term into the contribution to eikx or the x independent part. The former terms will be resonant and their sum, projected on expıkxgy gives the solvability condition, while the latter will determine A0.  c L 1 ψ 1 , x = 12 k A x x λ g 1 e i k x  c L 2 ψ 0 , x =  1 k c A x x λ g e i k x  c g L 0 ψ 1 , x =  1 c g c k A x x D 1 g 1 e i k x  c g  A  X 2 D 0 g 0  c g L 1 ψ 0 , x =  2 i k c g A x x λ g e i k x + β ψ 1 , x = 1 1 c k A x x β g 1 e i k x +  A  x 2 β g 0 + 1 c J x ( ψ 0 λ β ψ 0 ) = 1 c  A  x 2 ( λ β ) y g 2 + 1 c J ( ψ 1 , λ β ψ 0 ) = ∣ k c  A  2 A [ 1 c 2 ( β λ g g 2 ) y  β λ g g 0 , y ] e i k x  1 c 2  A  X 2 ( β λ g g 1 ) y + J ( ψ 0 , λ L 0 ψ 1 ) = ∣ k 2 c 2  A  2 A [  g ( λ D 2 g 2 ) y  2 g y λ D 2 g 2 + 2 c 2 g ( λ D 0 g 0 ) y ] e i k x + 1 c  A  x 2 ( g λ D 1 g 1 ) y + J ( ψ 0 , λ L 1 ψ 0 ) = 2 k 2  A  x 2 ( λ 2 g 2 ) y β c ψ 0 , T = 1 c A T β g e i k x The zero mode terms are all proportional to Ax2 and determine g0  c g D 0 g 0 + β g 0 + γ c g 2  1 c 2 β λ g g 1 y + 1 c g λ D 1 g 1 y + 2 k 2 λ 2 g 2 y = 0 Multiplying the eikx terms by the adjoint and integrating leads to A T g 2 c + ı k A x x 2 g λ g 1  c λ g 2  c g c k 2 g D 1 g 1  2 λ g 2 + 1 c k 2 β g g 1 + ι k c  A  2 A 1 c 2 g β λ g g 2 y  β λ g g 0 , y  1 2 c g 2 λ D 2 g 2 y  1 c g g y λ D 2 g 2 + c g 2 λ D 0 g 0 y = 0 We can do a few integrations by parts (noting that ∂∂yF=0 and find A T β g 2 c + ı A x x 2 g λ g 1  c λ g 2  c g c 2 k 2 g β g 1  2 c g λ g 2 + 1 c k 2 β g g 1 + ı k c  A  2 A 1 2 c 2 γ g 2 g 2  β λ g 2 g 0 , y  2 c g g y λ D 0 g 0 = 0 Expressing this as f 1 A T + i k f 2 A X X + i k f 3  A  2 A = 0 gives the values of fj in the text and leads to the sech solution. .
Solitary packet
Equation (11) is the nonlinear Schrödinger equation and has solitary solutions in which the packet dispersion is balanced by nonlinear steepening or flattening.
with the condition that $${f}_{2}/{f}_{3}>0$$. Fig. 7 shows K and $$\text{\Omega}$$ which determine the scale and change in propagation of the packet. We can regard the latter as a change in the carrier wave, which now looks like
Amplitudedependent packet $$K/\mathcal{A}=\sqrt{{f}_{3}/2{f}_{2}}$$ setting the inverse size and $$\text{\Omega}/{\mathcal{A}}^{2}$$ setting the frequency shift.
Since both $$\omega $$ and $$\text{\Omega}$$ are positive, the carrier wave moves slightly faster to the west, while the packet is propagating to the east.
Larger amplitude waves will both be associated with a shorter packet and a larger speedup of the carrier wave. Unlike the solitary Rossby waves in a channel (MalanotteRizzoli 1980MALANOTTERIZZOLI, P. 1980. Solitary Rossby waves over variable relief and their stability. Part II. Numerical experiments., Dyn. Atmos. Oceans 4, 261294.) which exist for the long waves with westward group velocity (and in the regime where the carrier wave is unstable to modulational perturbations, Plumb, 1977), these travel in the opposite direction as the carrier wave. But, from the nonlinear Schrödinger equation, modulation instability can indeed occur when $${f}_{2}/{f}_{3}>0$$, so the isolated packets can form from an longer wave train.
DISCUSSION
We have shown that the topographic betaeffect can support either cyclonic solitons or, for topogra phies with $${\partial}^{2}\left(1/{h}^{2}\right)/\partial {y}^{2}$$ negative, anticyclones. In the example, this topography has weak slope until near the outer boundary where it drops suddenly. However, the need for boundaries seems to make these solutions less useful than they might otherwise be.^{3} 3 Although a solution with just a boundary at the coast would be of interest. A model with stratification, on the other hand, could lead to more relevant solutions; unlike the barotropic model here, the long waves with limited crosstopography structure can become nondispersive (Allen 1975ALLEN, J. 1975. Coastal trapped waves in a stratified ocean, J. Phys. Oceanogr. 5, 300325.), opening up the possibility of interesting solitary solutions, However, it is likely that the barotropic mode will be excited on flat areas, so that multiple scales will appear in y ; we are currently studying this problem.
The nonlinear barotropic wave packets, on the other hand, have the nice feature that they can be confined over the topographic slope. Unlike betaplane analogues, the group velocity is opposite in direction to the phase speed, This may depend on the specifics of the topography, which can, as we saw in the solitary wave discussion, change the sign of the group velocity. But for the topography with a limited region of slope joining flat regions, energy will be moving with the shallow region to its left (in the northern hemisphere), with the phase translating in the opposite direction.
Although we have used various topographic shapes, the qualitative results will not be terribly sensitive to the exact structure. There are two requirements for these nonlinear solutions to exist. First the linear problem $$(\nabla \cdot \lambda \nabla \psi =\beta \psi /c$$ with either just the crosstopographic derivatives for solitary waves or both for the groups) must have a wellbehaved solution. Secondly, for solitary waves, the projection of the nonlinear terms on the lowest order crosstopographic structure must be nonzero. For groups, the requirement $${f}_{1}/{f}_{2}>0$$ is more difficult to evaluate.
Nonlinearity will be important when the steepness $$\leftu\right/c$$ is not very small. Observations, (e.g., (Johns & Watts 1986JOHNS, W. & WATTS, D. 1986. Time scales and structure of topographic Rossby waves and meanders in the deep Gulf Stream, J. Mar. Res. 44, 267290.)) suggest this ratio is indeed order one. It seems likely that the western boundary currents and associated eddies could provide a disturbance, but the solitary wave could be disrupted by encountering changes in the topographic slope or direction as well as by additional forcing before it really settles. However the calculations herein suggest that linear wave theory is only part of the story.
APPENDIX A: ALGEBRAIC DETAILS
Here, we give the termbyterm decomposition of the righthand side of the second order equation (8)
(with subscripts indication partial derivatives). We can split each term into the contribution to $${e}^{ikx}$$ or the x independent part. The former terms will be resonant and their sum, projected on $$\text{exp}\left(\text{\u0131}kx\right)g\left(y\right)$$ gives the solvability condition, while the latter will determine $${A}_{0}$$.
The zero mode terms are all proportional to $$A{}_{x}^{2}$$ and determine $${g}_{0}$$
Multiplying the $${e}^{ikx}$$ terms by the adjoint and integrating leads to
We can do a few integrations by parts (noting that $$\left.\frac{\partial}{\partial y}F=0\right)$$ and find
Expressing this as
gives the values of $${f}_{j}$$ in the text and leads to the sech solution.

1
The QG form will be the same except $$\zeta ={\nabla}^{2}\psi $$ and $$\gamma ={\partial}^{2}\lambda /\partial {y}^{2}$$; for the sample topographies in table 1 , the values can be quite different from our finitetopography case $${\mathcal{D}}_{0}\beta /{c}_{0}$$ is selfadjoint. The result is

2
Which is a solution to the zeroth order problem since $$c{\mathcal{L}}_{0}+\beta $$ is selfadjoint.

3
Although a solution with just a boundary at the coast would be of interest.
ACKNOWLEDGMENTS
It is a pleasure to contribute to a volume honoring Affonso Mascarenhas. He was a student working with Prof. MolloChristensen, and I served on his thesis committee. His love for the ocean was obvious, as was his realization of oceanography's importance for Latin and South America. Comments by the reviewers and Margaret Gregory helped improve the paper.
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Publication Dates

Publication in this collection
10 Oct 2022 
Date of issue
2022
History

Received
04 May 2022 
Accepted
31 Aug 2022