Nonlinear Disintegration of the Internal Tide

Karl R. Helfrich Department of Physical Oceanography, Woods Hole Oceanographic Institution, Woods Hole, Massachusetts

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Roger H. J. Grimshaw Department of Mathematical Sciences, Loughborough University, Loughborough, Leicester, United Kingdom

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Abstract

The disintegration of a first-mode internal tide into shorter solitary-like waves is considered. Since observations frequently show both tides and waves with amplitudes beyond the restrictions of weakly nonlinear theory, the evolution is studied using a fully nonlinear, weakly nonhydrostatic two-layer theory that includes rotation. In the hydrostatic limit, the governing equations have periodic, nonlinear inertia–gravity solutions that are explored as models of the nonlinear internal tide. These long waves are shown to be robust to weak nonhydrostatic effects. Numerical solutions show that the disintegration of an initial sinusoidal linear internal tide is closely linked to the presence of these nonlinear waves. The initial tide steepens due to nonlinearity and sheds energy into short solitary waves. The disintegration is halted as the longwave part of the solution settles onto a state close to one of the nonlinear hydrostatic solutions, with the short solitary waves superimposed. The degree of disintegration is a function of initial amplitude of the tide and the properties of the underlying nonlinear hydrostatic solutions, which, depending on stratification and tidal frequency, exist only for a finite range of amplitudes (or energies). There is a lower threshold below which no short solitary waves are produced. However, for initial amplitudes above another threshold, given approximately by the energy of the limiting nonlinear hydrostatic inertia–gravity wave, most of the initial tidal energy goes into solitary waves. Recent observations in the South China Sea are briefly discussed.

Corresponding author address: Karl R. Helfrich, Department of Physical Oceanography, Woods Hole Oceanographic Institution, Woods Hole, MA 02543. Email: khelfrich@whoi.edu

Abstract

The disintegration of a first-mode internal tide into shorter solitary-like waves is considered. Since observations frequently show both tides and waves with amplitudes beyond the restrictions of weakly nonlinear theory, the evolution is studied using a fully nonlinear, weakly nonhydrostatic two-layer theory that includes rotation. In the hydrostatic limit, the governing equations have periodic, nonlinear inertia–gravity solutions that are explored as models of the nonlinear internal tide. These long waves are shown to be robust to weak nonhydrostatic effects. Numerical solutions show that the disintegration of an initial sinusoidal linear internal tide is closely linked to the presence of these nonlinear waves. The initial tide steepens due to nonlinearity and sheds energy into short solitary waves. The disintegration is halted as the longwave part of the solution settles onto a state close to one of the nonlinear hydrostatic solutions, with the short solitary waves superimposed. The degree of disintegration is a function of initial amplitude of the tide and the properties of the underlying nonlinear hydrostatic solutions, which, depending on stratification and tidal frequency, exist only for a finite range of amplitudes (or energies). There is a lower threshold below which no short solitary waves are produced. However, for initial amplitudes above another threshold, given approximately by the energy of the limiting nonlinear hydrostatic inertia–gravity wave, most of the initial tidal energy goes into solitary waves. Recent observations in the South China Sea are briefly discussed.

Corresponding author address: Karl R. Helfrich, Department of Physical Oceanography, Woods Hole Oceanographic Institution, Woods Hole, MA 02543. Email: khelfrich@whoi.edu

1. Introduction

The generation of long internal solitary waves from barotropic tidal flow over localized topography has received much attention over the last few decades owing to the ubiquity of these waves in coastal seas and over continental shelves. The usual description involves a downstream lee wave that is released from the topography as the tide turns (e.g., Maxworthy 1979). Nonlinearity leads to steepening. Breaking is prevented by nonhydrostatic dispersion, and the balance between nonlinearity and dispersion results in the generation of a rank-ordered packet of solitary waves. Other possible mechanisms include upstream blocking (Lee and Beardsley 1974), transcritical generation (Grimshaw and Smyth 1986; Melville and Helfrich 1987), and the scattering of an internal tide beam at the base of the surface mixed layer (Gerkema 2001). Which of these mechanisms occurs depends on the details of the barotropic flow, stratification, and topography. The focus here is on the lee wave release, or more generally the radiation of a low-mode internal tide, which subsequently steepens and disintegrates.

A time scale for the emergence of the lead solitary wave from a localized disturbance has been developed by Hammack and Segur (1978) for single-layer flows [see Helfrich and Melville (2006) for the two-layer extension] based on weakly nonlinear Korteweg–de Vries (KdV) theory. The emergence time scale depends on the amplitude and length scale of the initial disturbance. Provided that the length scale is short compared to the internal deformation radius and that the amplitude is not too large, this estimate should be valid. However, when the initial disturbance length scale is comparable to the internal deformation radius, as is the case for a radiating internal tide, or if the emergence time scale is comparable to the local inertial period, the neglected effects of rotation may be important.

Individual solitary waves, because they are short compared to the deformation radius, are typically considered to be unaffected by rotation. However, even weak rotation eliminates permanent form solitary wave solutions in KdV-type theories (Leonov 1981). An initial solitary wave will decay by radiation damping due to resonance with long inertia–gravity (Poincaré) waves (Melville et al. 1989; Grimshaw et al. 1998a, b). The damping can be significant if the solitary wave propagates for a time comparable to the local inertial period. Continued propagation may result in the steepening of the inertia–gravity wave to produce a secondary, growing solitary-like wave behind the original one and the formation of “quasi-cnoidal” wave packets (Helfrich 2007).

The propagation of an internal tide, from which the solitary waves may emerge, is fundamentally affected by rotation at all times. The role of rotation in the disintegration of an internal tide into weakly nonlinear solitary waves has been considered by Gerkema and Zimmerman (1995), Gerkema (1996), Holloway et al. (1999), New and Estaban (1999), and Grimshaw et al. (2006). The first two studies are the most relevant. They undertook numerical studies based on the weakly nonlinear Boussinesq equations with rotation (bidirectional propagation versions of KdV). The numerical results showed that, unless the barotropic tidal flow over the topography produced an initial disturbance with sufficient nonlinearity, the disintegration of the internal tide into higher frequency solitary waves was inhibited. The radiated internal tide remained a coherent long tide. They argued that the coherence was due to the presence of long, weakly nonlinear periodic inertia–gravity wave solutions to the hydrostatic limit of the model. In contrast to solitary waves, which require a balance between nonlinearity and nonhydrostatic dispersion, the periodic inertia–gravity waves arise from the balance of nonlinearity with rotational dispersion. These nonlinear solutions were first found by Ostrovsky (1978) for the hydrostatic limit of the rotationally modified KdV equation. The hydrostatic solutions exist only up to a limiting amplitude, beyond which the rotational dispersion is not able to balance the nonlinearity. This led Gerkema (1996) to propose a threshold argument for the disintegration of the tide based on the nonlinearity of the initial tide (given by a measure of the forcing strength related to the barotropic tide, stratification, and topographic slope). If the initial amplitude (i.e., nonlinearity) was above the limiting amplitude, the nonlinear tide solutions were not generated. Instead, the tide would disintegrate with most of the initial tide evolving rapidly into shorter solitary waves. If less than the threshold, disintegration was strongly inhibited. They also noted that, for a given forcing amplitude and frequency, increasing rotation (i.e., latitude) inhibited the disintegration of the internal tide. New and Estaban (1999) noted that increasing the tidal frequency had the same effect as a decrease in latitude.

The goal of this paper is to explore further the role of rotation in the evolution of the low-mode internal tide and the production of shorter, nonhydrostatic solitary waves. This process can be viewed as a competition between high-frequency nonhydrostatic dispersion and low-frequency rotational dispersion to balance the nonlinear steepening of the tide. However, numerous recent observations show that both tides and solitary waves are quite large in many places (e.g., Stanton and Ostrovsky 1998; Ramp et al. 2004) so that a weakly nonlinear theory may be inadequate. Thus, the restriction to weak nonlinearity will be eliminated while retaining the assumption of weak nonhydrostatic effects in a two-layer model. The inclusion of full nonlinearity has the added benefit that the properties of the hydrostatic, nonlinear inertia–gravity waves that appear to play a substantial role the disintegration of the internal tide can be significantly different when compared to the weakly nonlinear models. Furthermore, no assumption about the strength of the rotational effects, as required by unidirectional KdV-type theories, is necessary.

The paper is organized as follows. Section 2 introduces the mathematical model for a rotating, two-layer flows with O(1) nonlinearity and weak nonhydrostatic effects. Properties of the solitary wave solutions to these equations in the absence of rotation are given. The nonlinear, hydrostatic inertia–gravity wave solutions are found and discussed. In section 3 numerical solutions to the full equations are explored for sinusoidal linear inertia–gravity wave (i.e., tide) initial conditions. In section 4 the results are summarized and recent observations of large-amplitude internal tides and solitary waves in the South China Sea are discussed.

2. The model

The situation under consideration is an inviscid, two-layer fluid with layer depths hi and velocity vectors ui. Here i = 1 and 2 refer to the upper and lower layers, respectively. The layer densities are ρ1 and ρ2 = ρ1 + Δρ. The system is rotating about the z axis with constant Coriolis frequency f (> 0). The gravitational acceleration g is directed in the negative z direction. In the absence of motion h1 = h0 and h2 = Hh0, where H is the total depth. The bottom is flat and the upper surface is rigid.

The propagation of nonlinear interfacial waves in this system will be studied in the limit of fully nonlinear, α = a/hs = O(1), and weakly nonhydrostatic long waves, β = (hs/l)2 ≪ 1. Here α and β are the usual parameters describing nonlinear and nonhydrostatic effects, respectively. The scaling depth hs is a measure of the vertical scale of the waveguide (taken below to be the total depth H) and a and l are, respectively, the wave amplitude and length scales. The KdV theory requires β = O(α) ≪ 1. No restriction is placed on the relative magnitude of the rotational effects. In the absence of rotation a set of equations for fully nonlinear, weakly nonhydrostatic interfacial waves has been developed by Miyata (1988) and Choi and Camassa (1999) (hereafter the MCC equations). The extension of the MCC theory to include rotation is discussed in (Helfrich 2007), so only the resulting equations are given below.

The waves will be taken to propagate in the x direction. With rotation, motion in the transverse y direction will occur; however, the flow will be taken to be independent of y. The continuity and vertically averaged x and y momentum equations for each layer are, respectively,
i1520-0485-38-3-686-e1
i1520-0485-38-3-686-e2
i1520-0485-38-3-686-e3
Here the overbar indicates a vertical average over a layer; ui and υi are, respectively, the velocities in the x and y directions; η(x, t) = h0h1 is the interface displacement; and P(x, t) is the pressure at the interface. The subscripts t and x indicate partial differentiation. The O(β) nonhydrostatic terms Di are (Choi and Camassa 1999)
i1520-0485-38-3-686-e4
In the weakly nonlinear limit (α ≪ 1),
i1520-0485-38-3-686-e5

Note that the nonhydrostatic terms (4) are independent of rotation even though no restriction has been placed on the relative strength of the rotation. This follows from the requirement that β ≪ 1 and the assumption that at t = 0 the lowest-order horizontal velocities (from the series ui = u(0)i + βu(1)i + · · ·) are independent of z. It is then straightforward to show from the vorticity equation that both u(0)iz and υ(0)iz then remain zero. Physically, the tilting of the planetary vorticity in the O(β) vertical shear (i.e., production of horizontal vorticity) does not alter the lowest-order horizontal vorticity. Therefore, u(0)i = u(0)i(x, t) and υ(0)i = υ(0)i(x, t). Choi and Camassa show that the O(β) nonhydrostatic terms are functions of u(0)i = ui + O(β) and thus are independent of rotation.

Equations (1)(3) are nondimensionalized using g′H, H, l, and l/g′H for (ui, i), hi, x, and t, respectively. The reduced gravity g′ = gΔρ/ρ1. The equations are simplified by eliminating Px between (2) with i = 1 and 2. The Boussinesq approximation Δρ/ρ1 ≪ 1 is also employed since it is reasonable in the oceanic context and consistent with the rigid-lid assumption. The nondimensional, rotating MCC equations (referred to as MCC-f ) are
i1520-0485-38-3-686-e6
i1520-0485-38-3-686-e7
i1520-0485-38-3-686-e8
i1520-0485-38-3-686-e9
Here s = u2u1 and υ = 21 are the vertical jumps between the layers of the horizontal velocities. Note that η (=h0h1) has been eliminated in favor of h1 and h2 = 1 − h1. The barotropic transports in the x and y directions are, respectively,
i1520-0485-38-3-686-e10
With the rigid lid U = U(t) must be specified; in (6)(9) it has been set to U = 0. The nonhydrostatic terms Di are given by (4) with u1 = s(h1 − 1) and u2 = sh1.
The two parameters that appear in (6)(9) are
i1520-0485-38-3-686-e11
where the scaling depth hs is set to H in β. The internal deformation radius LR = g′H/f. The parameter γ, essentially the inverse of a Rossby number, measures the relative effects of rotation. As discussed above, γ is not restricted to be small. This is particularly important in the present context since the internal tide has lLR [or ω/f = O(1), where ω is the frequency]. This is to be contrasted with the weakly nonlinear models of Ostrovsky (1978) and Gerkema and Zimmerman (1995), where rotation was taken to be weak and comparable to nonlinear effects. However, Tomasson and Melville (1992) derive a weakly nonlinear model [their Eqs. (2.19)–(2.21)] with no restriction on the rotation, as do New and Estaban (1999).
The MCC-f equations can be manipulated to obtain an energy equation (following Choi and Camassa 1999; Helfrich 2007). The total energy between x1 and x2 is
i1520-0485-38-3-686-e12

a. γ = 0: Nonrotating internal solitary waves

With no rotation, γ = 0, and β finite, (6) and (7) reduce to the (nonrotating) MCC equations. If υ = V = 0 at t = 0, they remain zero from (8) and (9). Solitary waves solutions to (6) and (7) of the form η = η(ξ), where ξ = xct, can then be found from (Miyata 1985, 1988; Choi and Camassa 1999)
i1520-0485-38-3-686-e13
with s(ξ) given by
i1520-0485-38-3-686-e14
The solitary wave phase speed
i1520-0485-38-3-686-e15
and η0 is the wave amplitude. For η0 → 0, c → ±c0, where
i1520-0485-38-3-686-e16
is the linear two-layer longwave phase speed.

As in weakly nonlinear theory, the solitary waves point into the deeper layer so that η0 < 0 (> 0) for h0 < 0.5 (> 0.5). There are no solitary wave solutions for h0 = 0.5. Solitary waves are limited to a maximum amplitude, η0max = h0 − 0.5, that reaches middepth. This limiting wave has infinite wavelength and is a smooth, dissipationless transition between two uniform (conjugate) states. These “table top” solitary waves and conjugate states do not occur in the KdV equation, but do appear when the cubic nonlinearity is included in the KdV-type models (Kakutani and Yamasaki 1978; Helfrich and Melville 2006), though with a different limiting amplitude. The MCC solitary wave properties, including the limiting conjugate state amplitude, agree quite well with fully nonlinear and nonhydrostatic theories, numerical calculations, laboratory experiments, and oceanic observations (e.g., Choi and Camassa 1999; Michallet and Barthélemy 1998; Ostrovsky and Grue 2003; Camassa et al. 2006).

The MCC equations, unlike their weakly nonlinear counterparts, do not filter out Kelvin–Helmholtz instability (Jo and Choi 2002). Linearization of (6)(9) about an initial uniform velocity jump s0 shows that for all s0 > 0 the flow is unstable for wavenumbers above a threshold. As a consequence, the velocity jump induced by a finite-amplitude solitary wave may lead to high-wavenumber instability. Numerical solutions of the MCC equations show that, if the grid resolution is too fine, unstable short waves first emerge near the wave crest and ultimately overwhelm the calculations (Jo and Choi 2002). The instability can be controlled in some cases by filtering out wavenumbers above a threshold (W. Choi 2007, personal communication).

b. β = 0: Hydrostatic nonlinear internal inertia–gravity (tidal) waves

In the limit β = 0 with γ finite, (6)(9) are simply the two-layer shallow-water equations, which do not have solitary wave solutions. However, periodic, finite-amplitude inertia–gravity wave solutions can be found (Plougonven and Zeitlin 2003). These solutions are the two-layer versions of the single-layer nonlinear inertia–gravity waves found by Shrira (1986). As in the weakly nonlinear limit, these periodic solutions arise from a balance between nonlinearity and low-frequency rotational dispersion. The analysis of the two-layer waves follows Plougonven and Zeitlin (2003); however, new properties of the solutions are discussed with an emphasis on them as models of the internal tide.

We seek steadily propagating periodic solutions of (6)(9) with β = 0 and the form of h1(ξ), where ξ = xct and c is the phase speed. Thus, the continuity equation (6) gives, after integration in x and requiring that s → 0 as h1h0,
i1520-0485-38-3-686-eq1
Using this result in (7) with β = 0 gives
i1520-0485-38-3-686-e17
where
i1520-0485-38-3-686-e18
Equations (1)(3) with β = 0 conserve the potential vorticity (scaled by f /H) in each layer,
i1520-0485-38-3-686-e19
With the potential vorticity set to the background value in each layer, q1 = h−10 and q2 = (1 − h0)−1, (19) can be used to find
i1520-0485-38-3-686-e20
This choice for qi is not arbitrary since (20) can be found directly from (8) and (9).
Using (20) to eliminate υ in (17) gives
i1520-0485-38-3-686-eq2
Multiplication by Fξ and integration gives
i1520-0485-38-3-686-e21
where D is a constant of integration, F′ = dF/dh1, and
i1520-0485-38-3-686-e22
For periodic waves continuity requires
i1520-0485-38-3-686-eq3
where λ is the wavelength. This implies that h1 oscillates between hm and hM, where 0 ≤ hm < h0 < hM ≤ 1. Discussion of solutions of (21) will be limited to h0 ≤ 0.5. Because of the symmetry inherent in the equations with the Boussinesq approximation, solutions for h0 > 0.5 can be obtained by taking h0 to be the mean depth of the lower layer and h2 given by the solution h1(ξ) of (21). The point where h1 = hm (hM) will be referred to as the trough (crest) of the wave, even though for h0 < 0.5 these points correspond to the maximum and minimum interfacial displacement η = h0h1, respectively. Similarly, hMh0 will be termed the wave amplitude since, as will be shown below, hMh0h0hm.
The nature of periodic solutions to (21) depends critically on F′(h1). Smooth solutions require F′(h1) ≠ 0 in hmh1hM. Should F′(h1) = 0 in this range, then h1x → ∞. For smooth periodic solutions h1x = 0 at h = hm and hM. This sets the constant of integration
i1520-0485-38-3-686-eq4
from which a relation for c can be found using (22),
i1520-0485-38-3-686-e23
It follows from (21) that Dg(h1) ≥ 0 for hmh1hM. For smooth solutions F′(h1) > 0 and, since hmh0hM, F′(h0) > 0. From (18) F′(h0) = c2/c20 − 1; therefore, c2 > c20.
If F′(h1) = 0, then the first zero of this function sets the upper bound for hM. As solutions approach this limit, they take on a “corner wave” shape, similar to what is found in the weakly nonlinear theory (Ostrovsky 1978; Grimshaw et al. 1998b) and the fully nonlinear single-layer model (Shrira 1986). Whether this limiting corner wave occurs (for a given h0) depends on c. From (18)
i1520-0485-38-3-686-e24
The minimum of F′ occurs at
i1520-0485-38-3-686-eq5
and F′(ĥ) = 0 for c = c*, where
i1520-0485-38-3-686-e25
The behavior of F′(h1) is illustrated in Fig. 1; F′(ĥ) > 0 (≤ 0) for c > c* (≤ c*). Thus, for c < c* a limiting corner wave occurs with the upper bound on hM given by the first zero of F′(h1).

These two types of waves are illustrated in Fig. 2. The scaling length l has been set to LR, giving γ = 1 and time scaled by f−1. The upper panel shows the interface η(ξ) = h0h1 over one wavelength for h0 = 0.25 and frequency ω = 1.4 (= ω/f in dimensional variables), found by numerical integration of (21). As the wave amplitude increases, the profiles take on a square, or lobate, shape but remain smooth. The maximum wave for this frequency is hM = 0.5504. The wave profiles are asymmetric with hMh0h0hm. The corner wave class is illustrated in the lower panel, where ω = 2. The crest becomes increasingly peaked until the limiting wave, at hM = 0.3061, is reached. The solution can then be constructed as a sequence of arcs in analogy with the sequence of parabolic arcs found in the weakly nonlinear theory (Ostrovsky 1978; Grimshaw et al. 1998b). Construction of these families with fixed ω is discussed below.

A convenient way to illustrate the solutions is to consider where in the hmhM plane, for a given h0, periodic solutions occur. An example is shown in Fig. 3a for h0 = 0.25. Periodic waves can be found everywhere within the indicated lines, which represent different limits of the possible wave speeds c. These boundaries are determined as follows: The limit c → ∞ is found by setting the denominator of (23) to zero. This gives a relation between hm and hM, shown by the lowest curve in Fig. 3a, and provides a lower bound on hM, given hm. A portion of the upper bound can be deduced from Fig. 1. To find solutions with hMĥ requires cc*. Thus, setting c = c* in (23) and solving the resulting equation for hM(hm) gives the curve extending from hM = 1 to the small solid circle where hM = ĥM = ĥ. The associated value of hm is denoted by ĥm. No solutions with c < c* and hm < ĥm are possible since then the first zero of F′(h1) occurs for h1 < ĥM. Thus, for hmĥm(hMĥM) the wave speed c*c ≤ ∞.

The c* curve extends to hm = h0; however, for hm > ĥm this curve no longer defines the upper bound on hM. In this range the upper bound on the wave speed, clim (c0climc*) is given by the first zero of F′(h1) from (24). These zeros give hM = hM(c), and (23) gives the corresponding hm. This curve is indicated by c = clim(hm) in Fig. 3a. Thus, for a fixed hm > ĥm, the waves approach a limiting corner wave shape as hM increases.

Figure 3b shows the lower-right portion of the hmhM plane for h0 = 0.25 with the loci of waves with several values of frequency ω = ck, where k = 2π/λ is the wavenumber. For ω = 2 (the semidiurnal tide at 30°), the maximum amplitude wave is a corner wave with c = 0.5088 at (hm, hM) = (0.2274, 0.3061). For the lower frequency ω = 1.4 (the semidiurnal tide at 44°) the amplitude is limited by the approach to c = c* = 0.5359 at (hm, hM) = (0.1475, 0.5504). Here the waves take on the lobate shape. The division between the cases occurs for ω = 1.796. The waves shown in Fig. 2 fall along these two lines. It is interesting that subinertial, ω < 1, waves can be found. They are in the lobate class and have large amplitudes. The dashed line shows the solutions with λ = 2.78, the wavelength of linear waves with ω = 1.4 from the linear inertia–gravity dispersion relation ω2 = γ2 + c20k2 with γ = 1.

The wavelength is found from the integration of (21),
i1520-0485-38-3-686-e26
The denominator is zero at both limits, so numerical integration employs a Taylor expansion at each end. Waves of a particular frequency or wavelength are found from tabulating the properties of solutions within the allowable hM range.

The effect of changing the stratification is shown in Fig. 4, where the regions of periodic solutions are plotted for several values of h0. Also shown for each case is the locus of solutions with ω = 1.4. As h0 decreases, the range of ω = 1.4 solutions decreases and the limiting wave changes from the lobate class to the corner wave. As in Fig. 3b, all solutions with ω > 1.4 fall to the right (with smaller amplitudes) of this curve. It is also interesting that for h0 = 0.5 the waves are symmetric. They all fall along the line hm = hM and are in the lobate class. The maximum amplitude wave with ω > 1.4 is indicated by the square.

Last, the dependence of the phase speed c on the amplitude, hMh0, for ω = 1.4 is plotted in Fig. 5. Finite amplitude can either increase or decrease c compared to the linear speed c0f = c0(1 − ω−2)−1/2, and the amplitude dispersion does not have to be monotonic.

c. Finite β and γ

When both nonhydrostatic effects and rotation are present, there are no known analytical solutions to the MCC-f Eqs. (6)(9). Numerical solutions have so far only examined the effect of rotation on the evolution of an initial nonrotating MCC solitary wave from (13) (Helfrich 2007). As in earlier weakly nonlinear studies (Grimshaw et al. 1998a), it was found that the solitary wave decayed by radiation of a longer inertia–gravity wave. On longer time scales, the inertia–gravity wave itself steepened to produce a secondary, growing solitary-like wave. This decay and reemergence process is repeated until a nearly localized wave packet is formed, consisting of a long envelop through which a train of faster solitary-like, or quasi-cnoidal, waves propagate.

Initial conditions given by a long, hydrostatic inertia–gravity wave, either a linear sinusoidal wave or one of the fully nonlinear solutions just described, have not been explored and are the focus of this study. In particular, parameters relevant for the propagation and possible disintegration of an internal tide will be considered.

In what follows the term inertia–gravity (IG) wave will refer to the long waves (from the previous section) that are fundamentally hydrostatic and are controlled by rotation, even though it is possible to have nonhydrostatic inertia–gravity waves. This nomenclature distinguishes these essentially hydrostatic, rotational waves from the shorter solitary-like waves that are controlled by nonhydrostatic effects and are only weakly affected by rotation.

3. Numerical solutions

To organize the presentation of results the length scale l will be set to LR in (11), giving γ = 1 and β = (H/LR)2 = Hf2/g′ in (6)(9). This choice is arbitrary and it is possible to take l = H, giving β = 1 and γ = H/LR. In either case, the sole remaining parameter is H/LR. Conversion between them amounts to a rescaling of x and t. In the former scaling the long, hydrostatic IG waves have lengths of O(1) and the nonhydrostatic, solitary-like waves are an order of magnitude shorter. Typical midlatitude values of H = 100–3000 m (coastal to deep ocean), f ≈ 10−4 s−1, and g′ ≈ 0.01 − 0.05 m s−2 give β ≈ (0.2–30) × 10−4. Thus, β1/2 = H/LR = (0.5–5) × 10−2 is representative of oceanic conditions.

a. Methods

The MCC-f Eqs. (6)(9) are solved with the numerical method developed by Wei et al. (1995) for a set of wave equations closely related to the nonrotating single-layer version of the MCC equations. The method uses centered, fourth-order finite differences for all x derivatives, except those in Di which are differenced using centered, second-order stencils. Temporal integration is a fourth-order Adams–Bashforth–Moulton predictor–corrector scheme with iteration on the corrector step. Only minor modifications of the scheme are necessary for the MCC-f equations. The scheme was successfully tested by checking solitary wave propagation in the absence of rotation. The solitary wave instability (Jo and Choi 2002) was found in some cases, and could be controlled by the periodic removal of high wavenumbers (W. Choi 2007, personal communication).

Solutions of the nonhydrostatic MCC-f equations are compared to numerical solutions in the hydrostatic limit β = 0. In this limit, though, the numerical scheme above fails if wave breaking occurs. Thus, for the hydrostatic runs the nonoscillatory central shock–capturing scheme of Jiang and Tadmor (1998) is used.

The numerical domain is periodic in x. The resolution in x varies from 600 to over 1000 grid points to ensure that any short, solitary-like waves are properly resolved.

b. Results

The theoretical nonlinear IG waves (nonlinear tides) are quite robust to weak nonhydrostatic effects. Figure 6a shows the evolution of a large-amplitude nonlinear tide solution of the lobate class for h0 = 0.25 and λ = 2.8 with hm = 0.18 (hM = 0.4229 and c = 0.6162) after eight periods of propagation for β1/2 = 0.02 and 0.04. The only appreciable effect of the nonhydrostatic terms is a weak decrease of the phase speed with increasing β. The wave shapes and amplitudes are essentially unchanged. A similar calculation for a corner-class wave is shown in Fig. 6b for h0 = 0.1, λ = 2.024, hm = 0.085, hM = 0.139, and c = 0.4513. This initial wave is quite close to the limiting corner wave at hM = 0.1401 for these values of h0, hm, and λ. Again, the weak nonhydrostatic dispersion leads to a decrease in wave speed that is slightly more pronounced than the previous example. The wave amplitude is affected with increasing β, leading to a smoothing of the wave crest and a small reduction of the amplitude. The corner shape of the wave induces the enhanced sensitivity to nonhydrostatic effects. Still, the overall wave shape is stable. The balance between nonlinearity and rotational dispersion that produces the nonlinear IG wave solutions is not broken by weak nonhydrostatic effects. These two examples are representative of other cases with similarly small, oceanographically realistic, values of β.

Because of their stability to weak nonhydrostatic effects, the fully nonlinear hydrostatic IG waves likely play an important role in the evolution of more general initial conditions, as suggested by the weakly nonlinear studies. This is explored by following the evolution of an initial condition given by a linear IG wave solution to (6)(9) with β = 0,
i1520-0485-38-3-686-eq6
and V = 0. The amplitude is a0 and ω2 = γ2 + c20k2. While very idealized, this initial condition is representative of a low-mode internal tide radiated from some localized topography.

An example of the evolution of a linear sinusoidal disturbance with amplitude a0 = 0.06 and wavelength λ = 2.8 with h0 = 0.25 and β1/2 = 0.02 is shown in Fig. 7. The linear internal tide has a frequency ω = 1.394. Nonlinear IG waves for these values of h0 and λ are of the lobate class (cf. Fig. 3b). The interface η is shown over four periods, λ/c0f , of the linear IG wave in a frame moving with the linear wave speed c0f (>0). Also shown in the figure is the evolution of the same initial condition from the hydrostatic (β = 0) numerical model. The initial sine wave steepens and by tc0f/λ = 1 the hydrostatic solution proceeds to breaking (shock formation). The nonhydrostatic dispersion of the MCC-f solution prevents breaking and gives the undular bore. For longer times the disintegration of the initial wave into short nonlinear waves is inhibited, and for tc0f/λ ≥ 2.5 the MCC-f solution consists of a spreading packet of short waves riding on a long, nonlinear tide that is very close to the hydrostatic model solution. By the end of the calculation one small solitary-like wave (at xc0ft ≈ 2.3) has separated from the packet. The short waves are not phase locked to the long wave and, in this example, they propagate slower than the long wave. The leading solitary-like wave remains close to the shock in the hydrostatic model solution.

Another example for the same parameters except for a larger initial amplitude, a0 = 0.1, is shown in Fig. 8. The increased amplitude leads to generation of many more, and larger, nonhydrostatic waves. However, the qualitative behavior is the same. The disintegration of the initial condition into short waves is ultimately inhibited, leaving behind a long nonlinear tide underneath the shortwave packet. Again, the longwave part of the MCC-f solution is very close to the hydrostatic model solution.

Recall that these “short” waves are still dynamically long and only weakly nonhydrostatic. The terms short and long are used to distinguish the relative scales of waves. The short waves depend fundamentally on the nonhydrostatic dispersion and typically only weakly on the rotation. The long waves are essentially hydrostatic and are strongly controlled by rotation. This dynamical separation is highlighted by examining the transverse velocity jump υ, which may act as a low-pass filter since the short solitary-like waves will typically have a weak expression in υ (Gerkema 1996; Gilman et al. 1996). The υ fields for the MCC-f and hydrostatic model solutions at tc0f/λ = 4 of the previous two figures are shown in Fig. 9. The hydrostatic and nonhydrostatic solutions are nearly identical for the a0 = 0.06 runs and quite close for the a0 = 0.1 case.

The initiation of breaking in the hydrostatic model solutions leads to the continual loss of energy until that solution settles onto a long wave. The total energy [given by (12) with β = 0, integrated over the wavelength] in the two hydrostatic model solutions is shown in Fig. 10. By the end of each run the energy reaches a new equilibrium. This new wave is very close to the “longwave” part of the MCC-f solutions. This is particularly clear in Fig. 7, but also true for the other example.

Also shown in Figs. 7 and 8 at tc0f/λ = 4 are the theoretical nonlinear IG wave solutions (from section 2b) with the same energy and wavelength as in the hydrostatic model solution at this time. The theoretical waves are reasonably close to the long-wave portion of the MCC-f and hydrostatic model calculations, suggesting that the two solutions have settled onto a nonlinear IG wave with the short nonhydrostatic waves superimposed. There will be nonlinear interaction between the short and long waves (Gilman et al. 1995, 1996; Helfrich 2007), and this may explain the differences between the exact nonlinear solution and the longwave part of the MCC-f and hydrostatic model solutions. Furthermore, at a given wavelength there is a continuum of nonlinear IG wave solutions up to the limiting amplitude (cf. Fig. 3b for λ = 2.78). Thus, it is quite possible that two or more of these solutions have been generated.

By inference, the energy in the short waves in the MCC-f solutions is given approximately by the loss of energy in the hydrostatic model solutions. For a0 = 0.06 and 0.1, about 2.6% and 10.2%, respectively, of the initial energy is goes into the short nonhydrostatic waves. The energy loss from the hydrostatic model solutions, or conversely the energy transfer to the short nonhydrostatic nonlinear waves, as a function of amplitude of the initial linear IG wave, a0, for h0 = 0.25 and λ = 2.8 is summarized in Fig. 11. Substantial loss of energy from the initial tide due to breaking (or into short nonhydrostatic waves) does not occur until a threshold amplitude a0 ≈ 0.04 is exceeded. Below the threshold, solutions from the hydrostatic model and the MCC-f equations are nearly identical. This threshold is an important result and is the consequence of the low-frequency rotational dispersion. Not all internal tides will steepen and produce short solitary-like waves. Calculations were not continued beyond a0 = 0.1 owing to the appearance of Kelvin–Helmholtz shear instability. The instability is precipitated by the strong velocity jump between the layer induced by the emerging solitary waves and the finite-amplitude long IG wave (cf. Jo and Choi 2002). The degree of numerical filtering required to keep the instability from overwhelming the calculations began to influence the evolution.

Changing β does not modify these basic results, as might be expected from the stability of the nonlinear IG waves to weak nonhydrostatic effects. However, the properties of short waves are affected. Figure 12 shows the MCC-f solutions for β1/2 = 0.01, 0.02, and 0.04 at tc0f/λ = 4 using the initial condition in Fig. 7. The amplitudes of the short waves increase with decreasing β and, as expected from the properties of MCC solitary waves in this amplitude range, the wave width decreases. The underlying long IG wave is essentially the same. The high wavenumber wiggles on the β1/2 = 0.04 solution near xc0ft = 1.2 are the beginnings of the Kelvin–Helmholtz shear instability, which is enhanced with increasing β.

The examples with h0 = 0.25 and λ = 2.8 lead to underlying nonlinear IG waves that are in the lobate class. Generally, the lobate-class waves for a fixed frequency or wavelength exist over a larger range of amplitudes then the corner-class waves. Thus, in the corner regime it is more likely that the initial sinusoidal wave may have more energy than the largest possible nonlinear IG wave. Figures 3b and 4 show that to move into the corner class either ω must be increased (or λ decreased) with h0 fixed, or h0 decreased with ω (or λ) fixed. Taking the latter approach and setting h0 = 0.1 and λ = 1.924 (linear wave frequency ω = 1.4) puts the limiting IG wave in the corner class. The limiting wave has hm = 0.0862 and hM = 0.1357 and has total energy per wavelength equal to a linear IG wave with amplitude a0 = 0.0194.

Figure 13 shows numerical solutions for a case with h0 = 0.1, λ = 1.924, and a0 = 0.015. The total energy of the initial wave is within the range of nonlinear IG wave solutions. The solutions of the MCC-f (with β1/2 = 0.02) and hydrostatic models are much the same as the previous examples. The production of short solitary-like waves in the MCC-f model is inhibited once the solutions settle on underlying long nonlinear IG wave, which is again almost identical to the hydrostatic model solution.

The solution becomes quite different when the initial amplitude is increased. An example with a0 = 0.025 is shown in Fig. 14. The other parameters are unchanged. The initial energy is now outside the range of nonlinear IG wave solutions. As before, the hydrostatic model solution proceeds to breaking and the MCC-f solution develops an undular bore that rapidly separates into a train of solitary waves. The breaking in the hydrostatic model diminishes with time until the solution settles on a long wave with total energy just slightly less than that of the maximum theoretical IG wave with this wavelength. The hydrostatic model solution at tc0f/λ = 4 is almost indistinguishable from the theoretical solution with the same energy.

The MCC-f solution, on the other hand, continues to disintegrate until it consists primarily of individual solitary waves. There is little indication from the η field of an underlying IG wave similar to the hydrostatic model solution. The rotational dispersion did not arrest the disintegration of the initial tide. One difference between this case and the others is the large amplitude of the emerging solitary waves. At tc0f/λ = 2.5 the largest wave has an amplitude η0 ≈ −0.16. Thus |η0|/h0 ≈ 1.6, compared to |η0|/h0 ≈ 0.4 in Fig. 8. With such large waves a significant fraction of the initial mass of the upper layer (=h0λ) is contained in the solitary waves. In the hydrostatic model solution, all of the mass remains in the long-wave part of the solution.

The transverse velocity jump υ for the MCC-f and hydrostatic model runs in Fig. 14 at the last time shown are plotted in Fig. 15. The hydrostatic model solution is comparable to those shown in Fig. 9. The MCC-f solution is quite different and is characterized by a large jump in υ at the location of the largest solitary wave. The magnitude of υ is of the same order as the hydrostatic solution. Thus, the υ field, which in the previous examples effectively filtered out the short waves, is now strongly affected by the short solitary wave(s). Indeed, a jump in υ is precisely what is predicted by weakly nonlinear theories for the radiation damping of a solitary wave in a rotating system (Grimshaw et al. 1998a). This implies a strong, continuing interaction between the solitary waves, or at least with the largest one, with the IG waves. The longwave component of the MCC-f solution is the result of the solitary wave(s) rather than an underlying long wave that remains after the disintegration. When the nonhydrostatic calculation is carried further, the large solitary wave decays and there is some evidence of the emergence of a quasiperiodic wave packet, as found in Helfrich (2007). However, the large amplitudes of the waves and the periodic domain make this difficult to identify unambiguously. It does show that the transfer of energy from the long, hydrostatic part of the solution to the short, nonhydrostatic waves can be, in part, reversed.

This last example shows that the energy loss in the hydrostatic model will not always provide a good estimate of the energy transferred to the short, nonhydrostatic waves. The hydrostatic model does, however, give an approximate lower bound on the energy transfer. The energy per wavelength of the hydrostatic model solution as a function of time for the two h0 = 0.1 calculations (Figs. 13 and 14) are shown in Fig. 10. The energy lost from the hydrostatic solution as a function of the initial amplitude a0 is given in Fig. 16 for a series of runs with the same parameters, h0 = 0.1 and λ = 1.924. As before, there is a threshold amplitude, now about 0.01, below which the rotational dispersion prevents breaking in the hydrostatic calculation. Also shown by the solid line is the energy of the initial condition in excess of the energy of the limiting theoretical nonlinear IG wave solution for these parameters. Energy loss above this curve for a0 > 0.0194 implies that the final state has less energy than the limiting wave, and vice versa. MCC-f runs with initial amplitudes a0 ≲ 0.18 behave as shown in Fig. 13, with the disintegration arrested by rotational dispersion and the longwave part of the solution close to the hydrostatic case. The energy transferred into the short waves is approximately the same as lost from the hydrostatic model solution. When a0 ≳ 0.2, the MCC-f solutions behave as shown in Fig. 14, suggesting that most of the energy goes into the shorter solitary-like waves. The transition in behavior occurs near a0 = 0.0194 where the initial wave has energy equal to the limiting nonlinear IG wave. Calculations with β1/2 = 0.01 yield the same results, though the solitary wave properties are modified.

Nearly complete disintegration of the initial internal tide does, for the examples in Figs. 13 and 14, require an initial energy greater than the energy of the limiting nonlinear IG wave. Whether this is always the case, or depends upon the particular parameters or the shape of the initial condition (e.g., departure from sinusoidal shape and proximity to a theoretical nonlinear IG wave), will require more study. However, some calculations have been made for h0 = 0.25 and λ = 1.571 (ω ≈ 2), which put the limiting wave in the corner class. A calculation with a0 = 0.05, shown in Fig. 17, has an initial energy well in excess of the limiting wave. The disintegration is not as complete as the earlier example. The longwave part of both the η and υ fields is close to the hydrostatic model solution. The largest solitary wave does cause the transverse velocity υ to depart from the hydrostatic model solution in the same sense as Fig. 15. One distinction between these two examples is the nonlinearity of the solitary waves. In Fig. 14 |η0|/h0 ≈ 1.8, and in Fig. 17 |η0|/h0 ≈ 0.7. Increasing nonlinearity of the solitary waves leads to a stronger interaction between the short and long waves. This apparently prevents the solution from settling into a state in which a packet of short waves is essentially superimposed on a stable, long IG wave.

4. Discussion

The fully nonlinear MCC-f model [(6)(9)] solutions show that the hydrostatic, nonlinear IG waves prevent the total disintegration of the initial internal tide, consistent with earlier studies of weakly nonlinear waves. However, the conditions for the disintegration of the initial internal tide into short solitary waves are further clarified. For the small, oceanographically reasonable, values of the nonhydrostatic parameter β1/2 = H/LR considered, there is a minimum amplitude of the initial tide below which the rotational dispersion is sufficient to prevent nonlinear steepening and breaking. It is, however, well below the limiting amplitude for the nonlinear IG waves. The initiation of breaking and energy loss in the hydrostatic model runs (Figs. 11 and 16) begins for a0/h0 ≈ 0.12 − 0.16, though it is not clear from these two choices of h0 and λ that this will always be the case. The degree of disintegration increases monotonically above this threshold, but is still incomplete. A growing fraction of the energy in the initial tide is shed into short solitary waves, leaving a long nonlinear tide on which packets of shorter waves propagate. The properties of the underlying nonlinear tide are well approximated by theoretical nonlinear IG waves. Initial internal tide energies above the energy of the limiting nonlinear IG wave may lead to a complete disintegration of the internal tide into shorter solitary waves, though the nonlinearity of the short waves and their interaction with the IG waves appears to be an important aspect of this process. In these cases the hydrostatic model does not give an accurate representation of the longwave part of the solution. Conditions that give underlying lobate-class IG waves are less likely to undergo significant disintegration since these waves generally exist over a larger range of amplitudes than the corner-class waves.

Numerical calculations (not shown) with weakly nonlinear versions of the equations [essentially those used by Gerkema (1996)] give similar results. However, the weakly nonlinear models can result in substantial quantitative differences for even moderate amplitudes. This is due largely to the differences in the hydrostatic, nonlinear IG waves in each system. For example, the Boussinesq set of equations from Gerkema and the related unidirectional propagation and rotating KdV equation (Ostrovsky 1978) have, in the hydrostatic limit, only the corner-class IG waves. The addition of cubic nonlinearity to the rotating KdV model gives rise to the lobate-class hydrostatic wave, but eliminates the corner class. In addition to expanding the applicability of these theories to O(1) nonlinearity, the MCC-f equations more accurately represent the important long nonlinear tides. A cost of using these equations is the possibility of high-wavenumber Kelvin–Helmholtz instability induced by the vertical shear of the finite-amplitude solitary and IG waves. This problem can, however, be eliminated by replacing the full dispersion operator D2D1 in (4) with its weakly nonlinear limit (5). This truncation causes minor changes in the solitary wave properties and results in equations that are close to those obtained when cubic nonlinear effects are included in the Gerkema (1996) model. It leaves the long, hydrostatic waves unaffected and gives good results for the radiation damping of solitary waves when compared to the full MCC-f equations (Helfrich 2007).

The MCC-f model is limited to two layers, though extension to multiple layers is possible. However, the system of governing equations grows and rapidly becomes very cumbersome. It is probably easier and more accurate to use a continuously stratified, nonhydrostatic numerical model, such as in Lamb (1994). The role of the nonlinear IG waves should be qualitatively unchanged since these wave solutions persist with similar properties for continuous stratifications (Helfrich 2007, manuscript submitted to J. Mar. Res.).

The model results can be applied to recent observations of large-amplitude internal waves in the northeastern South China Sea. Ramp et al. (2004) have shown that very large solitary waves with amplitudes greater than 100 m can be found in the western part of the South China Sea. From an analysis of the arrival of the packets of solitary waves Ramp et al. (2004) and Zhao and Alford (2006) determined that the waves were generated by tidal flow over the abrupt topography about 400 km to the east in Luzon Strait near the Batan Islands. Satellite observations indicate that the solitary waves first appear about 100 km west of the strait (Zhao et al. 2004), pointing to the emergence of the solitary waves from the radiating internal tide west of the strait (c.f. Lien et al. 2005). The tidal currents in the generation region are mixed, with both primary semidiurnal and diurnal constituents of comparable magnitude (Ramp et al. 2004; Zhao and Alford, 2006). At this latitude, 21°N, the diurnal (semidiurnal) frequency is 1.4f (2.7f ). An estimate of h0 can be found from the continuously stratified longwave eigenmode solution using a density profile in the midbasin. The lowest eigenmode has a linear, nonrotating longwave phase speed c0 = 2.85 m s−1 and the zero-crossing of the horizontal velocity is about 600 m deep in 3000 m of water. Thus h0 ≈ 0.2 in the two-layer approximation. For these tidal frequencies and this stratification, the semidiurnal internal tide is in the corner class and the diurnal tide is in the lobate class (c.f. Figs. 3b and 4). The latter will have nonlinear tide solutions over a large range of amplitudes [limiting wave at (hm, hM) = (0.125, 0.493)]. The semidiurnal nonlinear internal tide solutions will exist only over a limited range of amplitudes [limiting wave at (hm, hM) = (0.1926, 0.2159)].

As a test that the model results are relevant, a simple application to the South China Sea conditions has been made. The longwave phase speed c0 = 2.85 m s−1 from the density profile gives g′ = 0.017 m s−2 for a two-layer model with h0 = 0.2 and H = 3000 m. At 21°N, the deformation radius LR = g′H/f = 137 km. The linear diurnal and semidiurnal internal tides have nondimensional (by LR) wavelengths λ = 2.56 and 1.0, respectively. Figure 18 shows two runs with the MCC-f model for these wavelengths with initial amplitude a0 = 0.0167, or 50 m in dimensional units. The calculations were made with β1/2 = 3/137 ≈ 0.02. The diurnal tide (Fig. 18a) is shown at t = 6.17 and the semidiurnal tide (Fig. 18b) at t = 8.14. These nondimensional times correspond to propagation at the respective linear phase speeds c0f of about 480 km—the distance from Luzon Strait to the western side of the basin. As expected, the diurnal tide has remained intact, while the semidiurnal tide has begun to disintegrate, producing a lead solitary wave with amplitude η0 ≈ 0.05, or 150 m.

From these nonhydrostatic modeling results we expect that the internal tide (of a given amplitude) will remain largely intact at diurnal frequency and experience significant disintegration into short solitary waves at semidiurnal frequency. This is indeed what is observed. Figures 8 and 9 in Zhao and Alford (2006) show that the arrival of large internal solitary waves at the western side of the basin occurs when the generating tides at Luzon Strait have maximum western velocity (after taking into account a phase shift consistent with propagation times across the basin) and are dominantly semidiurnal in character. When the tides in Luzon Strait change to dominantly diurnal, the large solitary waves are eliminated. This happens even though the magnitude of the barotropic flow in the strait is comparable in both regimes. Furthermore, Zhao and Alford (2006, their Fig. 2) find that the amplitude of the low-pass filtered internal tide at the western slope mooring is smaller by a factor of ∼2 during the semidiurnal regime than during the diurnal phase. This is in rough agreement with Fig. 18. New measurements by D. Farmer (2007, personal communication) from moorings in midbasin, about 200 km west of the generation site, show the same relation between the dominant frequency of the Luzon Strait barotropic tide and the occurrence of large solitary waves at the moorings (D. Farmer and C. Jackson 2007, personal communication).

This simple comparison with the observational data qualitatively supports the theoretical and modeling conclusions, though more extensive and direct comparisons with the data are needed. This should include modeling of the generation of the radiated internal tide since that sets the initial condition for the disintegration. It will probably be necessary to also incorporate variations in topography along the propagation path that influence the evolution of both the internal tide and any solitary waves produced by the disintegration.

Acknowledgments

KRH was supported by a Woods Hole Oceanographic Institution Mellon Independent Study Award and ONR Grant N000140610798. The authors thank David Farmer and Chris Jackson for discussion of their data.

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Fig. 1.
Fig. 1.

Example of F′(h1) from (24) for h0 = 0.1. Zeros of F′(h1) occur for cc* = 0.625.

Citation: Journal of Physical Oceanography 38, 3; 10.1175/2007JPO3826.1

Fig. 2.
Fig. 2.

Examples of the two classes of hydrostatic, nonlinear inertia–gravity waves for h0 = 0.25. The interfacial displacement η = h0h1 is shown over one wavelength. (a) Lobate-class waves for ω = 1.4 and hM = 0.3, 0.375, 0.46, and 0.55 and (b) corner-class waves for ω = 2 and hM = 0.27, 0.29, and 0.306. Dots at the end of each wave profile indicate the wavelength.

Citation: Journal of Physical Oceanography 38, 3; 10.1175/2007JPO3826.1

Fig. 3.
Fig. 3.

(a) Periodic nonlinear inertia–gravity wave solutions for h0 = 0.25 are found everywhere within the region in the hmhM plane bounded by the labeled curves. The small dot indicates the transition point between limiting amplitude corner-class waves to lobate-class waves. (b) Close-up showing the loci of nonlinear solutions with frequencies ω = 1, 1.4, and 2 and wavelength λ = 2.78.

Citation: Journal of Physical Oceanography 38, 3; 10.1175/2007JPO3826.1

Fig. 4.
Fig. 4.

Regions of nonlinear inertia–gravity wave solutions for several values of h0. The solid line within each solution domain shows the locus of ω = 1.4 solutions. When h0 = 0.5, the ω = 1.4 solutions extend up to the square.

Citation: Journal of Physical Oceanography 38, 3; 10.1175/2007JPO3826.1

Fig. 5.
Fig. 5.

Phase speed c of the ω = 1.4 nonlinear inertia–gravity wave solutions in Fig. 4 as a function of wave amplitude, hMh0. The speed has been scaled by the linear inertia–gravity phase speed c0f at each h0.

Citation: Journal of Physical Oceanography 38, 3; 10.1175/2007JPO3826.1

Fig. 6.
Fig. 6.

Numerical solutions to the MCC-f equations showing the stability of an initial nonlinear inertia–gravity wave solution to weak nonhydrostatic effects; the solid line shows the initial interface displacement η over one wavelength. The solutions with β1/2 = 0.02 (dashed) and 0.04 (dash–dot) are shown after eight periods of propagation. (a) Initial lobate-class wave with h0 = 0.25, λ = 2.8, hm = 0.18, and hM = 0.4229. (b) Initial corner-class wave with h0 = 0.1, λ = 2.024, hm = 0.085, and hM = 0.139.

Citation: Journal of Physical Oceanography 38, 3; 10.1175/2007JPO3826.1

Fig. 7.
Fig. 7.

Evolution of an initial linear inertia–gravity wave for h0 = 0.25, λ = 2.8, and a0 = 0.06. The interfacial displacement η is shown over four periods of the linear wave in a frame moving at the linear phase speed c0f . The MCC-f solution for β1/2 = 0.02 is given by the solid line and the hydrostatic model solution by the dashed line. The dash–dot line at tc0f/λ = 4 is the theoretical nonlinear inertia–gravity wave solution with the wavelength and energy of the hydrostatic model solution at that time.

Citation: Journal of Physical Oceanography 38, 3; 10.1175/2007JPO3826.1

Fig. 8.
Fig. 8.

As in Fig. 7, but a0 = 0.1.

Citation: Journal of Physical Oceanography 38, 3; 10.1175/2007JPO3826.1

Fig. 9.
Fig. 9.

Transverse velocity jump υ of the MCC-f (solid) and hydrostatic (dashed) model solutions at t = 4λ/c0f from (a) Fig. 7 and (b) Fig. 8. The dash–dot line in (a) is the theoretical nonlinear inertia–gravity wave solution with the wavelength and energy of the hydrostatic numerical model solution at this time.

Citation: Journal of Physical Oceanography 38, 3; 10.1175/2007JPO3826.1

Fig. 10.
Fig. 10.

Energy per wavelength E of the hydrostatic numerical solutions in Figs. 7 and 8 are shown by the solid lines. The initial amplitude a0 is indicated and the energy is scaled by the initial energy E0. The results from Figs. 13 and 14 are shown by the dashed lines.

Citation: Journal of Physical Oceanography 38, 3; 10.1175/2007JPO3826.1

Fig. 11.
Fig. 11.

Energy per wavelength lost from the hydrostatic numerical solution, δE, after cessation of breaking vs the amplitude a0 of the initial sinusoidal wave for h0 = 0.25 and λ = 2.8.

Citation: Journal of Physical Oceanography 38, 3; 10.1175/2007JPO3826.1

Fig. 12.
Fig. 12.

The MCC-f solutions at t = 4λ/c0f for h0 = 0.25, λ = 2.8, a0 = 0.06, and β1/2 = 0.1 (dashed), 0.02 (solid), and 0.04 (dash–dot).

Citation: Journal of Physical Oceanography 38, 3; 10.1175/2007JPO3826.1

Fig. 13.
Fig. 13.

As in Fig. 7, but h0 = 0.1, λ = 1.924, and a0 = 0.015.

Citation: Journal of Physical Oceanography 38, 3; 10.1175/2007JPO3826.1

Fig. 14.
Fig. 14.

As in Fig. 7, but h0 = 0.1, λ = 1.924, and a0 = 0.025.

Citation: Journal of Physical Oceanography 38, 3; 10.1175/2007JPO3826.1

Fig. 15.
Fig. 15.

The transverse velocity jump υ of the MCC-f (solid) and hydrostatic (dashed) numerical solutions in Fig. 14 at t = 4λ/c0f .

Citation: Journal of Physical Oceanography 38, 3; 10.1175/2007JPO3826.1

Fig. 16.
Fig. 16.

The energy per wavelength lost from the hydrostatic numerical solution, δE, (circles) vs the amplitude a0 of the initial sinusoidal wave for h0 = 0.1 and λ = 1.924. The solid line is the energy in the initial condition in excess of the theoretical limiting nonlinear inertia–gravity wave.

Citation: Journal of Physical Oceanography 38, 3; 10.1175/2007JPO3826.1

Fig. 17.
Fig. 17.

(a) The interface η and (b) the transverse velocity jump υ at t = 4c0f/λ for an initial linear inertia–gravity wave for h0 = 0.2, λ = 1.57, and a0 = 0.05. The MCC-f solution with β1/2 = 0.02 is shown by the solid line and the hydrostatic model solution by the dashed line.

Citation: Journal of Physical Oceanography 38, 3; 10.1175/2007JPO3826.1

Fig. 18.
Fig. 18.

MCC-f model results with conditions approximating (a) the diurnal and (b) the semidiurnal tides in the South China Sea. The panels show the interface η at times equal to the time for the linear tides to propagate across the basin. See the text for details. The semidiurnal solution in (b) has been periodically extended in x to a distance equal to the wavelength in (a).

Citation: Journal of Physical Oceanography 38, 3; 10.1175/2007JPO3826.1

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  • Fig. 1.

    Example of F′(h1) from (24) for h0 = 0.1. Zeros of F′(h1) occur for cc* = 0.625.

  • Fig. 2.

    Examples of the two classes of hydrostatic, nonlinear inertia–gravity waves for h0 = 0.25. The interfacial displacement η = h0h1 is shown over one wavelength. (a) Lobate-class waves for ω = 1.4 and hM = 0.3, 0.375, 0.46, and 0.55 and (b) corner-class waves for ω = 2 and hM = 0.27, 0.29, and 0.306. Dots at the end of each wave profile indicate the wavelength.

  • Fig. 3.

    (a) Periodic nonlinear inertia–gravity wave solutions for h0 = 0.25 are found everywhere within the region in the hmhM plane bounded by the labeled curves. The small dot indicates the transition point between limiting amplitude corner-class waves to lobate-class waves. (b) Close-up showing the loci of nonlinear solutions with frequencies ω = 1, 1.4, and 2 and wavelength λ = 2.78.

  • Fig. 4.

    Regions of nonlinear inertia–gravity wave solutions for several values of h0. The solid line within each solution domain shows the locus of ω = 1.4 solutions. When h0 = 0.5, the ω = 1.4 solutions extend up to the square.

  • Fig. 5.

    Phase speed c of the ω = 1.4 nonlinear inertia–gravity wave solutions in Fig. 4 as a function of wave amplitude, hMh0. The speed has been scaled by the linear inertia–gravity phase speed c0f at each h0.

  • Fig. 6.

    Numerical solutions to the MCC-f equations showing the stability of an initial nonlinear inertia–gravity wave solution to weak nonhydrostatic effects; the solid line shows the initial interface displacement η over one wavelength. The solutions with β1/2 = 0.02 (dashed) and 0.04 (dash–dot) are shown after eight periods of propagation. (a) Initial lobate-class wave with h0 = 0.25, λ = 2.8, hm = 0.18, and hM = 0.4229. (b) Initial corner-class wave with h0 = 0.1, λ = 2.024, hm = 0.085, and hM = 0.139.

  • Fig. 7.

    Evolution of an initial linear inertia–gravity wave for h0 = 0.25, λ = 2.8, and a0 = 0.06. The interfacial displacement η is shown over four periods of the linear wave in a frame moving at the linear phase speed c0f . The MCC-f solution for β1/2 = 0.02 is given by the solid line and the hydrostatic model solution by the dashed line. The dash–dot line at tc0f/λ = 4 is the theoretical nonlinear inertia–gravity wave solution with the wavelength and energy of the hydrostatic model solution at that time.

  • Fig. 8.

    As in Fig. 7, but a0 = 0.1.

  • Fig. 9.

    Transverse velocity jump υ of the MCC-f (solid) and hydrostatic (dashed) model solutions at t = 4λ/c0f from (a) Fig. 7 and (b) Fig. 8. The dash–dot line in (a) is the theoretical nonlinear inertia–gravity wave solution with the wavelength and energy of the hydrostatic numerical model solution at this time.

  • Fig. 10.

    Energy per wavelength E of the hydrostatic numerical solutions in Figs. 7 and 8 are shown by the solid lines. The initial amplitude a0 is indicated and the energy is scaled by the initial energy E0. The results from Figs. 13 and 14 are shown by the dashed lines.

  • Fig. 11.

    Energy per wavelength lost from the hydrostatic numerical solution, δE, after cessation of breaking vs the amplitude a0 of the initial sinusoidal wave for h0 = 0.25 and λ = 2.8.

  • Fig. 12.

    The MCC-f solutions at t = 4λ/c0f for h0 = 0.25, λ = 2.8, a0 = 0.06, and β1/2 = 0.1 (dashed), 0.02 (solid), and 0.04 (dash–dot).

  • Fig. 13.

    As in Fig. 7, but h0 = 0.1, λ = 1.924, and a0 = 0.015.

  • Fig. 14.

    As in Fig. 7, but h0 = 0.1, λ = 1.924, and a0 = 0.025.

  • Fig. 15.

    The transverse velocity jump υ of the MCC-f (solid) and hydrostatic (dashed) numerical solutions in Fig. 14 at t = 4λ/c0f .

  • Fig. 16.

    The energy per wavelength lost from the hydrostatic numerical solution, δE, (circles) vs the amplitude a0 of the initial sinusoidal wave for h0 = 0.1 and λ = 1.924. The solid line is the energy in the initial condition in excess of the theoretical limiting nonlinear inertia–gravity wave.

  • Fig. 17.

    (a) The interface η and (b) the transverse velocity jump υ at t = 4c0f/λ for an initial linear inertia–gravity wave for h0 = 0.2, λ = 1.57, and a0 = 0.05. The MCC-f solution with β1/2 = 0.02 is shown by the solid line and the hydrostatic model solution by the dashed line.

  • Fig. 18.

    MCC-f model results with conditions approximating (a) the diurnal and (b) the semidiurnal tides in the South China Sea. The panels show the interface η at times equal to the time for the linear tides to propagate across the basin. See the text for details. The semidiurnal solution in (b) has been periodically extended in x to a distance equal to the wavelength in (a).