Transformation of internal breathers in the idealised shelf sea conditions

Continental Shelf Research 110 (2015) 60–71

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Research papers

Transformation of internal breathers in the idealised shelf sea conditions Ekaterina Rouvinskaya a,n, Тatyana Talipova a,b, Oxana Kurkina a, Tarmo Soomere c, Dmitry Tyugin a a

Nizhny Novgorod State Technical University n.a. R. Alekseev, Minin Street 25, Nizhny Novgorod 603950, Russia Institute of Applied Physics, Ulyanov Str. 46, Nizhny Novgorod 603950, Russia c Institute of Cybernetics at Tallinn University of Technology, Akadeemia tee 21, 12618 Tallinn, Estonia b

art ic l e i nf o

a b s t r a c t

Article history: Received 31 March 2015 Received in revised form 20 August 2015 Accepted 21 September 2015 Available online 28 September 2015

We address the propagation and transformation of long internal breather-like wave in an idealised but realistic stratification and in the conditions matching the average summer stratification in the southern part of the Baltic Sea. The focus is on changes in the properties of the breather when the water depth increases and the coefficient at the cubic nonlinear term changes its sign, equivalently, the breather cannot exist anymore. The simulations are performed in parallel in the framework of weakly nonlinear Gardner equation and using fully nonlinear Euler equations. The amplitudes of breathers in these frameworks have slightly different courses in idealised conditions (when Earth's rotation is neglected) whereas a decrease in the amplitude is faster in the fully nonlinear simulation. The impact of the background (Earth's) rotation substantially depends on the spectral width of the initial breather. The evolution of narrow-banded breathers is almost the same for rotating and non-rotating situations but amplitudes of breathers with a wide spectrum experience substantial changes in realistic situation with the background rotation. The propagation of a narrow-banded breather along a path in the Baltic Sea over a location where the cubic nonlinear term changes its sign reveals fast disintegration of the breather into a precursor soliton and a transient dispersive wave group. & 2015 Elsevier Ltd. All rights reserved.

Keywords: Internal waves Baltic Sea numerical simulations Gardner equation Euler equations impact of Earth's rotation

1. Introduction Many strongly stratified shelf seas support the propagation of nonlinear solitary waves that may generate unexpectedly intense sediment resuspension (Bourgault et al., 2014; Olsthoorn and Stastna, 2014), large horizontal velocities, vertical displacements, strong mixing (Xu et al., 2012; Lamb, 2014) and extensive transport (Pan and Jay, 2009; Ladah et al., 2012) of water masses. The largest potential for the development of various hazardous situations is associated with solitary waves (Hsu et al., 2013) and especially with solitons (Osborne, 2010; Xu et al., 2010, 2011, 2012) that survive in collisions with similar entities and may create exceptional instantaneous loads. The core factor that exerts a first-order control on the ability of internal waves of different kinds to propagate through continental shelf waters and in inland seas is the local hydrography. In particular, the presence of a deep pycnocline may substantially modify, even prevent or block (Kurkina et al., 2011), the propagation of large strongly nonlinear internal waves. While such situations are n

Corresponding author.

http://dx.doi.org/10.1016/j.csr.2015.09.017 0278-4343/& 2015 Elsevier Ltd. All rights reserved.

created on most of continental shelves for a limited time interval by, e.g., several sequential upwelling events (Cheriton et al., 2014), a deep pycnocline is permanently present in a large part of the Baltic Sea (Leppäranta and Myrberg, 2009) and exerts there primary control not only on the propagation but also transformations of solitary internal waves (Kurkina et al., 2015). The main properties of the dynamics, propagation and transformation of long solitary internal waves in stratified basins have been described in detail in the international literature (Vlasenko et al., 2005; Helfrich and Melville, 2006; Apel et al., 2007; Grimshaw et al., 2007; Da Silva et al., 2011, 2015). This pool of knowledge involves extensive theoretical information about the shape and properties of internal solitons as a steady-state waves obtained in various frameworks, involving approaches based on the weakly nonlinear long wave theory such as Korteweg–de Vries (KdV) equation (Benney, 1966; Grimshaw, 1983), strongly nonlinear models of Miyata (1985) or Camassa and Choi (1999), or exact solutions of Euler equations (Dubriel-Jacotin, 1932). Solitonic internal waves are present virtually in all regions of the World Ocean and their presence is clearly distinguishable on images creates by various remote sensing applications (Jackson, 2004). Perhaps the most fascinating phenomena among quasi steady-

E. Rouvinskaya et al. / Continental Shelf Research 110 (2015) 60–71

state waves are so-called breathers. Breather is the wave packet with stable or slowly changing in horizontally variable media envelope. Differently from the well-known (e.g., KdV) solitons of elevation or depression, a breather concentrates the wave energy in both localised and oscillatory manner. In essence, it is a wave packet with a stable (or slowly changing in a horizontally variable medium) envelope. Such structures are valid long-living localised solutions in several branches of the weakly nonlinear theory, including the Gardner equation (an extension of the KdV equation) that describes the dynamics of internal waves in a specific stratification (Pelinovsky and Grimshaw, 1997). The appearance of such disturbances may vary from a shape typical for envelope solitons (that are common solutions of nonlinear Schrödinger equations) up to pairs of solitons of opposite polarities that switch their phase in a synchronised manner after certain time intervals (Clarke et al., 2000; Grimshaw et al., 2001; Slyunyaev, 2001; Grimshaw et al., 2005). Numerical simulations demonstrate that such structures may persist as long-living internal waves also in fully nonlinear Euler equations (Lamb et al., 2007). Similar structures have been sometimes observed in the ocean, for example near the coasts of South Korea (Lee et al., 2006), in the Andaman Sea (Osborne, 2010) or in the Celtic Sea (Vlasenko et al., 2014). The generation of internal breathers is often associated with the phenomenon of modulational instability (that is commonly thought to be one of the major sources of surface rogue waves in the open ocean, Kharif and Pelinovsky (2003)). This mechanism may become active also for certain stratification patterns (Grimshaw et al., 1997, 2010; Talipova et al., 2011) that occur frequently in various shelf seas (Grimshaw et al., 2007, 2014). One of the mechanism of the internal breather generation is described in (Vlasenko and Stashchuk, 2015), where the role of solitary waves of the second, third and higher modes is studied. There are still major shortages in the understanding of the potential mechanisms of the generation of internal breathers and of the possible role of these structures in the dynamics of seas and oceans. Systematic research into these aspects has only started at the turn of the millennium (Grimshaw et al., 2003; Nakoulima et al., 2004; Brovchenko et al., 2014; Terletskaya, 2014). It is natural that many factors such as horizontal inhomogeneity of the hydrographic situation, the presence of realistic bathymetry or Earth's rotation may all substantially impact the field of such nonlinear phenomena. While in most occasions the conditions favourable for the existence of internal breathers are relatively limited in both time and space, most of the Baltic Sea has almost continuously favourable conditions not only for modulational instability of internal wave trains (Talipova et al., 2011) but also for the propagation of internal breathers (Kurkina et al., 2011). Several changes in the hydrographic properties that this water body has undergone during the last decades have increased the chances for the existence of longliving internal breathers. Major changes in the location patterns of low pressure systems in the North Atlantic (Lehmann et al., 2011) have led to radical alteration of the atmospheric forcing in the southern Baltic Sea (Soomere et al., 2015). A turn in the air-flow by some 40° at the end of the 1980s (Soomere and Räämet, 2014) has apparently decreased the frequency of sequences of pressure patterns and wind events that were able to bring large amounts of salty and oxygen-rich water from the North Sea into the Baltic Sea. The radical decrease in the frequency of the major salt water inflows has resulted in freshening of the surface waters of the Baltic Sea and in major changes in the entire stratification pattern (Väli et al., 2013). This process evidently alters the areas hosting internal solitons of different kinds (Kurkina et al., 2014). Moreover, a substantial deepening of the pycnocline in most of the Baltic Sea basin may have created favourable conditions for long-living breathers. This conjecture is supported by increasing evidence of traces of

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groups of solitonic internal waves in this water body (http://iki. rssi.ru/asp/iw_images/index.html#baltic) that apparently are generated by storm surges. In this paper we address the process of transformation of internal breathers in horizontally inhomogeneous water bodies with properties mirroring those of certain shelf seas. The test basin is the Baltic Sea where the hydrographic conditions are often favourable for such disturbances. Section 2 presents a short insight into the mathematical description of the transformation of an internal breather during its propagation over a sea with varying depth using both weakly nonlinear (Gardner equation) and fully nonlinear (Euler equations) approaches. This material includes the analysis of the impact of Earth's rotation on the transformations. This model is applied in Section 3 to study the process of transformation in the typical conditions of the Baltic Sea. Numerical simulations are performed using software package IGWResearch that has been developed in the Nizhny Novgorod State Technical University (Tyugin et al., 2011; Kurkina and Talipova, 2011). The main results are summarised in Section 4.

2. Transformation of breathers on a sloping bottom 2.1. Numerical replication of internal breathers in a three-layer water body We employ the conventional model of propagation of internal breathers based on Gardner equation (Holloway et al., 1999; Grimshaw et al., 2007; Talipova et al., 2013, 2014). For a basin of a constant depth (horizontal bottom) and hosting homogeneous hydrographic conditions in the horizontal direction it has the following classic appearance:

∂η ∂η ∂ 3η + (αη + α1η2) + β 3 = 0, ∂t ∂x ∂x

(1)

where η has the meaning of deviations of the interface between the layers of the fluid from an undisturbed position, x represents the spatial coordinate and t is time. The coefficients α , α1 and β depend on the particular stratification and their exact expressions are presented in Appendix. Gardner Equation (1) is integrable and has several kinds of classic soliton solutions (Grimshaw et al., 2007). If the coefficient at the cubic term, α1 > 0 , this equation has also breather solutions (Pelinovsky and Grimshaw, 1997): cosh χ cos ψ + cos θ cosh φ

A (x, t ) = 2

+ α a sin θ cosh φ + b sinh χ cos ψ ab b cosh χ sin ψ − a cos θ sinh φ α1 − a sin θ cosh φ + b sinh χ cos ψ

sinh χ sin ψ + sin θ sinh φ a cos θ sinh φ − b cosh χ sin ψ b sinh χ cos ψ + a sin θ cosh φ a cos θ sinh φ − b cosh χ sin ψ

, (2)

where

⎛x ⎞ a2 − 3b2 t χ = a⎜ − − xBR ⎟, ˜ ˜ V T ⎠ ⎝L 0 ⎛x ⎞ 3a2 − b2 t θ = b⎜ − − xph ⎟, V0 T˜ ⎝ L˜ ⎠ ϕ = Re (γδ ), ψ = Im (γδ ), L˜ =

6βα1 , V0 = α2

3 L˜ ˜ L˜ i+γ 1 ,T= ,δ= ln , γ = a + i⋅b. ˜ i−γ β 2γ T

(3)

Here xBR and xph are arbitrary real constants. The appearance of the breather depends on two free parameters a and b. An extended weakly nonlinear model that takes into account (slowly changing) spatial variations in the hydrographic conditions and Earth's rotation is presented in Appendix. This model adequately replicates propagation of solitons even in extreme regimes compared with Euler equations (Talipova et al.,

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Fig. 2. Variations in the coefficients of Gardner Eq. (1) and the basic parameters of linear wave motion along the numerical wave basin.

Fig. 1. Vertical density structure (a) and Väisälä frequency (b) in numerical simulations.

2015) for slow changes in the environment. It also reasonably replicates the dynamics of weakly nonlinear relatively small breathers with a wide enough envelope over a horizontal bottom (Lamb et al., 2007). This model, however, is not really applicable for the description of breathers in inhomogeneous environments because the width of the envelope several times exceeds the wavelength of the carrier wave and the model eventually fails when the envelope width is comparable with the typical scales of the spatial changes in the environment. Qualitatively, a breather – because of a combination of its long spatial extension and relatively small amplitude – is much more sensitive with respect to the inhomogeneities of the medium changes than a classic soliton. To understand how the outcome of the weakly nonlinear approach matches the realistic dynamics of breathers, we compare this outcome with results obtained using fully nonlinear Euler equations. Numerical simulations of breathers in both frameworks were performed in a two-dimensional model environment with a three-layer stratification and with a sloping plane bottom (Fig. 1). The layers' thicknesses and water densities in these layers have been adjusted according to (Grimshaw et al., 1997). The model water depth increases from 65 to 83 m over a distance of 25 km. The densities of water are constant in each layer (from the top to the bottom layer: 1010, 1020 and 1030 kg/m3). Such densities do not correspond to typical shelf sea conditions and the density jumps are about twice as large as observed in the open Baltic Sea (Leppäranta and Myrberg, 2009). Nevertheless, we choose these parameters in our simulations because a relatively strong stratification is convenient feature in experiments to minimise computational efforts: it allows to have faster propagation speed and larger values of nonlinearity parameters, which in turn means faster dynamics of initial disturbance. The model parameters and the limits of the basin are chosen so that none of the coefficients changes its sign in the entire basin. In particular, the coefficient at the cubic term remains positive along the entire numerical basin with gradually increasing depth of the bottom and constant depths of the pycnoclines. The thicknesses of jump layers (8 m) match those often

occurring in the Baltic Sea. The maximum gradients in these layers (associated with the boundaries between water masses of different densities) are located at depths of 20 and 50 m. Several experiments were performed using an equivalent basin of constant depth of 65 m. The dependence of the related values of the coefficients of Gardner equation and associated possible wave regimes in a three-layer fluid on density stratification were analysed in (Kurkina et al., 2015). Although the areas of sign change of the coefficients at the cubic term are rather different for a layered fluid and a fluid with a continuous density stratification mimicking the layers, the obtained results can be used as a first estimate of wave regimes in this case. The gentle variations in the total water depth cause large changes in the coefficients of Gardner equation along the numerical basin (Fig. 2). Even quite small changes in the water depth lead to a decrease in the linear coefficient of wave amplification Q (see Appendix) by about 12% whereas further changes in the water depth have almost no impact on the values of this parameter. The linear wave speed c (that represents both phase and group speed for long waves) monotonically increases by about 25% (from 1.25 to 1.55 m/s) with the increase in the water depth. Similarly, the coefficient β (called dispersion coefficient) at the linear term of Eq. (1) increases from 250 to 550 m3/s. Changes to the coefficients α and α1 at the nonlinear terms have more complicated and non-monotonic character. The coefficient α at the quadratic term is negative in the entire basin. It changes from
7  10
3 s
1 at the shallow margin to almost zero at a distance of 10 km and increases in magnitude up to almost the initial value at the deep margin of the basin. The coefficient α1 at the cubic term is positive in the entire basin, has a minor maximum at a distance of 2.5 km from the shallow margin, decreases monotonously along the basin and vanishes at the distance of about 25 km. The extension of the basin (25 km) is chosen so that α1 remains positive in the entire basin. This property allows for the existence of breather solution among long internal waves (Talipova et al., 1999). Negative values of α1 are to be avoided as the breather cannot exist for α1 < 0 . Importantly, the magnitudes of the coefficients α and α1 are comparable and thus the relative roles of the quadratic and the cubic terms are basically on the same level. As the numerical model used in the experiments is described in (Lamb, 1994), we only provide a short insight into its major features in Appendix. The initial disturbance was introduced as an internal breather, a solution of Gardner equation defined in Eqs.

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Fig. 3. (a) Numerically generated breather A in Euler equations (solid line) and its numerical counterpart in Gardner equation (dashed line) at the shallow margin of the numerical basin, (b) amplitude spectrum of breather A.

(2) and (3) with parameters a ¼1, b¼ 3. Its initial amplitude was 2.5 m, length of the carrier wave was about 700 m and the length of the envelope was almost 3 km (Fig. 3). The associated flow field was generated using the nonlinear model (see Appendix). Such a structure obviously is not (quasi) stationary in the framework of fully nonlinear Euler equations. It is thus natural that during the initial propagation the structure first radiated short waves. To reach a quasi-stationary breather in Euler equations, we first simulated its propagation over a water of constant depth of 65 m (matching the water depth at the shallow margin of the numerical basin). An x–t diagram of the interface displacement (not shown here) is analysed to understand the process of the breather adjustment (Lamb et al., 2007). After the initial adjustment, the breather became almost stationary in the numerical realisation of Euler equations. This structure we call breather A (Fig. 3a) in what follows. In terms of coefficients of a Fourier representation

Cj =

2 N

N

⎛ − 2π i ⎞ ⎟ N ⎠

∑ η (k) wN(k − 1)(j − 1), wN = exp ⎜⎝

k=1

(4)

using a discrete set of N harmonics η(k), its spectrum (Fig. 3b) has a clear maximum at wavelengths of about 700 m (wave numbers k0 ¼0.009 m
1). The spectral width of this maximum at a half of its amplitude is approximately Δk¼ 0.005 m
1 and thus the spectrum of the entire structure is relatively narrow. A minor maximum in the spectral representation at the longest reproduced components apparently reflects the presence of a stationary component that is characteristic to breather solutions of both Gardner and Euler equations

Fig. 4. Snapshots of breather propagation over a flat bottom in non-rotating numerical basin.

ε=

α η + α1η2 c

(5)

for this (Gardner) breather was relatively small, in the range of 4– 6% and its shape almost fully matched the shape of breather A at the shallow margin of the numerical basin (Fig. 3а). The two breathers behave in almost the same manner in basins of constant depth when the Earth's rotation is neglected (Fig. 4). The propagation speed of the Gardner breather slightly exceeds

2.2. Propagation of breathers in a non-rotating basin of constant depth As a next step in the comparison of the evolution of the initially equivalent breathers in the two frameworks we constructed a breather solution for Gardner equation, the parameters of which maximally matched the numerically obtained quasi-stationary breather A. The amplitude of the (Gardner) breather was also 2.5 m and the length of the envelope was about 3 km. The nonlinearity parameter

Fig. 5. Amplitudes of Euler (filled circles) and Gardner (solid line) breathers propagating in a non-rotating basin of constant depth.

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Fig. 7. Normalised amplitudes of Euler (filled circles, the initial amplitude 2.5 m) and Gardner (solid line, the initial amplitude 2.52 m) breathers propagating over the inclined bottom with a slope of 0.001 in a non-rotating basin.

Fig. 6. Snapshots of the propagation of Euler (solid line) and Gardner (dashed line) breathers over the inclined bottom with a slope of 0.001 in a non-rotating numerical basin of variable depth (depth variations in the bottom panel).

that of breather A. The amplitudes of both breathers vary with time. Their approximate values, evaluated as half-sum of the maximum elevation and the maximum depression, differ maximally by 3–5% (Fig. 5). These values are always evaluated with respect to the depth of 24 m, that is, at the location where the maximum of the vertical mode is located (see Appendix). Doing so makes it possible to conveniently compare the outcome of simulations using Euler equations with the outcome of the weakly nonlinear model. It is natural that instantaneous amplitudes of both breathers vary to some extent in time as the phase speed of the carrier wave differs from that of the envelope and the extremes of the envelope generally do not coincide with the troughs or crests of the carrier wave. 2.3. Difference in weakly and fully nonlinear propagation of breathers The propagation of both breathers in a non-rotating basin with a sloping bottom corresponds to wave evolution in very weakly changing environment. The slope is only 0.001, which is much smaller than typical slopes of seabed in shelf seas, e.g., to the north-west of Australia (Holloway et al., 1997). Such small slopes, however, are not uncommon in the central part of the Baltic Sea (Baltic proper) (Leppäranta and Myrberg, 2009) where the total depth varies between 100 and 200 m and thus is commeasurable with the numerical basin in use. The evolution and transformation of both breathers (presented for the depth of 24 m) are fairly similar (Fig. 6). Most of the variations in the instantaneous shapes of the breathers stem from different phase speeds of the carrier waves. The Gardner breather is faster in the initial stage of propagation in relatively shallow

water, until to a distance of about 10 km. Later on the Euler breather starts to move faster and catches its weakly nonlinear counterpart at the end of the numerical basin. The variations in the propagation speed of the Gardner breather are first of all governed by the magnitude of the coefficient α at the quadratic term in Eq. (1). The difference in the nature of transformations of breathers in the weakly and fully nonlinear environment becomes vividly evident in a comparison of the amplitudes of the breathers (Fig. 7). The amplitude of the Gardner breather always exceeds its initial value. It increases by about 19% at a distance of 13 km from the shallow margin. As separation of breather energy from the energy of the background wave field is nontrivial and numerical estimates of the internal breather wave energy in the Euler equations are fairly complicated (Lamb, 2007; Talipova et al., 2013), we rely here on qualitative estimate of the changes. Although both structures radiate a certain amount of energy, the intensity of radiation is fairly small and, qualitatively, both breathers seem to transform almost adiabatically. Although further on the amplitude of the Gardner breather decreases to some extent (to a level of 115% of the initial value, evidently because of reasons described above), it starts to increase again near the deep end of the numerical basin. Contrariwise, the amplitude of the Euler breather decreases monotonically and at the deep end of the basin it is about 34% smaller than the initial value. This result highlights a core difference between the dynamics of internal breathers in weakly and fully nonlinear environments from the motion of other solitonic waves. It is well known that at small levels of nonlinearity the dynamics of disturbances that match the classic exact solutions (e.g., non-oscillating solitons of elevation or depression) to weakly nonlinear equations in inhomogeneous medium is practically the same in weakly and fully nonlinear frameworks (Maderich et al., 2009, 2010, Talipova et al., 2015). The presented results signal that the dynamics of disturbances matching breather solutions to these equations may be substantially different depending on which framework is used for their analysis. The obvious reason for the differences is that the spatial extension of substantial motions of water parcels is relatively compact for classic solitons but much longer for breathertype disturbances. In other words, the breather 'feels' more strongly inhomogeneities of the medium because of its extensive length. 2.4. The impact of earth's rotation on fully nonlinear evolution of

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Fig. 8. Normalised amplitudes of breather A with a narrow amplitude spectrum in fully nonlinear simulations using Euler equations for a rotating (black circles) and non-rotating (grey circles). The initial amplitude was 2.5 m in both cases.

Fig. 10. Normalised amplitudes of breather B with a wide amplitude spectrum in fully nonlinear simulations using Euler equations for a rotating (black circles) and non-rotating (grey circles) situation. The initial amplitude was 2.85 m in both cases.

breathers

but with a wide spectrum of components. To show that, it is sufficient to adjust just one parameter of the original Gardner breather. Let us consider a breather that correspond to a ¼1, b¼0.3 in Eqs. (2) and (3). We proceed as above: let such a breather to propagate within the fully nonlinear Euler equations in a rotating basin with a constant depth until it radiates away part of its energy and becomes quasi-stationary. This structure, called breather B, consists of one strong signal of depression surrounded by much lower elevations. The initial amplitude is 2.85 m, carrier wavelength about 1200 m and the extension of the envelope about 4 km (Fig. 9a). These parameters are comparable with those of breather A. The amplitude spectrum of breather B has two clearly evident maxima. The main peak at k0 ¼0.005 m
1 (the corresponding wavelength about 1250 m) is accompanied by another peak of almost the same height corresponding to an elevation of the interface in the entire numerical basin. The presence of this peak is not unexpected as fully nonlinear breathers often possess a nonzero total elevation of the affected level. The main difference compared to breather A is that the spectrum of breather B is much wider (Δk ¼0.016, Fig. 9b). Interestingly, the behaviour of the amplitude of breather B in a

To evaluate the impact of Earth's rotation on the propagation and transformation of breathers we include a background rotation with a frequency of f¼ 0.00012 s
1. This frequency corresponds to the geographical latitude of φ ¼56°N, that is, to the conditions of the central Baltic proper. The propagation of breather A with a narrow spectrum (Fig. 3b) is almost insensitive with respect to the background rotation in the fully nonlinear environment. Its shape and propagation speed in the numerical basin practically coincide with those obtained for the non-rotating case. Even its amplitude (that shows substantial sensitivity with respect to the switch from the weakly nonlinear to a fully nonlinear environment) does not deviate from the values obtained for the non-rotating case (Fig. 8). As weakly nonlinear propagation is generally even less sensitive with respect to the presence of background rotation, this experiment suggests that, at least for narrow-banded breathers, Earth's rotation does not have any appreciable impact on their evolution. This conjecture, however, cannot be generalised to the entire pool of possible breathers in the marine environment. The potential impact of Earth's rotation can be vividly demonstrated by means of considering another breather with similar parameters

Fig. 9. а) Initial shape of breather B, b) amplitude spectrum of breather B.

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rotating basin (Fig. 10) is substantially different from the course of this quantity when the background rotation is switched off. The presence of rotation leads to a considerable increase (up to 40% at a certain location in the basin) in the amplitude of this structure. This feature signals that the sensitivity of breathers with various appearance and parameters may depend on some fine and possibly concealed aspects of their internal structure and that each case should be studied separately. However, generally speaking, the sensitivity of at least some solitons with respect to the impact of background rotation is well known and solitons may disintegrate even when propagating in a rotating horizontally homogeneous basin of constant depth (Grimshaw et al., 1998; Grimshaw and Helfrich, 2012). The same effect should work also for breathers, because Gardner breathers' and the resulting Euler breathers' spectra do not approach zero for vanishing wavenumbers and contain substantial levels of lowfrequency harmonics (Figs. 3 and 9). Nevertheless some breathers can have less expressed low-frequency components. Such breathers are more stable to rotational effects. So a possible reason why breathers with a wide spectrum are more sensitive with respect to the background rotation is the presence of a substantial level of disturbances that are coherent over very long (basin-long) distance in breather A: the longer the constituent is, the stronger is the impact of the background rotation on its evolution.

3. Transformation of breathers in the Baltic sea conditions There is increasing evidence that in many occasions internal waves in the Baltic Sea registered using various remote sensing techniques (http://iki.rssi.ru/asp/iw_images/index.html) often resemble the appearance of different kinds of solitary waves or their sequences. If the structure reflected in Fig. 11 would be interpreted as an internal breather, the length of its carrier wave would be about 900 m, which is close to the relevant wavelengths in the above simulations. The changes in the hydrographic conditions of the Baltic Sea over the last decades have led to the frequent presence of

Fig. 11. Satellite image of sea surface in southern part of the Baltic Sea with a signature typical for an envelope internal soliton or internal breathe. The centroid of the wave packet is at 54°57′57″N, 15 °46′58″E.

favourable conditions for the excitation and propagation of longliving solitary internal waves in the southern Baltic Sea (Kurkina et al., 2011, 2014). The relevant kinematic parameters (dictated by the properties of stratification and governing the appearance of internal solitary waves) are such that breather-type waves may be frequent and long-living. As mentioned above, such waves may only emerge if the coefficient at the cubic nonlinear term α1 > 0 . There are numerous extensive areas corresponding to α1 > 0 in the recent hydrographic conditions (Fig. 12). As the tidal range is very small in the Baltic Sea, the largest deviations of the nearshore water level are driven by atmospheric factors but the relevant wind directions may substantially vary (Leppäranta and Myrberg, 2009). It is thus likely that there exists no predominant internal wave propagation direction in the southern Baltic Sea. However, it is also likely that internal waves and solitons driven by relaxation of storm surges will mostly propagate away from the coast. For this reason we choose the direction of wave propagation to the north-east, away from the Polish and German coasts that often suffer from high storm surges (Wolski et al., 2014). The potential path of an internal wave was chosen as a straight line with a length of L ¼55 km from 18°54′E, 56°0′N to18°21′E, 55°40′N (Fig. 12). Part of this path lies in a region in which the coefficient of the cubic term in Gardner equation (evaluated based on the hydrologic database GDEM-V3.0, Teague et al., 1990) is, on average, positive during the summer season. The typical bottom slope along this path (Fig. 12b) roughly corresponds to the gently sloping bottom used in the above numerical simulations. We used average hydrological conditions over this stretch to create a model representation of the stratification for simulations in the framework of fully nonlinear Euler equations. To maximally replicate the model situations discussed above, the vertical structure of water masses (Fig. 13) was assumed to be constant along the path and only the variations of the seabed were included in the model. The water depth varies in the range of 89–108 m (Fig. 14). The coefficient at the quadratic term in Eq. (1) is positive along the entire path whereas it greatly (by almost ten times) exceeds the value of the coefficient at the cubic term. The latter parameter becomes negative at a distance of 24 km from the beginning of the path. Thus, a breather cannot exist after this point. The initial disturbance was an internal breather defined by Eqs. (2) and (3) for Gardner equation with the coefficients evaluated at x¼ 0 in Fig. 14 and with parameters a ¼1, b ¼0.3. This choice makes it possible to take the impact of Earth's rotation on the propagation of the breather into account. The initial amplitude of the Gardner breather was 7.6 m. This is about three times as high as the breathers used in the above simulations. The difference mostly stems from the difference in stratification properties of the numerical basin and the real sea. As above, we first launched the process of numerical relaxation of the Gardner breather in Euler equations over a 9 km long idealised basin of constant depth until it was transformed to a quasi-stationary Euler breather with an amplitude of 5.8 m. The radiated wave trains rapidly dispersed. They reflected from the boundary of the numerical basin but as they were moving much slower than the breather, there was virtually no interaction between the breather and the radiated wave field. The dynamics of the breather is illustrated using the displacements of water parcels that were initially located at the depth of 20 m, that is, just below the mid-depth of the main pycnocline (Fig. 15). The breather initially contains two large undulations. After travelling over about 24 km a third wave crest of comparable amplitude emerges. The maximum amplitude of the entire structure decreases to some extent but until this location the disturbance has a general appearance of a breather. At this point the

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Fig. 12. The model path of the propagation of internal breathers (bold line) in the Baltic Sea bathymetry (a) and on the map of the values of the coefficient of the cubic nonlinearity in the average summer conditions (b).

Fig. 13. Average vertical stratification (a) and Väisälä frequency (b) along the chosen path of propagation of the internal breather.

coefficient at the cubic term of Eq. (1) becomes negative and a breather cannot exist anymore. While within the first 24 km the breather experiences certain transformations, it remains compact and its length (  3 km) is almost constant even though the water depth changes considerably. After passing this point the structure rapidly disintegrated into a gradually lengthening group of almost sinusoidal waves. It is natural that the propagation in the framework of Euler equations does not exactly replicate the properties of Gardner breathers, in particular, the exact location at which the breather cannot exist anymore and is transformed into other kinds of disturbances. As the two frameworks are supposed to qualitatively represent what happens with internal waves of reasonable amplitude, it is likely that the point at which the Euler breather collapses is not far from the location where the Gardner breather cannot exist anymore. It is characteristic that after travelling over 24 km a sort of 'precursor' soliton starts to build up at the front of the entire structure (Fig. 15). This wave of elevation exhibits certain

Fig. 14. Coefficients of Gardner equation along the chosen path (Fig. 12) of propagation of internal solitons in the southern Baltic Sea. The coefficients are evaluated for characteristic hydrological properties in July. The vertical dash-dotted line indicates the location of the zero-crossing of the coefficient at the cubic nonlinear term (24 km to the north-east from the beginning of the path).

transformations evidently because of changes in the water depth and becomes almost separated from the rest of the wave group at a distance of 50 km from the initial location. The rest of the group gradually spreads over an ever longer distance and at the end of simulations it apparently becomes a classic transient group of dispersive internal waves. Consistently with the results presented above, the impact of Earth's rotation plays a substantial role in the transformation of the entire structure in the initial phase of its propagation (Fig. 16). The maximum difference in the amplitudes reaches about 30% at a

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distances in the southern Baltic Sea where the stratification is favourable for their motion. These structures always experience influence from the changing water depth and a part of them substantially 'feels' the Earth's rotation. The governing parameter is still the (sign of the) cubic nonlinear coefficient (the coefficient at the cubic term of Gardner equation) that defines the area where a breather may exist. After reaching the border of such an area, the structure eventually disintegrates into dispersive transient wave group, possibly preceded by a precursor soliton.

4. Concluding remarks

Fig. 15. Snapshots of the numerical propagation of a breather along the path indicated in Fig. 12 to the north-east towards the deepest area of the Baltic proper. The vertical dash-dotted line indicates the location of the zero-crossing of the coefficient at the cubic nonlinear term. A small rectangle indicates the emerging precursor soliton.

Fig. 16. Normalised amplitudes of a breather-like structure with an initial amplitude of 5.8 m and propagating along the path indicated in Fig. 12 to the north-east towards the deepest area of the Baltic proper in fully nonlinear simulations using Euler equations for a rotating (black circles) and non-rotating (grey circles) situation.

distance of 5 km from the beginning of the path. The typical difference 20% persists until the location where the breather cannot exist anymore. The qualitative changes to the entire structure are basically the same. Therefore, our simulations suggest that large and powerful breather-like structures may exist and propagate over substantial

The presented analysis first of all signals that breathers, a kind of oscillating wave packet, may often exist in semi-sheltered shelf seas where the combination of fresh water input from rivers into the surface layer and penetration of saltier oceanic water into the bottom layer naturally supports the presence of a strong basically three-layer stratification. In favourable conditions they may travel over many dozen of kilometres and thus effectively carry wave energy between different sea regions. Simulations with a stratification that match the typical southern Baltic sea conditions suggest that such structures also survive during considerable time intervals and cover substantial distances in realistic sea areas with inclined bottom. Therefore, breathers may serve as an additional channel of spatial redistribution of energy supplied to water masses, e.g., during storm surges, the relaxation of which may give rise to different kinds of internal waves. The areas of collapsing of breathers (equivalently, the regions where the coefficient at the cubic nonlinear term in Gardner equation vanishes) not necessarily match the areas of breaking of linear internal waves or weakly nonlinear classic internal solitons. Importantly, the collapse of an internal breather is not necessarily (at least in the typical conditions of the southern Baltic Sea) associated with its breaking and generation of patches of mixing. Instead, our simulations suggest that after passing a point where a breather cannot exist, it gives rise to an internal soliton of elevation and a transient group of internal waves. Therefore, the potential presence and possible collapse of breathers may provide not only intense mixing in certain regions but also a concealed way of multi-step excitation of internal wave trains and solitons even in regions with low energy levels. It is often assumed that weakly nonlinear structures are almost perfectly replicated using fully nonlinear equations. This expectation is, however, not always true. In our simulations weakly nonlinear (Gardner) and fully nonlinear (Euler) breathers only behave in a very similar manner in an idealised basin with a constant depth, in other words, in a non-rotating horizontally homogeneous situation. If the environment becomes inhomogeneous, for instance, if the bottom is inclined, the evolution of Gardner and Euler breathers starts to deviate. As the deviations become evident even for quite small levels of nonlinearity, it is likely that the weakly nonlinear Gardner model is unsuitable for the description of long-term evolution of internal breathers in horizontally inhomogeneous marine environments. The physical reason for such a failure is that the internal breathers are usually much longer than the 'classic' internal solitons of elevation and thus 'feel' much more strongly the inhomogeneities of the environment. The mismatch between the amplitudes of the breathers in the Gardner and Euler models, however, does not tell that the Gardner model is completely useless. Even if its predictions for the amplitudes of breathers are not realistic, the predictions for the locations where the breathers collapse are eventually still correct. An intriguing outcome is the sensitivity of some breathers on the presence of background rotation. This feature has been demonstrated in terms of breathers with different appearance and

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amplitude spectra but it may become evident also in terms of some other parameters. This sensitivity may substantially affect the regions of existence of long-living structure of this kind and/or select out different subsets of breathers for different basins. In this context, it is still interesting and instructive that breathers with a narrow amplitude spectrum remain stable at quite high latitudes (56°N) of the Baltic Sea and supposedly at even higher latitudes. In essence, this means that internal breathers may be a frequent constituent of the dynamics of multi-layered shelf seas whenever there exist mechanisms for their generation.

69

explicit expressions in terms of the modal function Φ and its derivatives (e.g., Holloway et al., 1999). The function Q 2 c03 ∫ (dΦ0/dz ) dz

Q=

c 3 ∫ (dΦ/dz )2dz

, (А5)

has the meaning of the (linear) amplification factor of the wave in a horizontally inhomogeneous medium. Here the subscript “0” corresponds to the initial position x0. The coefficients of Gardner equation are are expressed as follows: H

3 ⎛ 3c ⎞ ∫ (dΦ/dz ) dz α = ⎜ ⎟ 0H , ⎝ 2⎠ (dΦ/dz )2dz

Acknowledgements

∫0

The research was partially supported by National Targets of Scientific Research in Russia (Grant no. 5.30.2014/K), by the Organization of scientific research in Russia (Grant no. 2015/133), by the Russian Foundation for Basic Research (Grant no. 15-3520563), the institutional financing by the Estonian Ministry of Education and Research (Grant IUT33-3), by the Estonian Science Foundation (Grant no. 9125) and through support of the European Regional Development Fund (ERDF) to the Centre of Excellence in Non-linear Studies CENS.

⎛c⎞ ⎟ β= ⎝ 2⎠

H

∫0 Φ2dz

⎜

α1 =

H

∫0 (dΦ/dz )2dz

, (А6)

2⎤ 2 2 2 ⎡ 3c ∫ dz { ⎣ 3 (dT /dz ) − 2 (dΦ/dz ) ⎦(dΦ/dz ) − α (dΦ/dz ) +Π , 2 ∫ (dΦ/dz )2dz

}

Π = αc ⎡⎣ 5 (dΦ/dz )2 − 4dT /dz⎤⎦ dΦ/dz.

(А7)

Here the function T has the meaning of a nonlinear correction to the modal function Φ(z) and satisfies the following equation: Appendix. Models of dynamics of internal waves in a horizontally inhomogeneous rotating ocean

⎡ ⎤ ⎛ dΦ ⎞2⎤⎥ d ⎡ d ⎡ dΦ ⎤ 3 d ⎢ 2 dT 2 (c − U )2 ⎜ ⎟ ⎢ (c − U ) ⎥ + N T = − α ⎢ (c − U ) ⎥+ ⎝ dz ⎠ ⎥⎦ dz ⎣ dz ⎦ 2 dz ⎢⎣ dz ⎣ dz ⎦ (А8)

As detailed descriptions of the asymptotic model for the transformation of long weakly nonlinear waves in a rotating inhomogeneous ocean based on a modified Gardner equation for inhomogeneous medium have been provided, e.g., in (Holloway et al., 1999; Grimshaw et al., 2004, 2007), we bring here only the basic equations without their derivation. The equation for the deviations of the isopycnal level η from its undisturbed position is described by the following equation:

⎛ αQ ∂ς α Q2 ⎞ ∂ς β ∂ 3ς f2 + ⎜ 2 ς + 1 2 ς 2⎟ + 4 3 = ∂x ⎝ c 2c ⎠ ∂s c c ∂s ς (x, s ) =

∫ ςds,

η (x, t ) , Q (x)

, (A1)

where f is the Coriolis parameter f¼ (4π/Te)sinφ, Te is the Earth's rotation period 24 h and φ is the geographical latitude. The undisturbed level of η corresponds to the maximum of the modal function Φ(z) that is defined from the solution of the Sturm– Liouville problem

d2Φ N2 (x, z ) + 2 =0 2 dz c (εx)

(А2)

with zero boundary conditions at the bottom and using so-called rigid lig approximation at the surface:

Φ (0) = Φ (H ) = 0.

→ Vt +

(А3)

The modal function is normalised as

Φmax = 1

with homogeneous boundary conditions T(
H) ¼T(0)¼0 and a condition T(maxz)¼ 0 at the location zmax of the maximum of the modal function where the value Φ(zmax) ¼1 is achieved. For the analysis of the problem of transformations of solitonic structures addressed in the body of the paper it is therefore necessary to first select the parameters of the soliton described by Eqs. (2) and (3) or the breather described by Eqs. (4) and (5) at a certain initial location. To evaluate the propagation of the resulting disturbance governed by Eq. (A1) it is necessary to use a suitable numerical method for its tracking and to calculate the coefficients of Eq. (1) at each location along the propagation path of the solition. For calculations of weakly nonlinear evolution of the breathers we use the numerical model developed by a team involving one of the co-authors of this paper (Holloway et al., 1997; 1999). This model has been successfully used in a number of studies of dynamics of internal waves (e.g., Kit et al., 2000; Talipova et al., 2014). The nonlinear evolution of the breathers is simulated using Lamb's (Lamb, 1994) model for the generation and propagation of internal wave breathers. The model equations are the two-dimensional (vertical section) Boussinesq equations on a rotating f plane for inviscid, incompressible fluid:

(А4)

and H is the total water depth. The linear wave speed of internal waves c (εx ) (here ε is a small parameter) for the given mode is also found from Eq. (A2) as an eigenvalue for each location x of the wave propagation. The coefficients of Eq. (A1) – so-called dispersion coefficient β, the coefficient at the quadratic nonlinear term α and at the cubic nonlinear term α1 (also called quadratic parameter and cubic parameter) are also found for each location of the wave propagation using certain fairly complicated but basically

( →V ∇)→V − fV→ × →k = − ∇P − →k ρg,

(А9)

→ ρt + V ∇ρ = 0,

(А10)

→ ∇ V = 0,

(А11)

ρ=

ρf − ρ0 ρ0

,

(А12)

→ where V (u,v,w) is the velocity vector, ∇ is the three dimensional

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vector gradient operator, subscript t denotes the time derivative, ρf – the density of sea water, ρ0 – the average or characteristic density (introduced owing to the Boussinesq approximation that assumes that the density ρf = ρ0 (1 + ρ) only changes insignificantly in the basin and ρ is a nondimensional quantity that has a meaning of density anomaly), P is the pressure, g is acceleration ⇀ due to gravity, f is, as above, the Coriolis parameter and k is the unit vector along the z-direction. The waves propagate in the xdirection, y-axis is perpendicular to the wave motion and z is the depth. The normal to the wave propagation (cross-section) velocity is included in the model, but no variation along the y-coordinate is allowed. This is realised by neglecting the partial derivatives with respect to y in the basically three-dimensional Eqs. (A9–A11). The equations are transformed to a terrain-following coordinate system (so-called sigma-coordinates). The equations are solved over a domain bounded below by the topography h(x) (prescribed by the user) and covered by a rigid lid at the surface. To initialise the simulation, it is necessary to prescribe horizontally homogeneous density field of water masses ρmean(z), an initial disturbance of this density field ρ(x, z, t¼0) ¼ ρmean ( z − ξ (x, z, t = 0)) and the corresponding initial velocity field. The latter is constructed to match the velocity field in a linear internal wave that creates the prescribed density distribution. The pattern of disturbances is described as

ξ (x, z, t = 0) = η (x) Φ (z ).

(А13)

The simulated breathers were constructed, as described in the main body of the paper, from Gardner breathers expressed by Eq. (2). The corresponding velocity field was generated using the IGW Research software package and asymptotic theory in a linear approximation:

u (x, y , t = 0) = cη (x)

dΦ , dz

w (x, y , t = 0) = − cΦ

d η (x). dx

(А14)

(А15)

The steps of the numerical scheme in space and time are chosen to satisfy the Courant–Friedrich–Levy criterion for stability.

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Further reading Stastna, M., Lamb, K., 2002. Large fully nonlinear internal solitary waves: the effect of background current. Phys. Fluids 14, 2987–2999.