Pycnocline variations in the Baltic Sea affect background conditions for internal waves

O. Kurkina1, A. Kurkin1, T. Soomere2,3, A. Rybin1, D. Tyugin1

1 Department of Applied Mathematics, Nizhny Novgorod State Technical University n.a. R.E. Alekseev, Nizhny Novgorod, Russia

2 Institute of Cybernetics at Tallinn University of Technology, Akadeemia tee 21, 12618 Tallinn, Estonia 3 Estonian Academy of Sciences, Kohtu 6, 10130 Tallinn, Estonia

Abstract-Significant changes in the vertical structure of the Baltic Sea water masses within the latter decades may substantially influence the dynamics of this water body through changing the properties of single internal waves, their propagation pathways, regions of breaking and associated areas of intense mixing and resuspension of bottom sediments. We evaluate the average nonlinear parameters that govern the field of long high-frequency weakly nonlinear internal waves in the framework of Gardner’s equation. The calculations are performed using hydrographic data calculated by the Rossby Centre Ocean circulation model (RCO) for the entire Baltic Sea for 1961–2005 with a horizontal resolution of 2 nautical miles. The focus is on changes in the nonlinear wave regimes such as wave polarities and limiting amplitudes of solitary internal waves. The extension of the changes is demonstrated by comparison of the parameters in question in the 1960s (characterizing the situation when strong salt-water inflows were frequent) and 1990s (when inflows were weak and rare). The spatial and seasonal distributions of the listed parameters differ considerably for these years. The largest change is a shift in the possible nonlinear wave regime (e.g., (dis)appearance of probable breather generation, polarity change and/or transition from one family of soliton solutions to another) over the sea areas between Gotland and the Swedish mainland. The typical areas where the internal waves alter their appearance (through adjustment, transformation or breaking) and areas of intense breaking have also been shifted over the three decades. The overall geographical distribution of the listed parameters of internal waves is still quite stable despite the notable changes of pycnocline depth both in summer and in winter.

I. INTRODUCTION

Many details of the functioning of the Baltic Sea substantially rely on the mechanisms supporting strong vertical stratification in this water body. This structure expresses interplay of fresh water influx from the drainage area and water and salt exchange through the Danish straits and leads to the presence of strong halocline in most parts of the sea [1]. This jump layer (that usually lies at depths of about 60–80 m in its central part called Baltic Proper and also serves as the main pycnocline) is inter alia the most important waveguide for internal waves and thus a core channel of wave energy transfer between different parts of the sea. Its position indirectly affects the functioning of its medium-range depths through changing internal wave properties [2], breaking regions of internal waves and associated intense mixing, and possible resuspension of bottom sediments [3].

The vertical structure of the Baltic Sea water masses (and especially the main halocline and pycnocline depths) has undergone significant changes within the latter decades [4]. These variations in vertical stratification may substantially influence the dynamics of this water body through changing the conditions for the generation and propagation of internal waves. Changes in the pycnocline depth may substantially relocate the typical areas of impact of internal-wave-generated hydrodynamic loads [5]. They not only affect large-scale infrastructure at the bottom but may also considerably modify the fate of the entire Baltic Sea ecosystem [4,6]. The related consequences to the hydrodynamic drivers are particularly extensive in this water body where the usually existing three-layer structure supports specific types of nonlinear internal waves [7] and where changes to the near-bottom hydrodynamic loads may adversely affect, e.g., chemical munitions dumped there after World War II.

The purpose of this paper is to evaluate changes to the propagation environment for internal waves in the Baltic Sea associated with changes in the vertical stratification of its water masses since the beginning of the 1960s. The comparison is made in terms of average kinematic and nonlinear parameters that govern the field of long high-frequency weakly nonlinear internal waves in the framework of Gardner’s equation [8]. The analysis is based on hydrophysical properties of the Baltic Sea water masses that are numerically replicated using the Rossby Centre Ocean model RCO. We start from the description of the framework of Gardner’s equation and possible variations in the properties of its soliton solutions in different hydrophysical environments. The basic features of the RCO model and changes to the vertical structure of the Baltic Sea water masses in 1961–2005 are described next. The core message of the subsequent analysis, formulated in the Results and Discussion sections, is that changes to the vertical structure of water masses have substantially affected the propagation regime of nonlinear internal waves in this water body.

II. MODEL AND METHOD

1. Variations of Gardner’s solitons To properly describe the propagation and transformations of

unidirectional weakly nonlinear internal waves in a water body with complicated vertical structure, it is customary to use

the variable-coefficient Gardner’s equation that contains two nonlinear terms (quadratic and cubic nonlinearity) [8]:

0)( 3

32

1 =∂∂+

∂∂+++

∂∂

xxc

tηβηηααηη . (1)

Here η is the elevation of a water parcel from its equilibrium position and the coefficients c (which has the meaning of the linear wave speed), α , 1α and β depend on the vertical structure of the undisturbed density field at a particular location (see [9] for a detailed overview).

This equation, often called extended Korteweg–de Vries (eKdV) equation, is an example of integrable nonlinear evolutionary equations. It possesses stationary (in a suitably moving co-ordinate system) localized solutions (solitary waves) for all combinations of the signs of the nonlinearities (except when 01 == αα and one must move to a higher-order approximation [7]) as well as multi-soliton solutions [10]–[12]. It has one more type of localized solutions, the breather, which is a periodically pulsating, or oscillating, isolated wave form [13,14].

The quadratic and cubic nonlinear terms1 (α , 1α ) can have either sign, depending on the fluid stratification, while the dispersive coefficient is always positive. Specifically, the possibility of a sign change for the cubic nonlinearity results in a variety of wave regime.

The properties of solitons2 of the Korteweg–de Vries (KdV) equation (which contains only one quadratic nonlinear term) are well known (see [15] and references therein). The inclusion of a cubic nonlinearity into this equation substantially modifies the appearance of its soliton solutions [10]–[14]. The single-soliton solution of Gardner’ equation (1) is:

( ) ( )[ ]VtxBAtx

−+=

γη

cosh1, . (2)

The soliton velocity 2βγ+= cV is expressed through the inverse width γ of the soliton. The parameters A and B depend on the coefficients of Eq. (1)

2

212

2 61,6α

βγααβγ +== BA (3)

and determine the soliton amplitude ( )BAa += 1 , understood as the maximum of ( )tx,η in Eq. (2).

The possible combinations of the signs of the nonlinear coefficients in Gardner’s equation correspond to different wave propagation regimes (Fig. 1; see [9] for more detailed analysis). The simplest particular case occurs when the quadratic nonlinear term is nonzero and the cubic nonlinear term vanishes ( 01 =α ). In this case solution (2) reduces into the classical KdV soliton

1 For brevity, the expressions ’quadratic nonlinearity’ etc. are used to denote the values of coefficients at the relevant terms in various equations. 2 Below we shall frequently speak about solitons of certain equations, having in mind the relevant soliton-like solutions to these equations.

.312

sech),( 2

⎥⎥⎦

⎤

⎢⎢⎣

⎡⎟⎠⎞

⎜⎝⎛ −−= tactxaatx α

βαη (4)

Differently from the surface waves (where the quadratic nonlinearity coefficient is always positive and the KdV soliton is always a wave of elevation), an internal KdV soliton may have the shape of either wave of elevation or wave of depression. Its polarity is defined by the sign of the quadratic nonlinearity coefficient α .

When the quadratic nonlinear term vanishes ( 0=α ) and the cubic nonlinearity is positive, Eq. (2) reduces into a soliton solution of the modified KdV equation [16]:

.66

sech),(2

12

1

⎥⎥⎦

⎤

⎢⎢⎣

⎡

⎟⎟⎠

⎞⎜⎜⎝

⎛−−= tactxaatx α

βαη

(5)

It can have either polarity but no soliton solutions on nonzero pedestal exist when the coefficient of cubic nonlinear term in the mKdV equation is negative.

When the coefficient 1α at the cubic nonlinearity is negative (as in the case of a two-layer fluid, lower half-plane of Fig. 1), soliton solutions of a single polarity, with 0>αη , exist with amplitudes between zero and a limiting value

1lim αα−=a . This family corresponds to 10 << B . At small amplitudes ( 1→B ) the solution described by Eq. (5) transforms into a KdV soliton (4). If 0→B , the amplitude of the soliton approaches the limiting value 1lim αα−=a . Its width is not bounded and the higher solitary waves obtain a table-like shape, with a flat crest surrounded by shock-like kinks. Therefore, the inclusion of a cubic nonlinear term into the wave equation gives rise to a widening of the wave profile and the formation of a plateau-like entity with a steep front and back and a gently sloping upper part. Such appearance of large-amplitude solitary waves has been repeatedly observed in both laboratory and field conditions [8,17,18].

If 01 >α (upper half-plane of Fig. 1), solitons of either polarity exist. The amplitude of solitons with 0>αη is not

Figure 1. Shapes of soliton solutions to Gardner’s equation (1) and associated propagation regimes (1–4) in Figs. 6 and 7 for different

combinations of the signs of coefficients at its nonlinear terms.

bounded because ∞<< B1 . In the limit of small amplitudes they also become KdV solitons but for large amplitudes ( ∞→B ) they become similar to the solitons of the mKdV equation (5). The solitons with 0<αη ) correspond to

1−<<−∞ B . If −∞→B , they will transform into an mKdV soliton. However, the waves of this family have a minimum amplitude 1alg 2 αα−=a . If 1−→B , this solution develops power-law tails and tends to the so-called algebraic soliton, which is structurally unstable [19]. In the range 01 <<− B (when the polarity of the soliton is opposite to the sign of the quadratic nonlinearity; denoted as filled areas in the upper half-plane of Fig. 1), only breathers exist [14,19], with no limitations on their amplitudes.

2. Simulated stratification of the Baltic Sea Equation (1) is applicable if the horizontal nonuniformities

of the density stratification are smooth and slow. In such situations the linear wave speed )(xc changes slowly on a scale of the typical wavelength and wave reflection (that is intrinsically present in an inhomogeneous medium) can be neglected. The core condition is that the wave speed does not vanish. The wave speed may approach zero, e.g., when a wave propagates into unstratified waters or approaches the shore. In such situations Eq. (1) is no more valid. This means that the wave must either be reflected or destroyed analogously to surface wave reflection, run-up or breaking.

Importantly, all other coefficients of Gardner’s equation can change arbitrarily, even very rapidly, and also may vanish. The zero-crossing points of some coefficients are critical for the wave propagation since solitary waves can be destroyed at such points. The presence of such points does not affect the applicability of the variable-coefficient Gardner’s equation [9].

The pool of explicit analytical expressions for the coefficients of Gardner’s equation is very limited. Such cases mostly involve two-layer stratification without shear flow [20–22]. The analysis of a special case of a two-layer of Boussinesq fluid whose upper layer is moving with constant velocity in the rigid lid assumption is presented in [23]. A few advanced cases are given in [12,24]. In general, for an arbitrary stratification all the coefficients of Gardner’s equation have to be evaluated numerically; see, e.g., [9] for the detailed presentation of the relevant formulas. Spatial maps of these coefficients for climatological average stratification of the Baltic Sea are presented in [5]. In this paper we focus on the major changes to these maps since the 1960s.

To study the nature and extension of the coefficients of Gardner’s equation we use hydrographic data calculated by the Rossby Centre Ocean circulation model (RCO) for the entire Baltic Sea for 1961–2005 with a horizontal resolution of 2 nautical miles. The RCO is a Bryan–Cox–Semtner primitive equation circulation model in z-coordinates following [25] with a free surface and open boundary conditions with the North Sea in the northern Kattegat. It is coupled to a sea ice model with elastic-viscous-plastic

rheology. Subgrid-scale mixing is parameterized using a turbulence closure scheme of the k–ε type with flux boundary conditions to include the effect of a turbulence-enhanced layer due to breaking surface gravity waves [26]. The model run, results of which are used in this study, employed 41 vertical layers with a thickness between 3 m close to the surface and 12 m at a 250 m depth. The model was forced with atmospheric data derived from the ERA-40 re-analysis using a regional atmosphere model with a horizontal resolution of 25 km [27,28]. It also accounted for river inflow and water exchange through the Danish straits. As the atmospheric model tends to underestimate wind speed extremes, the wind was adjusted using simulated gustiness to improve the wind statistics [27]. For further details of the model set-up and an extensive validation of the model output the reader is referred to [29–33].

III. RESULTS

1. Decadal changes in the pycnocline depth To get some flavor about changes in the stratification, we

evaluated first the variations in the depth of the main pycnocline following similar calculations in [4]. The sea water density in the water column was calculated using the standard equation of state [34] from the modeled temperature, salinity and depth at a few dozens of locations (Fig. 2). The location of the main pycnocline was evaluated as the depth of the largest density gradient four times a day. The seasonal pycnocline in the upper layer of the sea (in which the local gradients may be even higher than in the main pycnocline) is

Figure 2. Locations of selected vertical profiles of density in the Baltic Sea.

eliminated by looking at depths >30 m. This constraint was not used in the analysis of the seasonal variations and the vertical structure in shallow coastal regions as we were interested in the location of the strongest gradient that largely governs the propagation and appearance of internal waves.

The procedure resulted in discrete values of this depth (corresponding to the relevant vertical level in the RCO model) that varied all the way from about 30 m depth level to

the near-bottom layers (Fig. 3). To eliminate synoptic-scale and seasonal variations, these values were first smoothed over one month and then over four years (cf. [35]). The resulting estimates for the pycnocline location for single time instants and months deviate to some extent from those for the halocline (Fig. 3) but reveal almost equivalent match of seasonal and interannual variations. The 4-year average depths for both these layers closely match each other.

Most of the Baltic Sea hosts significant decadal-scale variations in the pycnocline depth since the 1960s. The typical range of variations of the 4-year mean pycnocline depth is 15–20 m. With a few exceptions, the pycnocline is located relatively high in the 1960s and at largest depths in the mid-1990s [4]. The nature of its relocation is substantially different in different sub-basins and seasons (Fig. 4, Fig. 5). For example, almost no changes to the pycnocline depth have occurred in the south-western Baltic Sea to the west of the Stolpe Channel (profiles 27 and 32 in Fig. 2) where the vertical structure of water masses is largely controlled by bathymetry. This suggests that the level of saltier water in the Bornholm Basin and in the vicinity of the Slupsk Sill [1] is mostly defined by the flow of saltier water over this sill.

To the north-east and east of this region, in the southern part of the Eastern Gotland Basin and in the Bay of Gdansk, the variations in the main pycnocline depth are much larger both in summer and winter conditions. The range of deepening is typically about 10 m. More importantly, the area in which the strongest density gradient is located in deep waters has notably widened towards the north and southeast (Fig. 4).

In summertime in the rest of the Baltic Sea the seasonal pycnocline contains the largest density gradient and is interpreted as the main pycnocline as its properties mainly govern the propagation regime of internal waves. This jump

Figure 4. Average depth of the main pycnocline in summer 1962–1966 (left) and 1992–1996 (right).

1965 1970 1975 1980 1985 1990 1995 2000 2005

30

40

50

60

70

80

90

100

110

120

Dep

th, m

1965 1970 1975 1980 1985 1990 1995 2000 2005

30

40

50

60

70

80

90

100

110

120

Dep

th, m

Figure 3. Instantaneous (thin gray line), monthly mean (thick gray line) and

4-year mean pycnocline depth (upper panel) and halocline depth (lower panel) at station BY31. On average, the largest salinity gradient is located by 1.1 m

deeper than the largest density gradient.

layer is located at depths below 20 m (except for a few spots) and this depth has undergone no notable changes from the 1960s until the 1990s.

Much more pronounced changes to the pycnocline depth have occurred in wintertime. The range of its deepening is, similarly to the summertime changes, about 10 m in the southern part of the Eastern Gotland Basin and in the Bay of Gdansk. The deepening, however, reaches values >20 m in the northern Baltic Proper. As the vertical stratification is much weaker in the Sea of Bothnia than in the Baltic Proper, it

is natural that variations in the pycnocline depth are of the same magnitude in the Sea of Bothnia or even larger than in the Baltic Proper.

The process of relocation of the pycnocline apparently considerably affected the western Gulf of Finland (this gulf has no sill between the gulf and the Baltic Proper) but almost did not occur in the Gulf of Riga (that does have a sill separating this water body from the Baltic Proper).

Figure 5. Average depth of the pycnocline in winter 1962–1966 (left) and 1992–1996 (right).

Figure 6. Spatial distribution of the solution types 1–4 of Gardner’s equation (Fig. 1) in summer 1962–1966 (left) and 1992–1996 (right).

2. Changes in the signs at the nonlinear terms of Gardner’s equation

In this paper, we characterize the changes to the propagation conditions of internal waves in terms of the signs of the coefficients at the nonlinear terms in Eq. (1). This combination of signs of (α , 1α ) determines the key properties of the solutions to Gardner’s equation as described in Fig. 1. The largest changes to the field of such internal waves occur at the boundaries of the areas corresponding to different combinations of these signs. For example, if a solitary internal wave propagates over a location where 1α changes its sign, the wave has to change its polarity. As the largest changes in the hydrographic conditions apparently occurred in the 1970s–1980s (Fig. 3), we compare the signs of these parameters for the mid-1960s and mid-1990s.

Consistently with the discussion above, changes to the type of solutions in the south-western Baltic Sea are minor both in summer and winter conditions. The changes in summer conditions (Fig. 6) are concentrated in the Eastern Gotland Basin and Western Gotland Basin. In much of the area between the island of Gotland and the Swedish mainland the solution type 4 (which was predominant in the 1960s) has been replaced by the solution of type 1 in the 1990s. This means that relatively wide table-like waves of depression (that were typical in the 1960s) have been replaced by much narrower and possibly higher (and thus generally creating larger water velocities) waves in the 1990s. At the same time, solutions of type 4 appeared in a sea area to the north of Gotland.

The area to the east Gotland hosted mostly solutions of type 3 in the 1960 but they were largely replaced by solutions of type 4 in the 1990s. This change is opposite to the above-

described one. The area near the Polish coast that hosted solutions of type 4 in the 1960 had solutions of type 2 in the 1990s; therefore, the polarity was kept but other properties of the waves have apparently been changed as explained above. Also, a few minor changes are evident near the entrance to the Gulf of Riga and in the central area of this basin where area that hosted solutions of type 2 in the 1960 has disappeared by the 1990s. Interestingly, changes to the solution type are minor in the Gulf of Finland, Sea of Bothnia and Bay of Bothnia. The only noticeable change is that small areas hosting solution of type 1 have appeared in the central part of the Sea of Bothnia in the 1990s in the area that generally hosts solutions of type 2.

Changes to the internal wave propagation conditions in wintertime (Fig. 7) mostly become evident as a relocation of the boundaries between areas hosting different types of solutions. As mentioned above, almost no changes are evident in the south-western Baltic Sea where the stratification is less dependent on the major salt-water inflows. The area hosting solutions of type 4 has generally increased from the 1960s to 1990s on expense of the area hosting solutions of type 3. This change, however, has not affected the overall geometry of these areas. There is almost no changes in the areas hosting solutions of type 2 (note that the coverage of these areas in winter is much smaller than in summer).

IV. DISCUSSION

In essence, large variations in the main halocline and pycnocline depths mirror the changes in the amount of saltier water inflows into the Baltic Sea. The more or less constant level of pycnocline in the south-western Baltic Sea (and very minor changes in the conditions for the propagation of internal

Figure 7. Spatial distribution of the solution types 1–4 of Gardner’s equation (Fig. 1) in summer 1962–1966 (left) and 1992–1996 (right).

waves) is consistent with the overall functioning of such inflows. They first fill the Arkona Basin, then the Bornholm Basin, and only occasionally penetrate further into the Eastern Gotland Basin. The flow regime and stratification in this part of the sea are largely governed by bathymetry-driven hydraulics and apparently are mostly affected by the bottom topography and wind-forcing [36]. The overflow over the Slupsk Furrow (Slupsk Sill) can be treated as a subcritical, gravity current in a wide channel, with some features peculiar to frictionally controlled rotating flows. As a result, the stratification in „downstream” areas of saltier bottom water pathways is almost insensitive about the salt water inflows.

The situation is different to the east of the Stolpe Channel and to the north of the Hoburg–Midsjö bank. The water masses in the Baltic Proper (Eastern Gotland Basin, Bay of Gdansk, Northern Gotland Basin) are firmly interconnected by a large-scale circulation pattern, numerous meso-scale eddies and other features that together provide intense horizontal mixing and halocline ventilation. Only the Western Gotland Basin is to some extent separated from this circulation [30,37]. It is natural to assume that these processes efficiently shape the pycnocline depth in the entire Baltic Proper. This shaping does not happen instantaneously. The overall circulation scheme in the Baltics is cyclonic and it is thus expected that the pycnocline depth of the southern Proper (Eastern Gotland Basin and Bay of Gdansk) reacts first to the changes in the saltier water inflow, with obvious changes to the propagation conditions of internal waves. The major shift the pycnocline depth in the Baltic Proper can thus be related to the major change in the saltier water inflow in the mid-1980s [38,39]. Somewhat larger variations in the pycnocline depth in the northern Baltic Proper [4] and associated changes to the internal wave propagation in summertime evidently stem from the joint impact of weaker overall stratification, voluminous river runoff into the northern Baltics and possibly events of exporting of the salt wedge from the Gulf of Finland [40,41].

The Sea of Bothnia is connected with the Baltic Proper (Northern Gotland Basin) via a deep but relatively narrow channel. This bathymetric feature apparently blocks meso-scale activity of the Baltic Proper from entering into the Sea of Bothnia and renders the exchange of bottom waters to a certain flow along this connecting channel. As a result, halocline ventilation is much weaker in the Sea of Bothnia than in the Baltic Proper. This feature is consistent with the very minor changes to the internal wave propagation regime in the above analysis.

The performed analysis suggests that no large changes to the propagation and breaking conditions of internal waves along the main pycnocline should be expected in the south-western Baltic Sea in the Stolpe Channel and to the west of it. As this area is usually thought to be one of the major internal wave generation regions [42], the level of internal wave energy supplied from this area to the rest of the Baltic Sea evidently have not undergone any large changes.

To sum up, the largest change highlighted in this analysis is a shift in the possible nonlinear wave regime (e.g., (dis)appearance of the probable breather generation, polarity change and/or transition from one family of soliton solutions to another) over the sea areas between Gotland and the Swedish mainland and to the south-east of Gotland in summer conditions. The typical areas where the internal waves alter their appearance (through adjustment, transformation or breaking) are also different due to changed background conditions. Areas of intense breaking of internal waves have also been shifted to some extent over the two decades. The overall geographical distribution of the listed parameters of internal waves is still quite stable despite the notable changes in the pycnocline depth both in summer and in winter.

Finally, large variations of the pycnocline depth, especially the potential for the deepening of the pycnocline, signals that the internal wave energy was likely concentrated in deeper parts of the water column in the 1990s than during the rest of the simulation period. This means that internal waves in the main pycnocline were less visible via various surface phenomena and their role in the functioning of the marine environment may have been underestimated. The pycnocline deepening also means that areas where internal waves were breaking may have also been different for different decades. Finally, changes to the typical pycnocline depth in the often three-layer Baltic Sea environment may have amplified nonlinear effects in the internal wave field as described in [7,9]. These features definitely call for further studies of the properties and impact of internal waves in the Baltics, the knowledge of which is now quite poor [43].

ACKNOWLEDGMENT

The research was supported by the project TERIKVANT (managed by the Estonian Research Council in the framework of the Environmental Technology R&D Program KESTA and financed by the ERDF), targeted financing SF0140007s11 of the Estonian Ministry of Education and Research, grant 9125 by the Estonian Science Foundation and through support of the ERDF to the Centre of Excellence in Non-linear Studies CENS. The simulated hydrographic data were kindly provided by the Swedish Meteorological and Hydrological Institute in the framework of the BONUS BalticWay cooperation [44].

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