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Topographic Waves in Enclosed Basins: Fundamentals and Observations

  • Kolumban HutterEmail author
  • Yongqi Wang
  • Irina P. Chubarenko
Chapter
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Part of the Advances in Geophysical and Environmental Mechanics and Mathematics book series (AGEM, volume 2)

Abstract

In Sect. 11.2, the notions of first and second class waves were introduced. The former were said to be due to the action of the gravity force. These waves are therefore also called gravity waves. The latter are due to the rotation of the Earth and cease to exist when the frame of reference is inertial. These waves are alternatively also termed Rossby-, vortex-, geostrophic or gyration-waves, see Fig. 19.1.

Keywords

Wind Stress Gravity Wave Potential Vorticity Topographic Wave Baroclinic Effect 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

19.1 Review of Early Work

In Sect. 11.2, the notions of first and second class waves were introduced. The former were said to be due to the action of the gravity force. These waves are therefore also called gravity waves. The latter are due to the rotation of the Earth and cease to exist when the frame of reference is inertial. These waves are alternatively also termed Rossby-, vortex-, geostrophic or gyration-waves, see Fig. 19.1. In a homogeneous fluid, subject to the rigid lid assumption and in the limit of the shallow water approximation the second class linearized waves are described by the differential equation
$$\begin{array}{rcl} & & T[\Psi ] = \frac{\partial } {\partial t}E[\Psi ] + \mathcal{J}\left [\Psi, \frac{f} {H}\right ] = 0,\end{array}$$
(19.1)
in which
$$\begin{array}{rcl} & & E[\Psi ] := \bigtriangledown \cdot \left (\frac{\bigtriangledown \Psi } {H} \right ), \\ & & \\ & & \mathcal{J}\left [\Psi, \frac{f} {H}\right ] = \frac{\partial \Psi } {\partial x} \frac{\partial } {\partial y}\left ( \frac{f} {H}\right ) -\frac{\partial \Psi } {\partial y} \frac{\partial } {\partial x}\left ( \frac{f} {H}\right ).\end{array}$$
(19.2)
Ψ is the volume transport stream function, H the water depth and f the Coriolis parameter.1 When the frame of reference is inertial (f = 0), then T[Ψ] = 0 reduces to ∂E[Ψ] ∕ ∂t = 0. This implies that E[Ψ] does not depend on time which shows that second class waves cannot propagate in this case. Therefore, H = { const.} and f(x, y)≠{ const.} defines the planetary Rossby2 waves, whilst f = { const.} and H(x, y)≠{ const.} defines the topographic Rossby waves, also simply called topographic waves (TWs). On the f-plane these only exist when the basin has variable bathymetry. We shall confine attention to this case.
Fig. 19.1

Carl-Gustav Rossby (1898–1957), a Swedish-US meteorologist (portrait from http://www.villasmunta.it/Storia_della_meteorologia/) and in the left picture clouds along a jet stream over Canada (http://en.wikipedia.org/). Carl-Gustav Rossby (28 December 1898, Stockholm; 19 August 1957, Stockholm) was a Swedish-US meteorologist who first explained the large-scale fluid motion in the atmosphere, (ocean and lakes) due to the rotation of the Earth. He was studying meteorology and oceanography from 1919 under Vilhelm Bjerknes in Bergen (Norway), in Leipzig and the Lindenberg Observatory in Brandenburg. In 1921, he returned to Stockholm to join the Swedish Meteorological–Hydrological Service where he participated in a number of oceanographic expeditions at Stockholm University. While ashore, between expeditions, he studied mathematical physics at Stockholm University. In 1925, Rossby moved to the United States and joined the US Weather Bureau in Washington, DC, where he worked on atmospheric turbulence. In 1928, he became Associate Professor at the Aeronautics Department (renamed soon later Meteorology Department) of the Massachusetts Institute of Technology, Boston, and additionally joined the Woods Hole Oceanographic Institution as a Research Associate. His major interests at this time included thermodynamics, mixing, turbulence and atmosphere–ocean interactions. Rossby, after having assumed US citizenship in 1938, was appointed in 1940 the chairmanship of the Department of Meteorology of the University of Chicago. It is here, where he turned his attention to large-scale atmospheric motions, in particular the jet stream and large-scale planetary waves. During World War II he trained military meteorologists. In 1947, he returned to Sweden to become the founding director of the Institute of Meteorology in Stockholm, but he kept part-time his positions at Chicago and Woods Hole. From Stockholm, he renewed by visits his friendship with Prof. Hans Ertel in Berlin. Their cooperation led to the mathematical formulation of the so-called Rossby waves. In his later years, between 1954 and his death in 1958 Rossby championed the field of atmospheric chemistry. Originally, only large-scale planetary waves on the β-plane in a constant thickness atmosphere were called Rossby waves. Today, any gyroscopic wave on a rotating frame tends to be named after Rossby as topographic Rossby waves, equatorial Rossby waves, planetary Rossby waves, etc. The text is based on: http://en.wikipedia.org.wiki/Carl-Gustav_Rossby.

The interest in topographic waves in the ocean arose from the conspicuous long periodic wave signals that were observed along the continental shelves of our oceans. In lakes, a considerable number of temperature and current observations in various lakes also disclosed pronounced oscillations with a characteristic period of a few days that could not be explained by (external or) internal gravity waves. Poincaré (1910) [34] was the first to point out the existence of such long periodic oscillations in a rotating circular basin with parabolic depth profile, and Lamb (1932) [26] gives the first analytic solution of these topographic waves. Much later, Birchfield and Hickie (1977) [5] considered the transient wind generation of the Lamb modes and demonstrated, how the gravity oscillations and the TWs modulate the seiche response by a slow rotation in the cyclonic sense of the pattern of coastal jets and return flow across the lake centre. Free circular basin solutions for second class waves have been found by Saylor et al. [35] and Saylor and Miller [36] and Wenzel [49]. An analytical method for determining second class modes in an elliptical paraboloid was prescribed by Ball [2] and Mysak [30], and Johnson [20] constructed solutions of TWs in an elliptical basin whose shoreline and depth contours form a family of confocal ellipses. Using a semi-analytic semi-numerical approach Stocker and Hutter [414243] found solutions to the TW-equation in a rectangular basin with symmetric bathymetric chart and could identify three typical wave modes of the TW-equation: (1) basin wide modes with large wavelengths, (2) basin wide modes with shorter wavelengths and (3) bay modes, in which the major part of the energy is concentrated in a localized (bay) region. From this study sprang a series of localized TWs by Stocker and Johnson [4546].3

Most work on gyration waves is on barotropic TWs. However, these waves also exhibit a baroclinic coupling when they propagate in a stratified fluid. Mysak et al. [33] have shown that, in a two-layer basin of the size and depth scale of many intermontane lakes, barotropic TWs are only slightly affected by the interfacial motion. However, the interfacial oscillations are driven by the TWs. Thus, one can find the baroclinic TW-mode from the solution of the barotropic TW-equation subject to the mass flux condition through the lake shores yet is able to observe the wave from temperature measurements in the thermocline.

Identification of TW-signals in spectra of temperature and isotherm–depth–time series is difficult because their mode frequencies for enclosed basins are often clustered together with subinertial modal frequencies of gravity waves. This makes a clear separation of second class modes from their first class counterparts a challenging exercise not only of time-series analysis, but also of numerical identification of their eigenfrequencies and mode structures.

19.2 Some Observations and Proposed Interpretations

ln this section we describe a series of long periodic oscillations of temperature or current data indicating that the phenomenon which underlies these observations can probably be interpreted in terms of topographic waves. We heavily draw from two monographs by Stocker and Hutter [42] and Hutter [18]; both are out of print for more than a decade. Our approach is to present facts, including interpretations and to postpone detailed explanations in terms of models to later sections.

19.2.1 Lake Michigan

The analysis outlined here and the figures are due to Saylor et al. [35] and Huang and Saylor [13]. During spring, summer and fall 1976 current meters were deployed in Southern Lake Michigan as indicated in Fig. 19.2. Most of these current meters were positioned along the Eastern shore between Benton Harbor and Muskegon, but ten current meters were also deployed along a straight line connecting opposite points along the shores between Racine and Holland.

The bathymetry of the Southern Lake Michigan is simple and concise. An approximation of the topography by a circular basin with radial profile dependence, see Fig. 19.3, may not be an oversimplification of the situation at hand.

Figure 19.4 collects kinetic energy spectra of the Eastward (Fig. 19.4a) and Northward (Fig. 19.4b) velocity components (generally at the 25-m level) at the six stations indicated in Fig. 19.4. Vertical lines are drawn to accentuate the two conspicuous energy peaks at near inertial ( ∼ 17 h) and near 4-day periods. Thus, the lake responds distinctly at these periods. This is further substantiated by the graphs presented in Fig. 19.5. They show a progressive vector diagram of the hourly averaged low-pass filtered current velocities from the 32-m depth current meter at station 11 (Fig. 19.5a) and coherence and phase difference between the East and North velocities from the same current meter (Fig. 19.5b).

Fig. 19.2

Location map showing the bathymetry of Southern Lake Michigan and the positions of the 16 current meter moorings, adapted from Saylor et al. (1980) [35]. © American Meteorological Society, reprinted with permission

Fig. 19.3

Average amplitudes of the North and East velocity components for rotational oscillations on the southern Lake Michigan cross section during an episode of wave excitation, 1–15 July 1976. Basin approximations as a paraboloid or as an inverted cone that are referred to later are also shown, adapted from Saylor et al. (1980) [35]. © American Meteorological Society, reprinted with permission

Fig. 19.4

(a) Kinetic energy spectra of the Eastward velocity component (u) recorded at the 25 m level at six stations in Southern Lake Michigan. Vertical lines are drawn to accentuate the two conspicuous energy peaks at near inertial and near 4-day periods. (b) as in (a), except for the Northward velocity component (v), adapted from [35]. © American Meteorological Society, reprinted with permission

Fig. 19.5

(a) Hodograph of hourly averaged, lowpass filtered current velocities from the 32-m depth current meter at station 11 for the month of July 1976. (b) Coherence (lower) and phase between the East and North velocities at the 32 m level from station 11. Two broad areas of high coherence highlight the cyclonic rotating 4-day oscillation and the anti-cyclonic near inertial oscillation, adapted from [35]. © American Meteorological Society, reprinted with permission

The hodograph clearly reveals the oscillatory wave motion with a period of about 4 days, the rotation of the current vector being in a cyclonic (counterclockwise) direction for this station, which is in the centre of the lake basin. Coherence and phase plots in Fig. 19.5b for the East (u)- and North (v)-components for one current meter at position 11 accentuate this, as they show two highly correlated signals centred at the two indicated frequencies. At the 4-day period, the u-component leads the v-component by a phase angle of 90 ∘  which corresponds to a cyclonic rotation; at the near-inertial period the phase angle of the u-component lags that of the v-component by approximately 90 ∘ , indicating an anti-cyclonic oscillation.

Further scrutiny of the data shows that (1) the 4-day period motions are clockwise rotations at all stations except 10 and 11 in the centre of the lake, where they are counterclockwise and (2) currents at on-shore positions are primarily along-shore while those at off-shore stations may have appreciable amplitudes for both, the East- and North-components.

These observations can be interpreted in terms of free TWs in a circular basin with nearly conical profiles. For a depth profile of the form
$$\begin{array}{rcl} H = {H}_{0}\left (1 -{\left (\frac{r} {a}\right )}^{q}\right ),\,\,\,q\, >\, 0,& &\end{array}$$
(19.3)
(H 0 is the maximum depth at the centre, a the radius of the circular basin and q a parameter characterizing the profile), Saylor et al. [35] deduce the frequency relation
$$\begin{array}{rcl} \frac{f} {\omega } = \frac{3m + 2q} {m}.& &\end{array}$$
(19.4)
Here, ω is the frequency, f the Coriolis parameter and m = 1, 2, 3,  the radial mode number. Values for the frequencies or periods are summarized in Table 19.1 and the streamline pattern of the vertically integrated transport for the lowest two modes is sketched in Fig. 19.6. The fundamental mode (m = 1) enjoys all properties of the observations mentioned above. In particular for the conical profile, its period is close to the observed 90 h period. Table 19.1 and (19.4), however, also show that for each mode the periods depend strongly on the topography, but not on the size of the basin (H 0 and a do not enter into the frequency relation). Moreover, the same period arises for different modes and different bathymetries, indicating that the mode structure is very important in interpreting observations.
Table 19.1

Eigenfrequencies (-periods) of the first three TW-modes computed according to (19.4) using f = 2π ∕ 16. 9 h

Profile

q

m = 1

m = 2

m = 3

 
  

f ∕ ω

T [h]

f ∕ ω

T [h]

f ∕ ω

T [h]

 
 

1

5

84.5

4

67.6

3

50.7

 
 

2

7

118

5

84.5

4

67.6

 
 

3

9

152

6

101

5

84.5

 
 

0

0

0

 
Fig. 19.6

Schematic sketch of the mass transport stream line patterns of the fundamental and first ‘higher’ mode plotted for three instances during a quarter period. During one cycle, the system of gyres rotates counterclockwise (on the northern hemisphere) around the basin (From [42]). © Springer, Berlin, reprinted with permission

In summary, the data analysis of the current meters and the interpretation of the observed periods and velocity rotations in terms of free topographic waves in a circular basin seem to be in harmony in the sense that no discrepancies were discovered.

19.2.2 Lake of Lugano (North Basin)

Southern Lake Michigan is a large and relatively shallow lake with a width of approximately 100 km, whereas Lake of Lugano is small (17 km long, approximately 2 km wide and 300 m deep).

In summer 1979, thermistor chains and current meters were positioned at various locations in the stratified Northern basin of Lake of Lugano. Only temperature-time series could be analyzed (for a detailed description see [193233]). They disclose a moderate long-periodic signal with a period of app. 74 h.

Figure 19.7 shows the map of Lake of Lugano (North basin) with indicated stations where the wind and temperature (generally at the lower portion of the metalimnion) were measured in July/August 1979. Figure 19.8 summarizes wind data at stations 4 and 7 (top) and isotherm–depth–time series at the stations 1 and 8, 6 and 4 (lower part). These time series ‘demonstrate a strong component of the motion with a period of perhaps 74 h (marked by circles emphasize this wave the troughs of the isotherm depths have been brought into prominence by thick solid and dotted lines. The front arises first at station 4, propagates Southwards, reaches station 1 approximately 7 h later (indicated by the solid line connecting the troughs at Cassarate and Melide) and is ‘reflected’ at the Southern end of the lake. The ‘reflected’ wave passes through station 4 again (though split up into two smaller minima with an intermediate maximum, indicated by an arrow marked with an encircled 1), and after 37 h, can be recognized at station 8 with a conspicuous minimum (heavy dotted lines). It appears later at Cassarate, Melide and as a reflected wave at Cassarate (arrow marked by an encircled 2), Porlezza, etc.’ [19]. The corresponding wave speed of approximately 12 cm s − 1 is substantially lower than the wave speed of the internal gravity wave of the two-layer model; neither can it be explained as a higher order baroclinic Kelvin wave of, say, a three-layer model. Furthermore, direct wind forcing can be excluded as a likely cause of excitation [33]).
Fig. 19.7

Northern Lake of Lugano with indicated positions of moorings (1–8) equipped with anemometers, thermistor chains and current meters in the field campaign July–August 1979 (From [42]). © Springer, Berlin, reprinted with permission

Fig. 19.8

Time series of 2 h means of the longitudinal and transverse components of the wind energy ((u 2 + v 2)1 ∕ 2 u,  (u 2 + v 2)1 ∕ 2 v) with directions of u and v components as indicated in the upper left corner at the stations 7 (Porlezza) and 4 (Cassarate) and unfiltered time series of selected isotherm depths at the stations 1 (Melide, top solid curve), 8 (Porlezza, dotted, superposed on the Melide curve) and 4 (Cassarate, solid curve). Components of the motion with conspicuous periods are marked by special symbols: troughs in heavy solid and dotted lines and marked by circles are for 74 h; triangles for 24 h; arrows for 12 h. The time shown is from 9 to 19 July 1979 (From Hutter et al. [19]). © Taylor & Francis, http://www.informaworld.com, reprinted with permission

A statistical cross-correlation analysis of the temperature time series indicated for all station-pair-time series – except stations 3/8 – high coherence and strongly suggested a counter clockwise rotation of the wave signal around the basin.

Mysak et al. [33] have explained the 74-h oscillation and the anticlockwise propagation of the phase as the baroclinic trace of a barotropically driven TW. They explain all their observations (except the ‘discrepancy’ in the phase of the pair 3/8 of the mean temperature displacement function, which interrupts the anticlockwise increase of the phase difference) in terms of the fundamental mode of TW in an elliptical two-layer basin. The streamlines as constructed by [20] are sketched in Fig. 19.9.4
Fig. 19.9

Contours of the mass transport stream function of the fundamental mode for elliptical topographic waves (From Johnson (1987) [20]). © Taylor & Francis, http://www. informaworld.com, reprinted with permission

To round out this picture it must be mentioned that the TW-equation has been approximately and numerically solved by the finite element technique [47] with results which do not at all support the interpretation using elliptical TW. Trösch finds that solutions in the 65–95-h-period range are generally localized to the two narrow ends and the bay of Lugano. Figure 19.10 shows the streamline pattern of three such modes, having periods of 68.5, 80.5 and 91 h, respectively.
Fig. 19.10

Three modes of long periodic waves in Lake of Lugano obtained by the finite element technique.\(\left.\begin{array}{rcll} {T}_{1} & =&68.5&\mathrm{h} \\ {T}_{2} & =&80.5&\mathrm{h} \\ {T}_{3} & =&91\,\mathrm{h} &\end{array} \right \}\) for t = 0 (above) and t = T ∕ 4 (below).From Trösch (1986) [47], with alterations. © National Cheng Kung University Press, Tainan, Taiwan, reproduced with permission

This was the state of conflicting interpretation of the same observations in the mid-1980s of the last century. It led Stocker and Hutter to a research activity of about 5 years duration and culminated in the finding that eigenmodes of the TW-equation in a closed basin with periods close to one another can be basin filling with small or large scale structures or they can be localized in relatively small bays.

19.2.3 Other Lakes and Ocean Basins

Stocker and Hutter [42] present 10 ∘ C-isotherm–depth data at ten different moorings distributed in Lake Zurich and give evidence that the consistently observed high amplitudes in the period range 100–110 h give rise to the suspicion of a counterclockwise propagation of an oscillating signal. Furthermore, rotary spectra of the water current at the positions and at depth show that near the 100 h period (1) the rotation of the current vector at the mid-lake position is predominantly counterclockwise, but (2) it is predominantly clockwise at some near-shore positions. This gives support for the suspicion that the 100 h signal could be a TW.

Csanady’s [7] coastal strip model that is applied to Lake Ontario is based on Gill and Schumann’s [9] shelf wave analysis, but describes a local near shore wave, which, in the 25 day period range in later work by Marmorino [28], is interpreted as a free TW. These waves are no basin wide response and therefore do not fall into the category dealt with in this chapter.

There is ample further evidence of vorticity generated flow features reminiscent of TW-behaviour. Saylor and Miller [36] also analyze time series of water currents from instruments moored in Lakes Erie and Huron at several offshore positions. They find for these lakes that in the period band of 85–125 h kinetic energy associated with anticlockwise (clockwise) rotation of the current vector is accumulated for ‘mid-lake’ (near-shore) mooring stations, suggesting a fundamental TW-mode.

Numerical studies of the wind-induced circulation in the Baltic Sea [373925] and interpretation of infrared satellite imagery [24], indicate local current patterns with cyclonically rotating gyres that are reminiscent of TWs induced by winds. Figure 19.11 shows the observed and computed vertically integrated transport in the Bornholm basin located in the South-West Baltic Sea after a strong wind from the Northwest. This current pattern was established from a configuration of completely reversed flow before the onset of the wind, thus confirming the basic concept of large-scale vorticity generation by the interaction of wind stress and bottom slope. The figure also shows the topography and its approximation by a circular basin. Adopting this circular basin, Wenzel [49] demonstrates that there are good reasons to assume that the system of gyres is an excited higher mode of TWs with a period of approximately 5 days. Kielmann [24] reaches similar conclusions.
Fig. 19.11

Observed (dark arrows) and computed (light arrows) vertically integrated transports in the South–West Baltic Sea after a strong wind from the Northeast, i.e. from the upper right-hand corner (From Simons [37], with additions; see also [38]). © Wiley Blackwell, reproduced with permission

Similar vortex systems for the vertically integrated transport of the Gulf of Bothnia are also attributed to wind generated topographic response [25]. Bäuerle [3] models this gulf as a channel having a trench profile; he constructs numerical TW solutions having periods of 70–75 h, but abstains from a comparison of his results with observations.

We may conclude this brief overview with the following slightly simplified statements:
  1. 1.

    There exists a large amount of episodic and isolated observational evidence which suggests that long periodic oscillating responses in lakes and local areas of such basins may be explained in terms of TW-models.

     
  2. 2.

    Coherent temperature and current data covering an entire basin for a period of longer duration do not seem to be available in order to clearly identify the primary cause of the motion and to interpret the observations uniquely in terms of a model.

     
  3. 3.

    It appears that long periodic circulation features which are the immediate result of strong winds, can be explained by simple idealized or more realistic and complicated models and both yield very similar if not identical results. Long periodic features which are the direct result of a strong wind gust permit interpretation in terms of TWs.

     
  4. 4.

    On the other hand, inferences from simple (analytically accessible) models and more realistic (only numerically exploitable) models which attempt to explain basin-wide TW-behaviour are conflicting. Hence, interpretation of basin-wide long periodic oscillations remains an open problem, at least as long as one cannot assume with certainty that the numerically discretized models generate flows which are the approximation (in a well defined sense) of the original nondiscretized TW-equation to the real topography.

     

The above statements, which were made more than 20 years ago by Stocker and Hutter [42] are still correct today but verification by measurements is difficult. The reason is that basin- and bay-filling TWs often have eigenfrequencies which are very close to each other, whereas their eigen-functions are far apart. This fact suggests observationally that it is very difficult to separate the different modes from synoptically taken data, whilst numerically high accuracy is needed to capture modes which are close to each other. This is what we attempt to elucidate in this and the two following chapters.

19.3 Baroclinic Coupling: The Two-Layer Model

Having discussed the purely barotropic topographic Rossby waves already, we proceed directly to their description in a stratified fluid and commence with a two-layer configuration.

Motions occur in both layers and are subject to a coupling by the thermocline. As we shall show later on, this coupling mechanism is weak in the sense that it is mainly one-way, i.e. the motion of the thermocline is driven by the barotropic transport. If the velocity fields in the two layers are unidirectional the motion is barotropic, if they are in opposite directions it is baroclinic.

The configuration of the lake and the notation is summarized in Fig. 19.12. Important in the depicted geometry are the vertical side walls that extend beyond the thermocline well into the hypolimnion. Application must, therefore, be limited to lakes with steep shores.
Fig. 19.12

Side view of a cross section of the two-layer lake in its natural coordinate system (x, y, z). Upper and lower layer variables are denoted by an index 1 or 2, respectively. The lake is within a rotating system of spatially constant angular velocity (1 ∕ 2)f (From [40]). © Versuchsanstalt für Wasserbau, Hydrologie und Glaziologie an der ETH Zürich, reproduced with permission

Lake topography varies in space only in the lower layer, i.e. the upper layer is confined by two vertical side walls, which must exceed the depth of the thermocline, so \(H(\vec{x}) > {D}_{1}\) for all \(\vec{x}\). We accept the varying thickness of the side walls with \(\vec{x}\) because of analytical simplicity.

19.3.1 Two-Layer-Equations

Basic idea in obtaining a description of the physical behaviour of our system is to formulate equations which describe conservation of mass, momentum** and energy for the individual layers. Thermodynamic effects will be neglected in this study. The evolving nonlinear system is linearized by the assumption of small Rossby numbers. Furthermore, surface elevations ζ are thought to be small in comparison to the depth of the upper layer. Turbulence will be ignored but wind stress, distributed over the thin upper layer, and acting as a driving force will be considered. Under these approximations, the equations of motion in components of a Cartesian system take the forms (Mysak [31], p 87, Mysak et al. [33])
$$\begin{array}{rcl} & & {u}_{1t} - f{v}_{1} = -g{\zeta }_{1x} + {\tau }^{x}/({\rho }_{ 1}{D}_{1}), \\ & & {v}_{1t} +{ {\it { fu}}}_{1} = -g{\zeta }_{1y} + {\tau }^{y}/({\rho }_{ 1}{D}_{1}), \\ & & {D}_{1}({u}_{1x} + {v}_{1y}) = {\zeta }_{2t} -\underline{{\zeta }_{1t}}, \\ \end{array}$$
(19.5)
$$\begin{array}{rcl} & & {u}_{2t} - f{v}_{2} = -g{\zeta }_{1x} - g\prime ({\zeta }_{2x} -\underline{{\zeta }_{1x}}), \\ & & {v}_{2t} +{ {\it { fu}}}_{2} = -g{\zeta }_{1y} - g\prime ({\zeta }_{2y} -\underline{{\zeta }_{1y}}), \\ & & {({H}_{2}{u}_{2})}_{x} + {({H}_{2}{v}_{2})}_{y} = -{\zeta }_{2t}, \\ \end{array}$$
(19.6)
where g′ is the reduced gravity g′ = g2 − ρ1) ∕ ρ2. Subscripts denote differentiation with respect to the subscript-variable. Everything that follows can be directly derived from (19.5)–(19.6).

19.3.2 Approximations

We will now transform the above equations and introduce further approximations which will make it apparent why the conservation law of potential vorticity is still a reasonable approximation for vorticity waves when barotropic–baroclinic coupling is present.

19.3.2.1 Rigid-Lid Approximation

It is known that to every wave type of the above system there exists an internal and an external variant. The periods of the latter are generally much smaller than those of the former and, by applying the rigid lid approximation, the external modes are impeded. This means, that compared to the interface elevation any surface elevation can be neglected, i.e. the underlined terms in (19.5) and (19.6) are ignored. With this, it follows from the mass conservations (19.5)3 and (19.6)3 that the quasi-solenoidal velocity field can be replaced by the stream function through which the components of the integrated transport are given by
$$\begin{array}{rcl} -{\psi }_{y} = {D}_{1}{u}_{1} + {H}_{2}{u}_{2},\;\;\;{\psi }_{x} = {D}_{1}{v}_{1} + {H}_{2}{v}_{2}.& &\end{array}$$
(19.7)
ψ is called the barotropic or mass transport stream function. Equations (19.5) and (19.6) can be transformed into a compact system in the variables ψ and \({\zeta }_{2} \equiv \zeta \). Assuming a constant Coriolis parameter f the result reads
$$\begin{array}{rcl} & & \hspace{-6.0pt}\nabla \cdot ({H}^{-1}\nabla {\psi }_{ t}) + f(\nabla \psi \times \nabla {H}^{-1}) \cdot \hat{\vec{ k}} \\ & & \quad = -g\prime {D}_{1}(\nabla \zeta \times \nabla {H}^{-1}) \cdot \hat{\vec{ k}} + \frac{1} {{\rho }_{1}}[\nabla \times (\vec{\tau }{H}^{-1}) + \frac{H} {{D}_{1}}\vec{\tau } \times \nabla {H}^{-1}] \cdot \hat{\vec{ k}},\qquad \quad \end{array}$$
(19.8)
$$\begin{array}{rcl} & & H{\nabla }^{2}{\zeta }_{ t} - \frac{{H}^{2}} {g\prime {D}_{1}{H}_{2}}\mathcal{L}{\zeta }_{t} + \frac{{D}_{1}} {{H}_{2}}\nabla {\zeta }_{t} \cdot \nabla H -\frac{f{D}_{1}} {{H}_{2}} (\nabla \zeta \times \nabla H) \cdot \hat{\vec{ k}} \\ & & \qquad = \frac{1} {g\prime {H}_{2}}[\nabla (\mathcal{L}\psi ) \times \nabla H] \cdot \hat{\vec{ k}} - \frac{H} {{\rho }_{1}g\prime {D}_{1}}f(\nabla \times \mathcal{L}\vec{\tau }) \cdot \hat{\vec{ k}},\quad \end{array}$$
(19.9)
where \(\hat{\vec{k}}\) is a unit vector in the positive z-direction and the operator \(\mathcal{L} = {\partial }_{{\it { tt}}} + {f}^{2}\) has been introduced.

Problem 19.1

Derive ( 19.8 ) and ( 19.9 ) from ( 19.5 ) and ( 19.6 ) and show in the process of derivation that the formulae ( 19.10 ) below for the horizontal velocities can be derived.

Mysak et al. [33] give a detailed discussion of the physics of (19.8) and (19.9), which is now briefly summarized. In the absence of stratification (g′ = 0) and wind forcing (\(\vec{\tau } =\vec{ 0}\)), (19.8) reduces to the conservation law of potential vorticity. Wind is the external force; the second term on the right-hand side of (19.8) may therefore be interpreted as the supply of potential vorticity due to wind action. Stratification (g′≠0) in a basin with variable topography (\(\nabla H\neq \vec{0}\)) couples the barotropic part of (19.8), namely its left-hand side, with the baroclinic processes. The first term on the right-hand side of (19.8) is, therefore, the production of potential vorticity due to baroclinicity; it represents the influence of the baroclinic effects on the barotropic motion.

By the same argument, (19.9) describes the influence of the barotropic processes (terms involving ψ) and the wind (terms involving \(\vec{\tau }\)) on the baroclinic motion. Ignoring these barotropic terms results in an equation describing internal waves with a phase speed
$${c}_{\mathrm{int}}^{2} = g\prime {D}_{ 1}{H}_{2}/H.$$

When \(\nabla H =\vec{ 0}\) the third and fourth term on the left-hand side vanish, and the equation describes classical internal Kelvin and Poincaré waves. Thus, (19.8) and (19.9) exhibit in general a two-way coupling, a baroclinic–barotropic coupling and a barotropic–baroclinic coupling the strengths of which must be estimated by a scale-analysis.

When deriving (19.8) and (19.9) from (19.5) and (19.6) the layer velocities can be expressed in terms of ψ and ζ. The expressions are
$$\begin{array}{rcl} \hspace{-14.22636pt}\mathcal{L}\vec{{u}}_{1}& =& \frac{1} {H}\left [\hat{\vec{k}} \times \nabla (\mathcal{L}\psi ) + {H}_{2}g\prime (\nabla {\zeta }_{t} - f\hat{\vec{k}} \times \nabla \zeta ) + \frac{{H}_{2}} {{\rho }_{1}{D}_{1}}(\vec{{\tau }}_{t} - f\hat{\vec{k}} \times \vec{ \tau })\right ], \\ & & \\ \hspace{-14.22636pt}\mathcal{L}\vec{{u}}_{2}& =& \frac{1} {H}\left [\hat{\vec{k}} \times \nabla (\mathcal{L}\psi ) - {D}_{1}g\prime (\nabla {\zeta }_{t} - f\hat{\vec{k}} \times \nabla \zeta ) - \frac{1} {{\rho }_{1}}(\vec{{\tau }}_{t} - f\hat{\vec{k}} \times \vec{ \tau })\right ], \\ \end{array}$$
(19.10)
(see Problem 19.1) which are additively composed of three parts, i.e. a barotropic, a baroclinic and a wind force component. The first are the same (and unidirectional) in both layers, and the second are in opposite directions and add up to vanishing total transport, reminiscent of barotropic and baroclinic behaviour, respectively.

19.3.2.2 Low-Frequency Approximation

In (19.9), ζ appears with a third order time derivative. This means that (19.9) can contain three types of waves. In fact, a more precise analysis shows that there are two (internal) gravity waves and one topographic wave of which the latter has the longest period. Because we want to study here topographic waves, we will search for solutions of (19.8) and (19.9) with low frequency ω. For ω ≪ f we may therefore neglect ω in comparison to f. Thus, \(\mathcal{L}\) reduces to \(\mathcal{L}\approx {f}^{2}\). Such an approximation, however, requires that periods are substantially greater than about 17 h (the inertial period corresponding to f at 45 ∘  latitude).

Parenthetically, we might also mention that this approximation holds only for lakes in which the internal seiche period (of a gravity or Kelvin wave) is considerably smaller than the period of topographic waves. Since the former increases with the lake dimension and the latter is size-invariant, the frequencies of gravity waves in larger lakes become of comparable order to those of topographic waves. For Lakes Zurich and Lugano the approximation is appropriate, for Lake Geneva or larger lakes it may be dubious, see Table 19.2.
Table 19.2

The gap between the eigenperiods of internal gravity and topographic waves depends on the lake dimension

Lake

Surface

Period of internal

Period of

 
 

length

gravity waves

topographic waves

 
 

[km]

[h]

[h]

 

Lugano

17. 2a

 ≤ 28a

74b

 

Zurich

28a

 ≤ 45a

100b

 

Geneva

72c

 ≤ 78d

72–96b

 

aHutter, 1983 [15] bMysak, 1985 [31] cGraf, 1983 [10]dBäuerle, 1985 [4]

The situation is nevertheless not as limiting as this statement might let us surmise, because we shall prove below that for many situations the baroclinic–barotropic coupling term on the right-hand side of (19.8) may safely be ignored. In this case, the TW-equation (19.8) uncouples from (19.9). Since also boundary conditions will be shown to be free of this baroclinic–barotropic coupling, solutions to the TW-problem can be obtained without solving the inertial gravity wave problem. The assumption ∣ω∣ ≪ f need not necessarily be invoked.

19.3.3 Scale Analysis

Information about the orders of magnitude of the various coupling terms in (19.8) and (19.9) is obtained by constructing dimensionless counterparts of these equations via the introduction of scales.

19.3.3.1 Wind Forcing

The external forcing mechanism in (19.8) and (19.9) is the wind. To estimate its relative importance, consider the identity
$$\begin{array}{rcl} \nabla \times (\vec{\tau }{H}^{-1})+ \frac{H} {{D}_{1}}\vec{\tau } \times \nabla {H}^{-1} \equiv {H}^{-1}(\nabla \times \vec{ \tau })+(\nabla {H}^{-1}) \times \vec{ \tau } + \frac{H} {{D}_{1}}\vec{\tau } \times \nabla {H}^{-1}.& & \\ & &\end{array}$$
(19.11)
The first term on the right can be neglected in comparison to the others, because the atmospheric length scale is in general much larger than the lake dimensions. Such a statement is tantamount to ignoring spatial variations of wind stress over the domain of the lake. Further, comparing the last two terms it is seen that they differ by an order D 1 ∕ H which, in view of our basic assumption, is small (cf. Table 19.3). Consequently, only the last term of (19.11) survives. In a way, this is a strange result: As far as the barotropic contribution of the motion is concerned, only a lake with variable topography can be affected by the wind. This leads to the conclusion that the assumption on atmospheric length scales may be doubtful. Indeed, varying topography in the vicinity of a lake may play a significant role as it can modify regional winds with atmospheric length scales to local winds with smaller length scales. An example is the topography around Lake of Lugano; but experimental evidence for the wind stress curl to be significant is so far lacking.
Table 19.3

Properties of some Swiss lakes

Lake

D 1

D 2 mean

D 2 max

\(\frac{{D}_{1}} {{D}_{2}^{\mathrm{mean}}}\)

\(\frac{{\rho }_{2}-{\rho }_{1}} {{\rho }_{2}}\)

R i

Half length

S  − 1

ζ0

 
 

[m]

[m]

[m]

  

[km]

[km]

 

[m]

 

Lugano

10 a

183 a

278 b

0.055

1. 91 ⋅10 − 3 a

4.05

8.6

4.5

1. 8 e

 

Zurich

12 a

52 a

124 b

0.231

1. 75 ⋅10 − 3 a

4.13

14

11.5

2. 9 e

 

Geneva

15 d

153 d

310 c

0.098

1. 41 ⋅10 − 3 d

4.24

36

72.1

6. 9 e

 

a Hutter, 1984b [17], p. 78b Hutter, 1983 [15], p. 1088c Graf, 1983 [10], p. 64d Bäuerle, 1984 [3]e Computed using (19.21)

19.3.3.2 Gratton’s Scaling

Gratton [11] and Gratton and LeBlond [12] consider lake stratifications with D 1 ≪ D 2, i.e. a thin upper layer lies on top of a deep hypolimnion. For this case they found that the baroclinic effect on the barotropic motion is of order D 1 ∕ D 2 smaller than the barotropic effect on the baroclinic motion. So, to order D 1 ∕ D 2 the coupling only arises as a forcing of the baroclinic motion by the barotropic mass transport.

Before we demonstrate this result let us point out its significance. The one-way coupling means that traces of the topographically induced motion can be observed by measuring baroclinic quantities such as temperature-time series of thermistor chains, moored within the metalimnion. The description of the observations in Lakes of Lugano in Sect. 19.1 are based on such temperature-time series.

We now introduce the following set of nondimensional variables:
$$\begin{array}{rcl} & & \psi := {\psi }_{0}\psi \prime,\;\;\;\zeta := {\zeta }_{0}\zeta \prime,\;\;\;\vec{\tau } := {\tau }_{0}\vec{\tau }\prime, \\ & & (x,y) := L(x\prime,y\prime ),\;\;\;\;t := {f}^{-1}t\prime, \\ & & H := Dh\prime = ({D}_{1} + {D}_{2})h\prime,\;\;{H}_{2} := {D}_{2}{h}_{2}, \\ \end{array}$$
(19.12)
where the primed variables are non-dimensional; L is a typical length scale of the considered waves (e.g. half the lake length). Higher wave modes, where cross variations are important, may require a (x, y)-scaling which is different for each spatial direction, but this will not be considered. With (19.12) we obtain the scaled equations as
$$\begin{array}{rcl} & & \nabla \cdot \left ({h}^{-1}\nabla {\psi }_{ t}\right ) + \left (\nabla \psi \times \nabla {h}^{-1}\right ) \cdot \hat{\vec{ k}} \\ & & \qquad = -{C}_{1}\left (\nabla \zeta \times \nabla {h}^{-1}\right ) \cdot \hat{\vec{ k}} + \left ( \frac{LD{\tau }_{0}} {f{\rho }_{1}{D}_{1}{\psi }_{0}}\right )\left (h\vec{\tau } \times \nabla {h}^{-1}\right ) \cdot \hat{\vec{ k}}, \end{array}$$
(19.13)
$$\begin{array}{rcl} & & \frac{1} {h}\left ({\nabla }^{2}{\zeta }_{ t} - \frac{{L}^{2}h} {{R}_{\mathrm{int}}^{2}{h}_{2}}\mathcal{L}{\zeta }_{t}\right ) - \frac{{D}_{1}} {{D}_{2}{h}_{2}}\nabla {\zeta }_{t} \cdot \nabla {h}^{-1} + \frac{{D}_{1}} {{D}_{2}{h}_{2}}\left (\nabla \zeta \times \nabla {h}^{-1}\right ) \cdot \hat{\vec{ k}} \\ & & \qquad = -{C}_{2}{h}_{2}^{-1}\left (\nabla \psi \times \nabla {h}^{-1}\right ) \cdot \hat{\vec{ k}} - \frac{{\tau }_{0}L} {{\rho }_{1}g\prime {D}_{1}{\zeta }_{0}} \frac{1} {h}\left (\nabla \times \mathcal{L}\vec{\tau }\right ) \cdot \hat{\vec{ k}}, \end{array}$$
(19.14)
where now \(\mathcal{L} = {\partial }_{{\it { tt}}} + 1\) and the coupling coefficients are given by
$$\begin{array}{rcl}{ C}_{1} = \frac{g\prime {D}_{1}{\zeta }_{0}} {f{\psi }_{0}},\qquad {C}_{2} = \frac{f{\psi }_{0}} {g\prime {D}_{2}{\zeta }_{0}} = \frac{{D}_{1}} {{D}_{2}} \cdot \frac{1} {{C}_{1}},& &\end{array}$$
(19.15)
and R int = g′D 1 D 2 ∕ (Df 2)1 ∕ 2 is the internal Rossby radius. Note that in (19.13) and (19.14) we have dropped the primes on the scaled (nondimensional) variables.
Let us now suppose that (19.13) and (19.14) are strongly coupled, i.e. that C 1 and C 2 are both O(1). Then (19.15) implies that
$$\begin{array}{rcl}{ \zeta }_{0} = O\left (\frac{f{\psi }_{0}} {g\prime {D}_{1}}\right )\quad \mbox{ and}\quad {\zeta }_{0} = O\left (\frac{f{\psi }_{0}} {g\prime {D}_{2}}\right ),& &\end{array}$$
(19.16)
and we observe that independent of the ψ0-scale, (19.16)1, 2 are consistent only if D 1 ∕ D 2 = O(1). Since we are concerned with the case D 1 ≪ D 2, it follows that C 1 and C 2 cannot both be of order unity, i.e. that (19.13) and (19.14) are only weakly coupled. Suppose we assume that (19.16)1 applies and thus choose
$$\begin{array}{rcl} \frac{{\zeta }_{0}} {{\psi }_{0}} = \frac{f} {g\prime {D}_{1}}& &\end{array}$$
(19.17)
as the scaling for the ratio ζ0 ∕ ψ0. Then C 1 = 1 and C 2 = D 1 ∕ D 2 ≪ 1. Therefore, to O(D 1 ∕ D 2), (19.14) reduces to
$$\begin{array}{rcl} \left ({\nabla }^{2} -{\left ( \frac{L} {{R}_{\mathrm{int}}}\right )}^{2}\mathcal{L}\right ){\zeta }_{ t} = - \frac{{\tau }_{0}L} {{\rho }_{1}g\prime {D}_{1}{\zeta }_{0}}(\nabla \times \mathcal{L}\vec{\tau }) \cdot \hat{\vec{ k}},& &\end{array}$$
(19.18)
where we have used h ∕ h 2 = 1 + O(D 1 ∕ D 2). Equation (19.18) is a wave equation forced by the wind stress curl, but the scale choice (19.17) leads to an unrealistically large value for the ζ0 scale (about ζ0 ≤ 50 m, which is several times the upper layer depth for most lakes5).
Hence, we are compelled to choose the scaling (19.16)2 (Gratton’s choice, which was based on data from the Strait of Georgia, British Columbia). Putting
$$\begin{array}{rcl} \frac{{\zeta }_{0}} {{\psi }_{0}} = \frac{f} {g\prime {D}_{2}},& &\end{array}$$
(19.19)
we find C 2 = 1 and C 1 = D 1 ∕ D 2 ≪ 1, We choose the ζ0 scale by setting the coefficient of the wind stress term in (19.13) equal to unity, which gives
$$\begin{array}{rcl}{ \psi }_{0} = \frac{LD{\tau }_{0}} {f{\rho }_{1}{D}_{1}}.& &\end{array}$$
(19.20)
Substituting (19.20) into (19.19) gives the scale ζ0 in terms of the wind stress scale τ0:
$$\begin{array}{rcl}{ \zeta }_{0} = \frac{LD{\tau }_{0}} {{\rho }_{1}g\prime {D}_{1}{D}_{2}} = \frac{{\psi }_{0}f} {g\prime {D}_{2}},& &\end{array}$$
(19.21)
which yields a realistic order of magnitude.6 Using (19.20) and (19.21) in (19.13) and (19.14), we obtain, correct to O(D 1 ∕ D 2)
$$\begin{array}{rcl} & & \nabla \cdot ({h}^{-1}\nabla {\psi }_{ t}) + (\nabla \psi \times \nabla {h}^{-1}) \cdot \hat{\vec{ k}} = (h\vec{\tau } \times \nabla {h}^{-1}) \cdot \hat{\vec{ k}},\end{array}$$
(19.22)
$$\begin{array}{rcl} & & ({\nabla }^{2} - {S}^{-1}\mathcal{L}){\zeta }_{ t} = -(\nabla \psi \times \nabla {h}^{-1}) \cdot \hat{\vec{ k}} - (\nabla \times \mathcal{L}\vec{\tau }) \cdot \hat{\vec{ k}},\end{array}$$
(19.23)
as the appropriate non-dimensional equations for ψ and ζ. In (19.23), note that we have introduced the stratification parameter S (Burger number), defined as
$$\begin{array}{rcl} S = {({R}_{\mathrm{int}}/L)}^{2}.& &\end{array}$$
(19.24)
For the derivation of (19.23) it is important that h 2≠0 (as illustrated in Fig. 19.12). If h 2 = 0, the third and fourth terms on the left side of (19.14) are not uniformly O(D 1 ∕ D 2) and hence could not be neglected. If an elliptic paraboloid contained a two-layer fluid, then clearly h 2 = 0, where the interface touches the basin wall and (19.23) would not be valid. Thus, Ball’s [2] solution7 for second-class waves in an elliptic paraboloid could not be easily extended to the stratified case by our analysis.

The derivation of (19.22) and (19.23) follows Mysak et al. [33] but is more general in that the low frequency assumption has not been invoked and the wind stress curl has not been ignored. With these two additional assumptions \(\mathcal{L}\) would be replaced by \(\mathcal{L} = 1\) and the last term in (19.23) would be missing. As stated above these assumptions are not needed to achieve the decoupling of the barotropic motion from the baroclinic influence.

Substituting (19.12) and the scaling (19.16)2 into (19.10) and using the scale ψ0 = ULD, as before, we obtain the following formulae for the velocities:
$$\begin{array}{rcl} \mathcal{L}\vec{{u}}_{1}& =& \frac{1} {h}\left [\hat{\vec{k}} \times \nabla \mathcal{L}\psi + {h}_{2}\left ((\nabla {\zeta }_{t} -\hat{\vec{ k}} \times \nabla \zeta ) + \frac{{D}_{2}} {D} (\vec{{\tau }}_{t} -\hat{\vec{ k}} \times \vec{ \tau })\right )\right ],\qquad \qquad \end{array}$$
(19.25)
$$\begin{array}{rcl} \mathcal{L}\vec{{u}}_{2}& =& \frac{1} {h}\left [\hat{\vec{k}} \times \nabla \mathcal{L}\psi -\frac{{D}_{1}} {{D}_{2}}\left ((\nabla {\zeta }_{t} -\hat{\vec{ k}} \times \nabla \zeta ) + \frac{{D}_{2}} {D} (\vec{{\tau }}_{t} -\hat{\vec{ k}} \times \vec{ \tau })\right )\right ].\qquad \qquad \end{array}$$
(19.26)
To O(D 1 ∕ D 2) these can be approximated by
$$\begin{array}{rcl} \mathcal{L}\vec{{u}}_{1}& =& \frac{1} {h}\left [\hat{\vec{k}} \times \nabla \mathcal{L}\psi + h(\nabla {\zeta }_{t} -\hat{\vec{ k}} \times \nabla \zeta +\vec{ {\tau }}_{t} -\hat{\vec{ k}} \times \vec{ \tau })\right ],\end{array}$$
(19.27)
$$\begin{array}{rcl} \mathcal{L}\vec{{u}}_{2}& =& \frac{1} {h}\hat{\vec{k}} \times \nabla \psi. \end{array}$$
(19.28)
Thus, for deep lakes, the lower layer current associated with topographic waves is essentially barotropic, whereas the upper layer current consists of a barotropic part, a baroclinic part and a contribution directly forced by the wind. Hence, we conclude that the current motions are generally surface intensified.

In Table 19.3, we collect some data pertinent to the above estimates. Values are given for the layer thickness and density difference of the summer stratification for three Swiss lakes from which Rossby radii, stratification parameters and typical values of the thermocline elevation can be computed. Accordingly, neglecting O(D 1 ∕ D 2)-terms is certainly justified for Lake of Lugano and still reasonable for the other lakes. Moreover, the thermocline-elevation amplitude ζ0 is smaller than D 1 in all three cases, a fact which gives some confidence in the scaling procedure.

19.3.4 Boundary Conditions

To solve (19.22) and (19.23) in some domain \(\mathcal{D}\) for a given \(\vec{\tau }\), we have to prescribe initial values for ψ and ζ and the boundary conditions on \(\partial \mathcal{D}\). The first boundary condition which we impose is that the total mass flux normal to \(\partial \mathcal{D}\) must vanish: in non-dimensional variables this can be written as \(\hat{\vec{n}} \cdot ({D}_{1}\vec{{u}}_{1} + {D}_{2}{h}_{2}\vec{{u}}_{2}){D}^{-1} = 0\) on \(\partial \mathcal{D}\), where \(\hat{\vec{n}}\) is a unit vector perpendicular to \(\partial \mathcal{D}\). On substituting for \(\vec{{u}}_{1}\) and \(\vec{{u}}_{2}\) from (19.25) and (19.26), this reduces to8
$$\begin{array}{rcl} \hat{\vec{n}} \cdot (\hat{\vec{k}} \times \nabla \psi ) = 0,\;\;\mbox{ on}\;\partial \mathcal{D}.& &\end{array}$$
(19.29)
Since \(\hat{\vec{n}} \cdot (\hat{\vec{k}} \times \nabla \psi ) = (\hat{\vec{n}} \times \hat{\vec{ k}}) \cdot \nabla \psi =\hat{\vec{ s}} \cdot \nabla \psi \), where \(\hat{\vec{s}}\) is a unit vector tangential to \(\partial \mathcal{D}\), (19.29) implies ψ ∕ ∂s = 0 on \(\partial \mathcal{D}\) and hence ψ = constant on \(\partial \mathcal{D}\). Thus, in a simply connected domain, without loss of generality, we take
$$\begin{array}{rcl} \psi = 0,\;\;\;\mbox{ on}\;\partial \mathcal{D}.& &\end{array}$$
(19.30)
Next we require \(\hat{\vec{n}} \cdot \vec{ {u}}_{i} = 0\) on \(\partial \mathcal{D}\) for each layer i. Upon again using (19.25) and (19.26), together with (19.29), we find
$$\begin{array}{rcl} \frac{\partial } {\partial n} \frac{\partial \zeta } {\partial t} -\frac{\partial \zeta } {\partial s} = -\frac{{D}_{2}} {D} (\hat{\vec{n}} \cdot \vec{ {\tau }}_{t} -\hat{\vec{ s}} \cdot \vec{ \tau }) = -\hat{\vec{n}} \cdot \vec{ {\tau }}_{t} +\hat{\vec{ s}} \cdot \vec{ \tau },\;\;\mbox{ on}\;\;\partial \mathcal{D},& &\end{array}$$
(19.31)
to O(D 1 ∕ D 2).

The boundary condition (19.29) applies, whether the simplifying assumptions D 1 ≪ D 2 and ω2 ≪ f 2 are imposed or not. Because (19.22) supposes D 1 to be small in comparison to D 2 we conclude that the barotropic part of the motion can be determined without simultaneously also determining the baroclinic response. However, if the corresponding barotropically driven baroclinic currents or thermocline elevations are to be determined, then (19.23) subject to the boundary condition (19.31) must also be solved. Since (19.23) is a forced wave equation, this by itself is a formidable problem. For weak stratification (S small) simplifications are possible. This is the case for most Swiss lakes (compare Table 19.3).

To introduce this additional simplification we note that our scales have been chosen such that dimensionless gradients are order unity. Hence, we expect ∇ 2 to be O(1) whereas S  − 1 is large. On the left-hand side of (19.23) we may thus ignore ∇ 2 in comparison to \({S}^{-1}\mathcal{L}\), implying
$$\begin{array}{rcl}{ \zeta }_{t} = S\{{\mathcal{L}}^{-1}(\hat{\vec{k}} \times \nabla \psi ) \cdot \nabla {h}^{-1} + (\nabla \times \vec{ \tau }) \cdot \hat{\vec{ k}}\},& &\end{array}$$
(19.32)
where \({\mathcal{L}}^{-1}\) is the inverse operator of \(\mathcal{L}\). Equation (19.32) can be described as the geometric optics approximation for ζ. Along the shore \(\partial \mathcal{D}\) we may assume a constant depth; then ∇ h  − 1 is parallel to \(\hat{\vec{n}}\), the unit normal vector along \(\partial \mathcal{D}\), and the first term in the curly bracket vanishes.9 With nonvanishing wind stress the emerging equation is not consistent with (19.31). For the unforced problem, however, (19.32) implies
$$\zeta (\vec{x},t) = 0,\;\;\mbox{ along}\;\partial \mathcal{D},$$
which is consistent with (19.31) provided that the term 2ζ ∕ ∂n∂t is ignored. This omission is justified in the low-frequency approximation.

We conclude: the geometric optics approximation is only consistent in the low-frequency limit. In all other cases, the baroclinic coupling should be computed with the full equations (19.22), (19.23) and (19.31).

19.4 Continuous Stratification

19.4.1 Modal Equations

The two-layer model can only nearly approximate the internal dynamics of a lake that permits a clear distinction of an epi- and a hypolimnion. It is important to investigate to which extent inferences from the pure barotropic or two-layer model of Sect. 19.3 can be carried over to a lake with continuous stratification. The oceanographic literature is rich in studies of low-frequency processes in a stratified medium. A general result, common to all studies, is that increasing the stratification increases the frequencies of the considered long-period motion. This has been shown by Wang and Mooers [48], Clarke [6] and Huthnance [14]. A review can be found in Mysak [29].

Here, in this section, our aim is to analyze, how and how strongly the individual baroclinic modes are coupled among each other and how the baroclinic part of the motion couples with the barotropic processes.

These equations are given below as (19.43) and (19.44). To derive them, we start from the continuity equation
$$\nabla \cdot \vec{v} = {\nabla }_{H} \cdot {\vec{v}}_{H} + \frac{\partial \,w} {\partial \,z} = 0$$
(19.33)
and integrate it over depth to obtain
$$\begin{array}{rcl} & & 0 ={ \int \nolimits \nolimits }_{-H}^{\zeta }\left ({\nabla }_{ H} \cdot {\vec{v}}_{H} + \frac{\partial w} {\partial z} \right )\mbox{ d}z \\ & & \stackrel{\mathrm{Leibniz}}{=}{\nabla }_{H} \cdot {{\int \nolimits \nolimits }_{-H}^{\zeta }{\vec{v}}_{ H}\mbox{ d}z }_{:=\vec{V }} -{{\left [{\vec{v}}_{H} \cdot \nabla \zeta - w\right ]}_{z=\zeta } }_{=-\partial \zeta /\partial t} -{{\left [{\vec{v}}_{H} \cdot {\nabla }_{H}H + w\right ]}_{z=-H} }_{=0} \\ & & \,\,\,\,= \frac{\partial \zeta } {\partial t} + {\nabla }_{H} \cdot \vec{V } = 0, \\ \end{array}$$
which agrees with (19.43)1 below. In the above, the bracketed terms are so identified because of the kinematic boundary conditions at the free and the basal surfaces. In a similar fashion, the horizontal components of the momentum equations are treated. To this end, we write the stress tensor as
$$\vec{t} = -p(z)\vec{I} +{ \vec{t}}_{\mathrm{dyn}},$$
(19.34)
in which p(z) is the hydrostatic pressure, given by
$$p(z) ={ {p}_{\mathrm{atm}} + {\rho }_{{_\ast}}g(\zeta - z)}_{{p}_{e}} +{ {\rho }_{{_\ast}}g{\int \nolimits \nolimits }_{z}^{\zeta }\sigma ({z}^{{\prime}})\,\mbox{ d}{z}^{{\prime}}}_{{ p}_{i}}$$
(19.35)
and p e and p i are the external and internal pressures, respectively. \({\vec{t}}_{\mathrm{dyn}}\) is the dynamic stress contribution, mostly due to turbulence; it will be decomposed as
$${ \vec{t}}_{\mathrm{dyn}} = \left (\begin{array}{cc} \; - {\rho }_{{_\ast}}\vec{\Gamma }\;&\; - {\rho }_{{_\ast}}\vec{\gamma }\; \\ \;\;{t}_{13}\,{t}_{23}\;\; & \;{t}_{33}\; \end{array} \right ),\quad \vec{\Gamma } = \left (\begin{array}{cc} {\Gamma }_{11}\, & \,{\Gamma }_{12} \\ {\Gamma }_{21}\, & \,{\Gamma }_{22} \end{array} \right ),\quad \vec{\gamma } = \left (\begin{array}{c} {\gamma }_{31} \\ {\gamma }_{32} \end{array} \right ).$$
(19.36)
With (19.35) and (19.36), the horizontal momentum equation takes the form
$$\frac{\mathrm{d}{\vec{v}}_{H}} {\mathrm{d}t} + f\hat{\vec{k}} \times {\vec{v}}_{H} + \frac{1} {{\rho }_{{_\ast}}}{\nabla }_{H}({p}_{e} + {p}_{i}) + {\nabla }_{H} \cdot \vec{\Gamma } + \frac{\partial \vec{\gamma }} {\partial z} = \vec{0},$$
(19.37)
in which
$$\begin{array}{rcl} \frac{\mathrm{d}{\vec{v}}_{H}} {\mathrm{d}t} & =& \frac{\partial {\vec{v}}_{H}} {\partial t} + \left ({\nabla }_{H}{\vec{v}}_{H}\right ){\vec{v}}_{H} + \frac{\partial {\vec{v}}_{H}} {\partial z} w +{ \left ({\vec{v}}_{H}{\nabla }_{H} \cdot {\vec{v}}_{H} +{ \vec{v}}_{H}\frac{\partial w} {\partial z} \right )}_{=\vec{0}} \\ & =& \frac{\partial {\vec{v}}_{H}} {\partial t} + {\nabla }_{H} \cdot \left ({\vec{v}}_{H} \otimes {\vec{v}}_{H}\right ) + \frac{\partial } {\partial z}\left ({\vec{v}}_{H}w\right ). \end{array}$$
(19.38)
Substituting (19.38) in (19.37), rearranging the emerging equation and integrating this equation from z =  − H to z = ζ yields
$$\begin{array}{rcl} \vec{0}& =& {\int \nolimits \nolimits }_{-H}^{\zeta }\bigg\{\frac{\partial {\vec{v}}_{H}} {\partial t} + f\hat{\vec{k}} \times {\vec{v}}_{H} + \frac{1} {{\rho }_{{_\ast}}}{\nabla }_{H}({p}_{e} + {p}_{i}) \\ & & \quad +\,\, {\nabla }_{H} \cdot \left ({\vec{v}}_{H} \otimes {\vec{v}}_{H} + \vec{\Gamma }\right ) + \frac{\partial } {\partial z}\left ({\vec{v}}_{H}w + \vec{\gamma }\right )\bigg\}\,\mbox{ d}z \\ & =& \frac{\partial \vec{V }} {\partial t} + f\hat{\vec{k}} \times \vec{V } + \frac{\zeta + H} {{\rho }_{{_\ast}}} {\nabla }_{H}{p}_{e} + \frac{1} {{\rho }_{{_\ast}}}{\int \nolimits \nolimits }_{-H}^{\zeta }\nabla {p}_{ i}\,\mbox{ d}z \\ & & +\,\,{\nabla }_{H} \cdot {\int \nolimits \nolimits }_{-H}^{\zeta }\left ({\vec{v}}_{ H} \otimes {\vec{v}}_{H} + \vec{\Gamma }\right )\,\mbox{ d}z +{ \left [\vec{\gamma } -\vec{\Gamma }\nabla \zeta \right ]}_{z=\zeta } -{\left [\vec{\gamma } + \vec{\Gamma }\nabla H\right ]}_{z=-H}\qquad \\ & & -\,\,{\vec{v}}_{H}(\zeta ){{\left [\frac{\partial \zeta } {\partial t} +{ \vec{v}}_{H} \cdot \nabla \zeta - w\right ]}_{z=\zeta } }_{=0} -\,\,{\vec{v}}_{H}(-H){{\left [{\vec{v}}_{H} \cdot \nabla H + w\right ]}_{z=-H} }_{=0}.\end{array}$$
(19.39)
Here, the step from the first line to the remaining lines involved application of Leibniz’ rule when interchanging the order of differentiation and integration. Moreover, the bracketed terms in the last line vanish in view of the kinematic boundary conditions on the free surface and the base. On the other hand, with
$$\vec{n} = \frac{(-\nabla \zeta,1)} {\sqrt{1 + \vert \vert \nabla \zeta \vert {\vert }^{2}}},\quad {\vec{t}}_{\mathrm{dyn}}\vec{n} = \left (\begin{array}{cc} \; - {\rho }_{{_\ast}}\vec{\Gamma }\;&\; - {\rho }_{{_\ast}}\vec{\gamma }\; \\ \;\;{t}_{13}\,{t}_{23}\;\; & \;{t}_{33}\; \end{array} \right ),\left (\begin{array}{c} - {(\nabla \zeta )}^{\mathrm{T}} \\ 1 \end{array} \right )$$
we may write at z = ζ(x, y, t)
$$\frac{1} {{\rho }_{{_\ast}}}{\left ({\vec{t}}_{\mathrm{dyn}}\vec{n}\right )}_{H} = \frac{\vec{\Gamma }\nabla \zeta -\vec{\gamma }} {\sqrt{1 + \vert \vert \nabla \zeta \vert {\vert }^{2}}} = \frac{1} {{\rho }_{{_\ast}}}{\left ({\vec{t}}_{\mathrm{atm}}\vec{n}\right )}_{H} = \frac{1} {{\rho }_{{_\ast}}} \frac{{\vec{\tau }}_{\mathrm{wind}}} {\sqrt{1 + \vert \vert \nabla \zeta \vert {\vert }^{2}}}$$
or
$$\frac{1} {{\rho }_{{_\ast}}}{\vec{\tau }}_{\mathrm{wind}} = \vec{\Gamma }\nabla \zeta -\vec{\gamma }.$$
(19.40)
Analogously for the stress boundary condition at the basal surface,
$$-\frac{1} {{\rho }_{{_\ast}}}{\vec{\tau }}_{\mathrm{bottom}} = \vec{\Gamma }\nabla H + \vec{\gamma }.$$
(19.41)
If we now substitute (19.40) and (19.41) into (19.39), the bracketed terms in the second last line of (19.39) can be replaced by
$$\frac{1} {\rho {_\ast}}\left ({\vec{\tau }}_{\mathrm{bottom}} -{\vec{\tau }}_{\mathrm{wind}}\right ).$$
(19.42)
In summary, we therefore obtain from (19.34) and (19.39)–(19.41):
$$\begin{array}{rcl} \frac{\partial \zeta } {\partial t} + \nabla \cdot \vec{ V } = 0,\qquad \frac{\partial \vec{V }} {\partial t} + f\hat{\vec{k}} \times \vec{ V } = -\frac{H + \zeta } {\rho {_\ast}} \nabla {p}_{e} +\vec{ F},\qquad & &\end{array}$$
(19.43)
in which ζ is the surface elevation, \(\vec{V }\) the transport, \(\hat{\vec{k}}\) a unit vector pointing upwards, ρ ∗  a constant reference density and
$$\begin{array}{rcl} \hspace{-17.07164pt}&&{p}_{e }= {p}_{\mathrm{atm}} + {\rho }^{{_\ast}}g(\zeta - z), \\ \hspace{-17.07164pt}&&\vec{F} = -\frac{1} {{\rho }^{{_\ast}}}{\int \nolimits \nolimits }_{-H}^{\zeta }\nabla {p}_{ i}\:\mathrm{d}z -\nabla \cdot {\int \nolimits \nolimits }_{-H}^{\zeta }(\vec{v} \otimes \vec{ v} +\vec{ \Gamma })\:\mathrm{d}z + \frac{\vec{{\tau }}_{\mathrm{wind}} -\vec{ {\tau }}_{\mathrm{bottom}}} {{\rho }^{{_\ast}}},\qquad \\ \hspace{-17.07164pt}&&{p}_{i }= {\rho }^{{_\ast}}g{\int \nolimits \nolimits }_{z}^{\zeta }\sigma (z\prime )\mathrm{d}z\prime + p\prime,\;\;\;\;\sigma \equiv \frac{{\rho }_{0} - {\rho }^{{_\ast}}} {{\rho }^{{_\ast}}}. \end{array}$$
(19.44)
This derivation follows Hutter [16], p. 18–21. Here, p e is the external and p i the internal pressure. The latter consists of the dynamic baroclinic pressure p′ and the quasistatic baroclinic pressure due to the density anomaly σ(z), which is referred to a reference profile of density ρ0(z) of a stably stratified state at rest. The hydrostatic pressure assumption has not been made in (19.44); this is important. The force \(\vec{F}\) consists of a contribution of the baroclinic pressure p i , a term involving advection (\(\vec{v} \otimes \vec{ v}\)) and turbulent diffusion (\(\vec{\Gamma }\)), the wind stress and the bottom shear stress. In ensuing developments we ignore turbulent diffusion (\(\vec{\Gamma } \simeq \vec{ 0}\)) and advection (\(\vec{v} \otimes \vec{ v} \simeq \vec{ 0}\)), omit bottom shear (\(\vec{{\tau }}_{\mathrm{bottom}} \simeq \vec{ 0}\)) and spatial variations of the atmospheric pressure (\(\nabla {p}_{\mathrm{atm}} \simeq \vec{ 0}\)) and neglect non-linear terms such as ζ ∇ ζ. If we also invoke the rigid-lid assumption and thus eliminate surface gravity waves, (19.43) and (19.44) reduce to
$$\begin{array}{rcl} & & \nabla \cdot (H\bar{\vec{v}}) = 0, \\ & & \\ & & \frac{\partial \bar{\vec{v}}} {\partial t} + f\hat{\vec{k}} \times \bar{\vec{ v}} = -g\nabla \zeta - \frac{1} {\rho {_\ast} H}{\int \nolimits \nolimits }_{-H}^{0}\nabla p\prime \:\mathrm{d}z + \frac{\vec{\tau }} {{\rho }^{{_\ast}}H}, \\ \end{array}$$
(19.45)
where \(H\bar{\vec{v}} =\vec{ V }\) has been introduced and where \(\vec{\tau } =\vec{ {\tau }}_{\mathrm{wind}}\). In the next step we introduce the mass transport stream function ψ by setting
$$H\bar{u} = -{\psi }_{y},\;\;H\bar{v} = {\psi }_{x}$$
(19.46)
and eliminate ∇ ζ from (19.45)2 by taking the curl of this equation. This transforms (19.45) to the single equation
$$\begin{array}{rcl} & & \nabla \cdot \left ( \frac{1} {H}\nabla {\psi }_{t}\right ) + \mathcal{J}\left (\psi, \frac{f} {H}\right ) \\ & & \quad = - \frac{\partial } {\partial x}\left [ \frac{1} {{\rho }^{{_\ast}}H}{\int \nolimits \nolimits }_{-H}^{0}\frac{\partial p\prime } {\partial y}\:\mathrm{d}z\right ] + \frac{\partial } {\partial y}\left [ \frac{1} {{\rho }^{{_\ast}}H}{\int \nolimits \nolimits }_{-H}^{0} \frac{\partial p\prime } {\partial x}\:\mathrm{d}z\right ] + \nabla \times \left ( \frac{\vec{\tau }} {{\rho }^{{_\ast}}H}\right )\!. \\ & & \end{array}$$
(19.47)
This equation is analogous to (19.8). Notice that it involves no term due to the stratification of the state defined by ρ0(z). Baroclinic effects are all contained in the dynamic pressure p′, but the terms involving p′ may also describe deviations from the hydrostatic pressure distribution. We shall interpret p′ as being due to baroclinic effects. Thus, in the absence of stratification and without wind forcing, (19.47) reduces to the conservation law of potential vorticity. Stratification and external winds (the terms of the right-hand side of (19.47)) act as supplies of potential vorticity. Thus, the first term on the right-hand side couples the barotropic part of (19.47) to the baroclinic processes. Our experience with the two-layer model suggests that this baroclinic coupling is small and can be ignored to lowest order. If this is correct, the barotropic motion can be fully determined from equations (19.45)–(19.47) by simply omitting the p′-dependent terms.
To complete the formulation we still need a system of equations that describes the baroclinic processes and is coupled to the barotropic motion. To deduce it let us consider the Boussinesq approximated adiabatic equations of motion (see e.g. Chap. 4, (4.235)–(4.237) in Volume I)
$$\begin{array}{rcl} \begin{array}{ll} {u}_{t} -\mathit{fv} = -\frac{1} {{\rho }^{{_\ast}}}{p}_{x}\prime, & \\ \\ \\ &{\rho }_{t}\prime + \frac{\mathrm{d}{\rho }_{0}} {\mathrm{d}z} w = 0,\\ \\ \\ {v}_{t} + \mathit{fu} = -\frac{1} {{\rho }^{{_\ast}}}{p}_{y}\prime, & \\ \\ \\ &{w}_{t} + \frac{g} {{\rho }^{{_\ast}}}\rho \prime + \frac{1} {{\rho }^{{_\ast}}}{p}_{z}\prime = 0, \\ \\ \\ {u}_{ x} + {v}_{y} + {w}_{z} = 0,\qquad &\end{array} & &\end{array}$$
(19.48)
in which subscripts denote differentiation with respect to the subscripted variable. The first two of these are the horizontal momentum equations, the third expresses continuity, the fourth derives from the adiabatic heat equation, dT(ρ) ∕ dt = 0. Finally, the last equation is the vertical momentum equation; if w t were ignored, the hydrostatic pressure assumption would result from it.
The two equation sets, (19.47), describing the transport, and (19.48), representing the vertical details, suggest to split the velocity field into two parts,
$$\begin{array}{rcl} (u,v) = (\bar{u},\bar{v}) + (\tilde{u},\tilde{v}),& &\end{array}$$
(19.49)
such that the total transport of (u, v) is incorporated in the barotropic field and hence
$$\begin{array}{rcl} {\int \nolimits \nolimits }_{-H}^{0}\tilde{u}\:\mathrm{d}z ={ \int \nolimits \nolimits }_{-H}^{0}\tilde{v}\:\mathrm{d}z = 0.& &\end{array}$$
(19.50)
Substitution of (19.49) in (19.48) and use of (19.45) yields the new set of equations
$$\begin{array}{rcl} & & \tilde{{u}}_{t} - f\tilde{v} + \frac{1} {{\rho }^{{_\ast}}}{p}_{x}\prime - \frac{1} {{\rho }^{{_\ast}}H}{\int \nolimits \nolimits }_{-H}^{0}{p}_{ x}\prime \:\mathrm{d}z = g{\zeta }_{x} - \frac{{\tau }_{1}} {{\rho }^{{_\ast}}H}, \\ & & \tilde{{v}}_{t} + f\tilde{u} + \frac{1} {{\rho }^{{_\ast}}}{p}_{y}\prime - \frac{1} {{\rho }^{{_\ast}}H}{\int \nolimits \nolimits }_{-H}^{0}{p}_{ y}\prime \:\mathrm{d}z = g{\zeta }_{y} - \frac{{\tau }_{2}} {{\rho }^{{_\ast}}H}, \\ & & \tilde{{u}}_{x} +\tilde{ {v}}_{y} + {w}_{z} = \frac{{H}_{x}} {H} \bar{u} + \frac{{H}_{y}} {H} \bar{v}, \\ & & {\rho }_{t}\prime + \frac{\mathrm{d}{\rho }_{0}} {\mathrm{d}z} w = 0, \\ & & {w}_{t} + \frac{\rho \prime } {{\rho }^{{_\ast}}}g + \frac{1} {{\rho }^{{_\ast}}}{p}_{z}\prime = 0, \\ \end{array}$$
(19.51)
in which τ1 and τ2 are the x- and y-components of the wind stress. These have been written, so that the external wind forcing and the barotropic contributions of the motion appear on the right hand sides of the equations.

19.4.2 Spectral Decomposition of the Baroclinic Fields

We shall now demonstrate how (19.51) can be reduced to a set of spatially two-dimensional equations by using a vertical shape function expansion of the velocity, pressure and density fields. To this end, let us now introduce the following expansions of the baroclinic fields:
$$\begin{array}{rcl} & & \tilde{u}(x,y,z,t) ={ \sum \nolimits }_{n=1}^{N}{U}_{ n}(x,y,t){\phi }_{n}\left ( \frac{z} {H}\right ), \\ & & \tilde{v}(x,y,z,t) ={ \sum \nolimits }_{n=1}^{N}{V }_{ n}(x,y,t){\phi }_{n}\left ( \frac{z} {H}\right ), \\ & & w(x,y,z,t) ={ \sum \nolimits }_{n=1}^{N}{W}_{ n}(x,y,t){\Xi }_{n}\left ( \frac{z} {H}\right ), \\ & & p\prime (x,y,z,t) ={ \sum \nolimits }_{n=1}^{N}{P}_{ n}(x,y,t){\psi }_{n}\left ( \frac{z} {H}\right ), \\ & & \rho \prime (x,y,z,t) ={ \sum \nolimits }_{n=1}^{N}{R}_{ n}(x,y,t){\chi }_{n}\left ( \frac{z} {H}\right ).\end{array}$$
(19.52)
Here, {ϕ n , Ξ n , ψ n , χ n }, n = 1, 2, 3, , N are treated as known and their determination will be explained below. To find the evolution equations for U n , V n , W n , P n  and R n the Principle of Weighted Residuals is used.10 It essentially amounts to evaluating the integrals
$${\int \nolimits \nolimits }_{-H}^{0}\delta {\phi }_{ m}^{M}{(19.51)}_{ 1,2}\mathrm{d}z,\;\;\int \nolimits \nolimits \delta {\phi }_{m}^{C}{(19.51)}_{ 3}\mathrm{d}z,\;\;\int \nolimits \nolimits \delta {\phi }_{m}^{V }{(19.51)}_{ 4,5}\mathrm{d}z,\;\;$$
for arbitrary weighting functions δϕ m L (L = M, C, V ). Inserting (19.52) into these expressions, (19.51) can be reduced to the following spatially two-dimensional system of differential equations for the coefficient functions U n , V n , W n , P n and R n (for details of the derivation see the Appendix at the end of this chapter).
$$\begin{array}{rcl} & & {A}_{\mathit{mn}}^{M}\left (\frac{\partial {U}_{n}} {\partial t} - f{V }_{n}\right ) - \frac{1} {{\rho }^{{_\ast}}}\frac{\partial H/\partial x} {H} {B}_{\mathit{mn}}{P}_{n} + \frac{{C}_{\mathit{mn}}} {{\rho }^{{_\ast}}} \frac{\partial {P}_{n}} {\partial x} = \left (g\frac{\partial \zeta } {\partial x} - \frac{{\tau }_{1}} {{\rho }^{{_\ast}}H}\right ){D}_{m}^{M}, \\ & & {A}_{\mathit{mn}}^{M}\left (\frac{\partial {V }_{n}} {\partial t} + f{U}_{n}\right ) - \frac{1} {{\rho }^{{_\ast}}}\frac{\partial H/\partial y} {H} {B}_{\mathit{mn}}{P}_{n} + \frac{{C}_{\mathit{mn}}} {{\rho }^{{_\ast}}} \frac{\partial {P}_{n}} {\partial y} = \left (g\frac{\partial \zeta } {\partial y} - \frac{{\tau }_{2}} {{\rho }^{{_\ast}}H}\right ){D}_{m}^{M}, \\ & & {A}_{\mathit{mn}}^{C}\left (\frac{\partial {U}_{n}} {\partial x} + \frac{\partial {V }_{n}} {\partial y} \right ) - {K}_{\mathit{mn}}\left (\frac{\partial H/\partial x} {H} {U}_{n} + \frac{\partial H/\partial y} {H} {V }_{n}\right ) - {L}_{\mathit{mn}}\frac{{W}_{n}} {H} \\ & & \;\;\;\;= \left (\bar{u}\frac{\partial H/\partial x} {H} +\bar{ v}\frac{\partial H/\partial y} {H} \right )({D}_{m}^{C} - \delta {\phi }_{ m}^{C}(\xi = 0)), \\ & & {E}_{\mathit{mn}}\frac{\partial {R}_{n}} {\partial t} -\frac{{\rho }^{{_\ast}}{N}_{\mathrm{max}}^{2}} {g} {F}_{\mathit{mn}}{W}_{n} = 0, \\ & & {G}_{\mathit{mn}}\frac{\partial {W}_{n}} {\partial t} + \frac{g} {{\rho }^{{_\ast}}}{E}_{\mathit{mn}}{R}_{n} + \frac{1} {{\rho }^{{_\ast}}H}{H}_{\mathit{mn}}{P}_{n} = 0, \\ & & \mbox{ Summation over $n$, }\;\;(m,n = 1,2,3,\ldots,N).\end{array}$$
(19.55)
The last two of these equations can also be replaced by the second order (in time) equation
$$\begin{array}{rcl} & & {G}_{\mathit{mn}}\frac{{\partial }^{2}{W}_{n}} {\partial {t}^{2}} + {N}_{\mathrm{max}}^{2}{F}_{\mathit{ mn}}{W}_{n} + \frac{{H}_{\mathit{mn}}} {{\rho }^{{_\ast}}H} \frac{\partial {P}_{n}} {\partial t} = 0, \\ & & \mbox{ Summation over $n$, }\;\;(m,n = 1,2,3,\ldots,N).\end{array}$$
(19.56)
The various coefficient matrices are collected in Table 19.4. For (19.56) to hold true, E mn must be invertible, which according to Table 19.4 is the case. The matrices A mn M , etc., are all expressible in terms of the inner product
$$\langle f(\xi ),\;g(\xi )\rangle ={ \int \nolimits \nolimits }_{0}^{1}f(\xi )g(\xi )\mathrm{d}\xi.$$
Furthermore, \(\hat{{N}}^{2}\) in Table 19.4 is the normalized Brunt–Väisälä frequency
$$\hat{{N}}^{2} = \frac{{N}^{2}(\xi ;H)} {{N}_{\mathrm{max}}^{2}} = -\frac{\mathrm{d}{\rho }_{0}/\mathrm{d}z(\xi,H)} {\mid \mathrm{d}{\rho }_{0}/\mathrm{d}z{\mid }_{\mathrm{max}}} \:,\;\;\qquad \xi = \frac{z} {H} + 1.$$
The first two of (19.55) derive from the horizontal momentum equations (19.51)1, 2, the third corresponds to the continuity equation (19.51)3 and the fourth and fifth are obtained from the adiabaticity statement (19.51)4 and the vertical momentum balance (19.51)5. It is also evident that (19.55) constitute 5N equations for the 5N unknowns {U n , V n , W n , P n , R n }, provided the barotropic quantities \(\zeta,\bar{u},\bar{v}\) are known. If they are not, (19.55) must be complemented by (19.45) which, with the use of (19.52), take on the form
$$\begin{array}{rcl} & & \frac{\partial (H\bar{u})} {\partial x} + \frac{\partial (H\bar{v})} {\partial y} = 0, \\ & & \frac{\partial \bar{u}} {\partial t} - f\bar{v} + g\frac{\partial \zeta } {\partial x} = -\frac{1} {{\rho }^{{_\ast}}}\left ({M}_{m}\frac{\partial {P}_{m}} {\partial x} + {N}_{m}\frac{\partial H/\partial x} {H} {P}_{m} + \frac{{\tau }_{1}} {H}\right )\!,\qquad \quad\\ & & \frac{\partial \bar{v}} {\partial t} + f\bar{u} + g\frac{\partial \zeta } {\partial y} = -\frac{1} {{\rho }^{{_\ast}}}\left ({M}_{m}\frac{\partial {P}_{m}} {\partial y} + {N}_{m}\frac{\partial H/\partial y} {H} {P}_{m} + \frac{{\tau }_{2}} {H}\right )\!\end{array}$$
(19.57)
The vectors M m , N m (m = 1, 2, 3, , N) are also defined in Table 19.4.
Table 19.4

Matrix elements for the expansion of the field variables in terms of the flat-bottom buoyancy modes or Jacobi polynomials. ⟨a, b⟩ is the inner product ∫ \nolimits \nolimits 0 1 abdξ, λ n is the eigenvalue defined in (19.58)

Elmt

Definition

Buoyancy mode set

Jacobi polynomial set

 

A mn L

⟨δϕ m L , ϕ n ⟩, L ∈ { M, C}

λ n δ mn

δ mn

 

B mn

⟨δϕ m M , (ξ − 1)\frac{dΦ n } {dξ} ⟩ − 

​⟨\frac{dΞ m } {dξ}, (ξ − 1)\frac{d2 Ξ m } {dξ2} ⟩​

0 (n ≤ m),   b n − 1, m  (n > m)

 
 

⟨1, (ξ − 1)\frac{dψ n } {dξ} ⟩⟨1, δϕ m M

   

C mn

​⟨δϕ m M , ψ n ⟩ − ⟨1, ψ n ⟩⟨1, δϕ m M ⟩​

λ n δ mn

δ m, n − 1

 

D m L

⟨1, δϕ m L

0

0

 

E mn

⟨δϕ m V , χ n

δ mn

δ mn

 

F mn

⟨δϕ m V , \hat{N}2 Ξ n

δ mn

G m − 1, \hat{N}2 G n − 1

 
   

( = \hat{ N}0 2δ mn ,cont. strat.)

 

G mn

⟨δϕ m V , Ξ n

Ξ m , Ξ n

δ mn

 

H mn

⟨δϕ m V , \frac{dψ n } {dξ} ⟩

λ n δ mn

0 (n ≤ m),  h n − 2, m − 1 (n > m)

 

K mn

⟨(ξ − 1)\frac{dϕ n } {dξ}, δϕ m C

B mn  − \frac{dΞ m } {dξ} \frac{dΞ n } {dξ} \big{ | }0

0 (n < m),  b mn (n ≥ m)} − 

 
 

 − δϕ m C (0)ϕ n (0)

 

( − 1) n + m ⋅\sqrt{(2n + 1)(2m + 1)}

 

L mn

⟨\frac{dδϕ m C } {dξ}, Ξ n

λ n δ mn

h m − 1, n − 1 (n ≤ m),  0 (n > m)

 

M m

⟨1, ψ m

0

δ1m

 

N m

⟨1, ψ m ⟩ − ψ m (0)

 − \frac{dΞ m } {dξ} \big{ | }ξ = 0

δ1m  + ( − 1) m \sqrt{2m − 1}

 
It is our contention that, by accordingly selecting the shape functions, the barotropic modes and the baroclinic modes can almost completely be separated. This orthogonalization is exactly possible in stratified basins with constant depth; selecting the shape functions from this set will nearly achieve the uncoupling in the case of a variable bottom. Towards a motivation, consider (19.48) and ignore the vertical acceleration terms, i.e. restrict considerations to quasi-static pressure conditions. For this case (19.48) may be reduced to the single partial differential equation for w
$${N}^{2}{\nabla }^{2}w + \left ( \frac{{\partial }^{2}} {\partial {t}^{2}} + {f}^{2}\right )\frac{{\partial }^{2}w} {\partial {z}^{2}} = 0.$$
Subject to the boundary conditions
$$w = 0,\;\mbox{ at}\;z = 0,-H,$$
we know that this equation permits separation of variable solutions w(x, y, z, t) = W n (x, y, t)Z n (z), where Z n (z) satisfies the eigenvalue problem
$$\begin{array}{rcl} & & {Z}_{n}\prime \prime (z) + \frac{{N}^{2}(z)} {g{H}_{n}} {Z}_{n}(z) = 0,\;\;\;-H < z < 0, \\ & & {Z}_{n} = 0,z = -H,0, \\ \end{array}$$
with the eigenvalue gH n . Introducing the transformation z = H(ξ − 1) this becomes
$$\begin{array}{rcl} & & \frac{\mathrm{{d}}^{2}{Z}_{n}(z)} {\mathrm{d}{\xi }^{2}} + {\lambda }_{n}\hat{{N}}^{2}(\xi ;H){Z}_{ n}(\xi ) = 0,\;\;\;0 < \xi < 1, \\ & & {Z}_{n}(\xi ) = 0,\xi = 0,1, \\ & & {\lambda }_{n} = \frac{{N}_{\mathrm{max}}^{2}{H}^{2}} {g{H}_{n}}, \\ \end{array}$$
(19.58)
where λ n is the eigenvalue. This is the classical eigenvalue problem of internal waves in a basin of constant depth. It is selfadjoint, and so λ n is real and positive for all n = 1, 2, ; furthermore, the eigenfunctions form a complete set and can be normalized to satisfy the orthogonality relations
$$\begin{array}{rcl} \langle \hat{{N}}^{2}(\xi ;H){Z}_{ m}(\xi ),\;\;{Z}_{n}(\xi )\rangle = {\delta }_{\mathit{mn}}.& &\end{array}$$
(19.59)
We conjecture that by selecting shape functions and weighting functions from this set or from derivatives of them we will achieve a weak coupling of the essentially barotropic-TW motion with the internal wave motion. The arguments are:
  1. 1.
    If we choose Ξ m  = Z m the vertical velocity profiles are those of the internal wave motion of a fluid with constant depth (Fig. 19.13). Actual boundary conditions at the bottom are not satisfied by these functions. This will result in a coupling of the different internal modes.
    Fig. 19.13

    Typical vertical distribution of the Brunt–Väisälä frequency N (left) and the four lowest baroclinic modes (qualitative). Solid curves show the distribution of the vertical velocity component, dashed curves indicate the distribution of the longitudinal velocity component (when f = 0), Most energy is usually concentrated in the first baroclinic mode, the exact distribution must, however, be determined by continuous profiles of horizontal velocities (from [16]). © Springer, Vienna, reproduced with permission

     
  2. 2.
    If we further select ϕ m  = dΞ m  ∕ dξ we will exactly match the vertical distribution of the horizontal velocity profiles for internal waves in a basin with constant depth. To be consistent with (19.50) the function set {ϕ m } must be orthogonal to the constant function. One can easily verify that
    $$\langle 1,{\phi }_{m}\rangle =\langle 1, \frac{\mathrm{d}{\Xi }_{m}} {\mathrm{d}\xi } \rangle ={ \int \nolimits \nolimits }_{0}^{1}\frac{\mathrm{d}{Z}_{m}} {\mathrm{d}\xi } \:\mathrm{d}\xi = {Z}_{m}(1) - {Z}_{m}(0) = 0,$$
    by virtue of the boundary condition in (19.58)2.

    *

     
  3. 3.
    The momentum equations (19.48)1, 2, 5, the continuity equation (19.48)3 and the adiabaticity equation (19.48)4 now suggest that we should choose
    $$\{{\psi }_{m}\} =\{ {\phi }_{m}\} =\{\mathrm{ d}{\Xi }_{m}/\mathrm{d}\xi \}\;\;\mbox{ and}\;\{{\chi }_{m}\} =\{\hat{ {N}}^{2}{\Xi }_{ m}\}.$$
     
  4. 4.
    We weigh the horizontal momentum equations with the same weighting function as the shape functions of the horizontal velocity. Similarly, the shape functions and the weighting functions in the continuity equation should be chosen from the same function set. This yields
    $$\{\delta {\phi }_{m}^{M}\} =\{ \delta {\phi }_{ m}^{C}\} =\{ \frac{\mathrm{d}{\Xi }_{m}} {\mathrm{d}\xi } \}.$$
    Finally, the adiabatic equation and the vertical momentum equation then suggest that
    $$\{\delta {\phi }_{m}^{V }\} =\{ {\Xi }_{ m}\}$$
     
With these choices the matrix elements can readily be calculated; they are listed in column 3 of Table 19.4. This table also gives the elements for an alternative selection of basis functions. Leading idea in postulating (19.59) was to incorporate into the function set as much as possible of the particular physics under consideration. Consequently, the complete function set was that of the eigenfunctions for buoyancy waves in a stratified basin of constant depth. From a computational point of view this approach implies that the eigenvalue problem (19.58) must be solved in advance in order to obtain the required basis functions Ξ n . Alternatively, we could expand the functions \(\tilde{u},\tilde{v},w,p\) and ρ in terms of special orthogonal polynomials. Indeed, the scalar product ⟨θ, ψ⟩ =  ∫0 1θψdξ suggests the use of Jacobi polynomials G n (1, 1, ξ) (see [1]), which are orthonormal in the interval [0, 1] with respect to the weighting function 1 in the scalar product. They are defined by
$$\begin{array}{rcl} & & {G}_{n}(1,1,\xi ) = \frac{\sqrt{2n + 1}} {n!} {\sum \nolimits }_{k=0}^{n}{(-1)}^{n-k}\left (\begin{array}{c} n\\ k \end{array} \right )\frac{(n + k)!} {k!} {\xi }^{k}, \\ & & {G}_{0} = 1, \\ & & {G}_{1} = \sqrt{12}\left (\xi -\frac{1} {2}\right ), \\ & & {G}_{2} = \sqrt{180}\left ({\xi }^{2} - \xi + \frac{1} {6}\right ),\;\ldots \\ \end{array}$$
and satisfy the orthogonality relations
$${\int \nolimits \nolimits }_{0}^{1}{G}_{ n}{G}_{m}\mathrm{d}\xi =\langle {G}_{n},{G}_{m}\rangle = {\delta }_{\mathit{mn}}.$$
Note that the constant function G 0 is the first basis function and all G n for n > 0 are orthogonal to it, i.e. they span a vanishing vertical area. This is well in accord with condition (19.50). The derivative of G n + 1 is a polynomial of degree n and can be expressed as a linear combination of G k , k = 1, 2, , n. For later use we define
$$\begin{array}{rcl} (\xi - 1)\frac{\mathrm{d}{G}_{n}} {\mathrm{d}\xi } & \equiv & {\sum \nolimits }_{k=0}^{n}{b}_{ nk}{G}_{k}, \\ \frac{\mathrm{d}{G}_{n+1}} {\mathrm{d}\xi } & \equiv & {\sum \nolimits }_{k=0}^{n}{h}_{ nk}{G}_{k}.\end{array}$$
The advantage of this polynomial set is its easy accessibility that frees us from solving a problem oriented eigenvalue problem. Its likely disadvantage is a slower convergence in comparison to the ‘physical set’ of internal eigenfunctions
We now select
$$\begin{array}{rlrlrl} \left.\begin{array}{rcl} \{{\phi }_{m}\}& =&\{{G}_{m}\}, \\ \{{\Xi }_{m}\} =\{ {\psi }_{m}\} =\{ {\chi }_{m}\}& =&\{{G}_{m-1}\}, \end{array} \right \}\;\;m = 1,\ldots,N & & \end{array}$$
and the weighting functions
$$\begin{array}{rlrlrl} \left.\begin{array}{rcl} \{\delta {\phi }_{m}^{M}\} =\{ \delta {\phi }_{m}^{C}\}& =&\{{G}_{m}\}, \\ \{\delta {\phi }_{m}^{V }\}& =&\{{G}_{m-1}\}, \end{array} \right \}\;\;m = 1,\ldots,N. & & \end{array}$$
The corresponding matrix elements are listed in column 4 of Table 19.4. We now see the distinct properties of the two alternative approaches: When non-hydrostatic terms are ignored an expansion in terms of buoyancy modes uncouples the individual baroclinic modes in (19.56). The barotropic–baroclinic coupling arises in the horizontal momentum equation (19.55)1, 2, 3 and (19.57)2, 3 only in conjunction with topographic gradients. For a flat bottom all barotropic and baroclinic modes are uncoupled. On the other hand, the Jacobi set, even though it is more easily accessible, does not follow the physics so closely. Notice also that the buoyancy equations (involving E mn and F mn ) are strongly coupled in this case whereas the coupling due to topography gradients in the horizontal momentum equations (involving B mn ) is weaker.

Summarizing the governing equations, we obtain for an expansion in buoyancy modes

$$\begin{array}{rcl} & & \frac{\partial (H\bar{u})} {\partial x} + \frac{\partial (H\bar{v})} {\partial y} = 0, \\ & & \frac{\partial \bar{u}} {\partial t} - f\bar{v} + g\frac{\partial \zeta } {\partial x} = \underline{-\frac{1} {{\rho }^{{_\ast}}}\frac{\mathrm{d}\Xi } {\mathrm{d}\xi } (0)\frac{\partial H/\partial x} {H} {P}_{m}} + \frac{{\tau }_{1}} {{\rho }^{{_\ast}}H}, \\ \end{array}$$
$$\begin{array}{rcl} & & \frac{\partial \bar{v}} {\partial t} + f\bar{u} + g\frac{\partial \zeta } {\partial y} = \underline{-\frac{1} {{\rho }^{{_\ast}}}\frac{\mathrm{d}\Xi } {\mathrm{d}\xi } (0)\frac{\partial H/\partial y} {H} {P}_{m}} + \frac{{\tau }_{2}} {{\rho }^{{_\ast}}H}, \\ & & \frac{\partial {U}_{m}} {\partial t} - f{V }_{m} + \frac{1} {{\rho }^{{_\ast}}}\frac{\partial {P}_{m}} {\partial x} - \frac{1} {{\rho }^{{_\ast}}}{\lambda }_{m}^{-1}{B}_{ ml}\frac{\partial H/\partial x} {H} {P}_{l} = 0, \\ & & \frac{\partial {V }_{m}} {\partial t} + f{U}_{m} + \frac{1} {{\rho }^{{_\ast}}}\frac{\partial {P}_{m}} {\partial y} - \frac{1} {{\rho }^{{_\ast}}}{\lambda }_{m}^{-1}{B}_{ ml}\frac{\partial H/\partial y} {H} {P}_{l} = 0, \\ & & \frac{\partial {U}_{m}} {\partial x} + \frac{\partial {V }_{m}} {\partial y} -\frac{{W}_{m}} {H} - {\lambda }_{m}^{-1}{K}_{ ml}\left [\frac{\partial H/\partial x} {H} {U}_{l} + \frac{\partial H/\partial y} {H} {V }_{l}\right ] \\ & & \;\;\;= \underline{-{\lambda }_{m}^{-1}\frac{\mathrm{d}{\Xi }_{m}} {\mathrm{d}\xi } (0)\left [\frac{\partial H/\partial x} {H} \bar{u} + \frac{\partial H/\partial y} {H} \bar{v}\right ]}, \\ & & {G}_{\mathit{mn}}\frac{{\partial }^{2}{W}_{n}} {\partial {t}^{2}} + {N}_{\mathrm{max}}^{2}{W}_{ m} + \frac{1} {\rho {_\ast} H}{\lambda }_{m}\frac{\partial {P}_{m}} {\partial t} = 0, \\ & & \;\;\;(m,n,l = 1,2,3,\ldots,N).\end{array}$$
(19.60)
Boundary conditions which must be satisfied are
$$\begin{array}{rcl} \left.\begin{array}{l} \bar{\vec{u}} \cdot \vec{ n} = 0,\\ \vec{{U}}_{ m} \cdot \vec{ n} = 0,\quad m = 1,2,\ldots N, \end{array} \right \}\mbox{ along the shore,}& &\end{array}$$
(19.61)
where \(\vec{n}\) is the unit normal vector and it is assumed that the depth H does not vanish along the shore. The underlined terms in (19.60) describe the barotropic–baroclinic coupling. All these terms involve the gradient of H.

19.4.3 Scale Analysis

To estimate the significance of the barotropic–baroclinic coupling, let us non-dimensionalize equations (19.60). To this end we introduce the following scales and dimensionless variables:
$$\begin{array}{rcl} & & (x,y,H) = ([L]{x}^{{_\ast}},[L]{y}^{{_\ast}},[H]{H}^{{_\ast}}), \\ & & \zeta = [\zeta ]{\zeta }^{{_\ast}},\;\;t = [{f}^{-1}]{t}^{{_\ast}}, \\ & & (\bar{u},\bar{v},{U}_{m},{V }_{m}) = [U](\bar{{u}}^{{_\ast}},\bar{{v}}^{{_\ast}},{U}_{ m}^{{_\ast}},{V }_{ m}^{{_\ast}}), \\ & & {W}_{m} = [W]{W}_{m}^{{_\ast}}, \\ & & {P}_{m} = [P]{P}_{m}^{{_\ast}}, \\ & & \vec{{\tau }}_{\mathrm{wind}} = [{\tau }_{\mathrm{wind}}]\vec{{\tau }}^{{_\ast}}\end{array}$$
(19.62)
Bracketed quantities are orders of magnitude of the variables in question and variables having an asterisk are dimensionless. With (19.62), (19.60) becomes (asterisks are consistently omitted):
$$\begin{array}{rcl} & & \frac{\partial } {\partial x}(H\bar{u}) + \frac{\partial } {\partial y}(H\bar{v}) = 0, \\ & & \frac{\partial \bar{u}} {\partial t} -\bar{ v} + \mathbb{A} \frac{\partial \zeta } {\partial x} = \underline{-\mathbb{B}\frac{\mathrm{d}{\Xi }_{m}(0)} {\mathrm{d}\xi } \frac{\partial H/\partial x} {H} {P}_{m}} + \mathbb{C}\frac{{\tau }_{1}} {H}, \\ & & \frac{\partial \bar{v}} {\partial t} +\bar{ u} + \mathbb{A}\frac{\partial \zeta } {\partial y} = \underline{-\mathbb{B}\frac{\mathrm{d}{\Xi }_{m}(0)} {\mathrm{d}\xi } \frac{\partial H/\partial y} {H} {P}_{m}} + \mathbb{C}\frac{{\tau }_{2}} {H}, \\ & & \frac{\partial {U}_{m}} {\partial t} - {V }_{m} + \mathbb{B}\frac{\partial {P}_{m}} {\partial x} - \mathbb{B}{\lambda }_{m}^{-1}{B}_{ ml}\frac{\partial H/\partial x} {H} {P}_{l} = 0, \\ & & \frac{\partial {V }_{m}} {\partial t} + {U}_{m} + \mathbb{B}\frac{\partial {P}_{m}} {\partial y} - \mathbb{B}{\lambda }_{m}^{-1}{B}_{ ml}\frac{\partial H/\partial y} {H} {P}_{l} = 0, \\ & & \frac{\partial {U}_{m}} {\partial x} + \frac{\partial {V }_{m}} {\partial y} - \mathbb{D}\frac{{W}_{m}} {H} - {\lambda }_{m}^{-1}{K}_{ ml}\left [\frac{\partial H/\partial x} {H} {U}_{l} + \frac{\partial H/\partial y} {H} {V }_{l}\right ] \\ & & \;\;\;= \underline{-{\lambda }_{m}^{-1}\frac{\mathrm{d}{\Xi }_{m}(0)} {\mathrm{d}\xi } \left [\frac{\partial H/\partial x} {H} \bar{u} + \frac{\partial H/\partial y} {H} \bar{v}\right ]}, \\ & & \mathbb{E}{G}_{\mathit{mn}}\frac{{\partial }^{2}{W}_{n}} {\partial {t}^{2}} + {W}_{m} + \mathbb{F}{\lambda }_{m} \frac{1} {H} \frac{\partial {P}_{m}} {\partial t} = 0, \\ \end{array}$$
(19.63)
where
$$\begin{array}{rcl} \begin{array}{lll} \mathbb{A} = \frac{g[\zeta ]} {f[L][U]}, &\quad \mathbb{C} = \frac{[\tau ]} {{\rho }^{{_\ast}}f[H][U]},&\quad \mathbb{E} = \frac{{f}^{2}} {{N}_{\mathrm{max}}^{2}}, \\ \mathbb{B} = \frac{[P]} {{\rho }^{{_\ast}}f[L][U]},&\quad \mathbb{D} = \frac{[L][W]} {[H][U]}, &\mathbb{F} = \frac{f[P]} {{\rho }^{{_\ast}}[H][W]{N}_{\mathrm{max}}^{2}}. \end{array} & &\end{array}$$
(19.64)
Choosing the scales according to11
$$\begin{array}{lll} [L] = 1{0}^{4}\,\mathrm{m}, &[H] = 1{0}^{2}\,\mathrm{m}, &[\zeta ] = 1{0}^{-1}\,\mathrm{m}, \\{} [{f}^{-1}] = 1{0}^{4}\,\mathrm{s}, &[U] = 1\,\mathrm{{ms}}^{-1}, &[W] = 1{0}^{-2}\,\mathrm{{ms}}^{-1}, \\ {}[P/\rho {_\ast}] = 1{0}^{-1}\,\mathrm{{m}}^{2}\,\mathrm{{s}}^{-2},&[\tau /\rho {_\ast}] = 1{0}^{-2}\,\mathrm{{m}}^{2}\,\mathrm{{s}}^{-2},&{N}_{\mathrm{max}}^{2} = 1{0}^{-3}\,\mathrm{{s}}^{-2},\end{array}$$
the orders of magnitude of (19.64) are
$$\begin{array}{lll} \mathbb{A} = O(1), &\quad \mathbb{C} = O(1),&\quad \mathbb{E} = O(1{0}^{-5}), \\ \mathbb{B} = O(1{0}^{-1}),&\quad \mathbb{D} = O(1),&\quad \mathbb{F} = O(1).\end{array}$$
Important in the following argument are only the values for \(\mathbb{A}\), \(\mathbb{B}\) and \(\mathbb{C}\). Thus the barotropic–baroclinic coupling terms (underlined) in the momentum equations of the barotropic motion are small in comparison to the remaining terms of this equation, but this cannot be said about the baroclinic–barotropic coupling term (underlined) in the baroclinic continuity equation, because these terms do not contain a factor \(\mathbb{B}\) while at least some of the remaining terms in the equation are order unity.12 This argument demonstrates that the barotropic–baroclinic coupling is weak in the sense that to lowest order the barotropic motion is unaffected by the baroclinic processes. On the other hand, a baroclinic trace of the barotropic motion can be discerned, because to the same order of accuracy the barotropic flow serves as an input to the baroclinic response.
This is then the approximate solution procedure: We solve in a first step the TW-equation
$$\begin{array}{rlrlrl} \begin{array}{ll} \nabla \cdot \left (\frac{\nabla {\psi }_{t}} {H} \right ) + \mathcal{J}\left (\psi, \frac{f} {H}\right ) = 0,\;&\mbox{ in}\;\mathcal{D}, \\ \psi = 0,\;\; &\mbox{ on}\;\partial \mathcal{D}, \end{array} & & \end{array}$$
evaluate \bar{u} and \(\bar{v}\) according to (19.46) and substitute them into (19.63)6.

The internal wave problem ((19.63)4, 5, 6, 7) is then solved in a second step. Structurally, this is analogous to the two-layer case studied before.

The above calculations have not been used in a concrete computation for the evaluation of the effects of the barotropic motion in the baroclinic motion due to TWs. However, the computations have nevertheless been useful from a viewpoint of measurements. The scale analysis has shown for an arbitrary stratification that the barotropic–baroclinic coupling is weak in the sense that it is one-sided from the barotropic processes to the baroclinic processes, but not vice versa. This implies that traces of TWs can be observed in isotherm–depth–time series and compared with solutions of the barotropic TW-equation. The solutions of the baroclinic part of the equations do not have to be determined to perform a comparison between theory and measured quantities.

19.5 Discussion

In this chapter, an introduction was given to topographic Rossby waves. The starting point was the linearized wave equation for barotropic oscillations in a rotating container of which the vertical motion of the free surface was suppressed by the rigid lid assumption. The emerging equation subject to the no flux boundary condition at the circular container wall was solved for a power law radial bathymetry profile. It led to the following qualitative results of the associated eigensolutions:
  • Countable subinertial eigenfrequencies, whose values depend significantly upon the bathymetric variation. With the variable depth profile tending to constant depth the eigenfrequency (-period) approaches zero (infinity).

  • The eigenmodes, expressed as graphs of streamlines, consist of systems of gyres which rotate counter clockwise (on the N. H.) and may for elongated basins also structurally change within a period.

  • The fundamental mode consists of a pair of gyres; its scrutiny showed that the horizontal velocity vector close to the container centre rotates counter clockwise (on the N. H.), whilst the horizontal velocity vector at points close to the shore rotates in the opposite direction.

These results are in conformity with observations of velocity measurements in Southern Lake Michigan.

A similar analysis for the Northern basin of Lake of Lugano indicated that whole basin eigenmodes may not be the sole qualitative structure of TW-modes in elongated basins. As shown by Trösch, whose finite element bay mode solutions contrasted with Mysak et al.’s elliptical whole-basin solutions, TW eigensolutions could also arise as bay resonances. It could not be decided, which of the two mode types were suggested by the Lake of Lugano data.

Analysis of the two-layer equations for a small depth epilimnion and a deep hypolimnion (Gratton scaling) showed that the barotropic–baroclinic scaling is one-sided in this case to the extent that the barotropic TW drives the baroclinic response and not vice versa. This implies that TW signals may be extracted from thermistor chain data. That this qualitative behaviour prevails for a general vertical density profile was corroborated by employing a spectral analysis based on the application of the principle of weighted residuals.

19.6 Appendix

In this Appendix, we demonstrate how (19.51) can be reduced to a set of spatially two-dimensional equations by using a vertical shape function expansion of the velocity, pressure and density fields as given in the formulae (19.52), in which {ϕ n , Ξ n , ψ n , χ n }, n = 1, 2, N are a set of known functions of the independent variable z ∕ H; through H = H(x, y) they depend implicitly on x and y. It will be explained in the main text, from which function set they will be chosen. The coefficient functions U n , V n , W n , P n , R n depend on the spatial horizontal coordinates only and on the time. It is our goal to use the Principle of weighted residuals to deduce field equations for these quantities.

Let ⟨⟨f m , g n ⟩⟩ and ⟨f m , g n ⟩ be the following inner products:
$$\begin{array}{rcl} \langle \langle {f}_{m},{g}_{n}\rangle \rangle & =& {\int \nolimits }_{-H}^{0}{f}_{ m}\left ( \frac{z} {H}\right ){g}_{n}\left ( \frac{z} {H}\right )\mathrm{d}z, \\ & & \\ \langle {f}_{m},{g}_{n}\rangle & =& {\int \nolimits }_{0}^{1}{f}_{ m}(\xi ){g}_{n}(\xi )\mathrm{d}\xi. \end{array}$$
(19.65)
They can be connected by the transformation
$$\begin{array}{rcl} z = H(\xi - 1),\quad \quad \mathrm{d}z = H\mathrm{d}\xi.& &\end{array}$$
(19.66)
We mention that it would be more adequate to define ⟨⟨, ⟩⟩ as the integral from z =  − H to z = ζ and dropping terms involving ζ afterwards. This definition will be used in the transformation of (19.75) below.
With the aid of (19.65) and (19.66) the reader may easily deduce the following properties:
$$\begin{array}{rcl} \langle \langle {f}_{m},{g}_{n}\rangle \rangle & =& H\langle {f}_{m},{g}_{n}\rangle, \\ \left \langle \left \langle \frac{\mathrm{d}{f}_{m}} {\mathrm{d}z},{g}_{n}\right \rangle \right \rangle & =& \left \langle \frac{\mathrm{d}{f}_{m}} {\mathrm{d}\xi },{g}_{n}\right \rangle, \\ & & \\ \left \langle \left \langle \frac{\partial {f}_{m}} {\partial x},{g}_{n}\right \rangle \right \rangle & =& -\frac{\partial H} {\partial x} \left \langle \frac{\mathrm{d}{f}_{m}} {\mathrm{d}\xi } (\xi - 1),{g}_{n}\right \rangle, \\ \left \langle \left \langle \frac{\partial {f}_{m}} {\partial y},{g}_{n}\right \rangle \right \rangle & =& -\frac{\partial H} {\partial y} \left \langle \frac{\mathrm{d}{f}_{m}} {\mathrm{d}\xi } (\xi - 1),{g}_{n}\right \rangle. \end{array}$$
(19.67)
Consider now the momentum equations (19.51)1, 2 first. Substitute the expansions (19.52) for \(\tilde{u},\tilde{v}\) and p and form the following inner products: ⟨⟨ 19.51 1, 2, δϕ m M ⟩⟩; this yields
$$\begin{array}{rcl} & & {A}_{\mathit{mn}}^{M}\left [\frac{\partial {U}_{n}} {\partial t} - f{V }_{n}\right ] - \frac{1} {{\rho }_{\star }} \frac{\partial H\!/\partial x} {H} {B}_{\mathit{mn}}{P}_{n} + \frac{{C}_{\mathit{mn}}} {{\rho }_{\star }} \frac{\partial {P}_{n}} {\partial x} \\ & & \qquad = \left (g\frac{\partial \zeta } {\partial x} - \frac{{\tau }_{1}} {{\rho }_{\star }H}\right ){D}_{m}^{M}, \\ & & \\ & & \qquad {A}_{\mathit{mn}}^{M}\left [\frac{\partial {V }_{n}} {\partial t} + f{U}_{n}\right ] - \frac{1} {{\rho }_{\star }} \frac{\partial H\!/\partial y} {H} {B}_{\mathit{mn}}{P}_{n} + \frac{{C}_{\mathit{mn}}} {{\rho }_{\star }} \frac{\partial {P}_{n}} {\partial y} \\ & & \qquad = \left (g\frac{\partial \zeta } {\partial y} - \frac{{\tau }_{2}} {{\rho }_{\star }H}\right ){D}_{m}^{M},\hspace{28.45274pt} (m = 1,2,3,\ldots,N), \\ \end{array}$$
(19.68)
where
$$\begin{array}{rcl}{ A}_{\mathit{mn}}^{M}& =& \langle \delta {\phi }_{ m}^{M},{\phi }_{ n}\rangle, \\ {B}_{\mathit{mn}}& =& \left \langle \delta {\phi }_{m}^{M},(\xi - 1)\frac{\mathrm{d}\psi } {\mathrm{d}\xi } \right \rangle -\left \langle 1,(\xi - 1)\frac{\mathrm{d}{\psi }_{n}} {\mathrm{d}\xi } \right \rangle \langle 1,\delta {\phi }_{m}^{M}\rangle, \\ & & \\ {C}_{\mathit{mn}}& =& \langle \delta {\phi }_{m}^{M},{\psi }_{ n}\rangle -\langle 1,{\psi }_{n}\rangle \langle 1,\delta {\phi }_{m}^{M}\rangle, \\ {D}_{m}^{M}& =& \langle 1,\delta {\phi }_{ m}^{M}\rangle. \end{array}$$
(19.69)
Here and henceforth, functions carrying a prefix δ are weighting functions in the expansions (19.52). Moreover, in (19.68) summation over repeated indices n is understood. In an analogous manner, (19.51)4, 5 can be treated: the inner products ⟨⟨ (19.51)4, δϕ m V ⟩⟩ and ⟨⟨  (19.51)5, δϕ m V ⟩⟩ are formed and yield the equations
$$\begin{array}{rcl} & & {E}_{\mathit{mn}}\frac{\partial {R}_{n}} {\partial t} -\frac{{\rho }_{\star }{N}_{\mathrm{max}}^{2}} {g} {F}_{\mathit{mn}}{W}_{n} = 0, \\ & & \\ & & {G}_{\mathit{mn}}\frac{\partial {W}_{n}} {\partial t} + \frac{g} {{\rho }_{\star }}{E}_{\mathit{mn}}{R}_{n} + \frac{1} {{\rho }_{\star }H}{H}_{\mathit{mn}}{P}_{n} = 0, \\ & & \hspace{56.9055pt} (m = 1,2,3,\ldots,N), \\ \end{array}$$
(19.70)
in which
$$\begin{array}{rcl} \begin{array}{ll} {E}_{\mathit{mn}} =\langle \delta {\phi }_{m}^{V },{\chi }_{n}\rangle,\;\;\;\; &{G}_{\mathit{mn}} =\langle \delta {\phi }_{m}^{V },{\Xi }_{n}\rangle, \\ {F}_{\mathit{mn}} =\langle \delta {\phi }_{m}^{V },\hat{{N}}^{2}{\Xi }_{n}\rangle,\qquad &{H}_{\mathit{mn}} =\langle \delta {\phi }_{m}^{V }, \frac{\mathrm{d}{\psi }_{n}} {\mathrm{d}\xi } \rangle,\end{array} & &\end{array}$$
(19.71)
and where
$$\begin{array}{rcl} \hat{{N}}^{2} =\hat{ {N}}^{2}(\xi ) = \frac{{N}^{2}(\xi )} {{N}_{\mathrm{max}}^{2}} = - \frac{\frac{\mathrm{d}{\rho }_{0}} {\mathrm{d}z} } {{\left \vert \frac{\mathrm{d}{\rho }_{0}} {\mathrm{d}z} \right \vert }_{\mathrm{max}}}.& &\end{array}$$
(19.72)
Clearly, in order that these relations are meaningful, the boundary conditions p (ξ = 1) = 0 must be satisfied. Hence we must request
$$\begin{array}{rcl}{ \psi }_{n}(\xi = 1)\, =\, 0.& &\end{array}$$
(19.73)
With our choice of ψ n this condition will be nearly satisfied. Equations (19.70) can be combined to yield the single equation
$$\begin{array}{rcl} \begin{array}{r} {G}_{\mathit{mn}}\frac{{\partial }^{2}{W}_{n}} {\partial {t}^{2}} + {N}_{\mathrm{max}}^{2}{F}_{\mathit{ mn}}{W}_{n} + \frac{{H}_{\mathit{mn}}} {{\rho }_{\star }H} \frac{\partial {P}_{n}} {\partial t} = 0, \\ \\ (m = 1,2,3,\ldots,N).\end{array} & &\end{array}$$
(19.74)
Equations (19.68) and (19.74) are 3N equations for the unknowns U n , V n , P n , W n . The remaining N equations follow from the continuity equation (19.51)3. Because kinematic boundary conditions at the upper and lower surfaces must be incorporated when employing the Principle of weighted residuals, we shall go into greater details. Forming ⟨⟨ (19.51)3, δϕ m C ⟩⟩, it follows that
$$\begin{array}{rcl} \begin{array}{l} \langle \langle \tilde{{u}}_{x},\delta {\phi }_{m}^{C}\rangle \rangle +\langle \langle \tilde{ {v}}_{y},\delta {\phi }_{m}^{C}\rangle \rangle +\langle \langle {w}_{z},\delta {\phi }_{m}^{C}\rangle \rangle \\ \\ \quad = \left (\frac{{H}_{x}} {H} \bar{u} + \frac{{H}_{y}} {H} \bar{v}\right )\langle \langle 1,\delta {\phi }_{m}^{C}\rangle \rangle.\end{array} & &\end{array}$$
(19.75)
Using the definition of the inner product ⟨⟨, ⟩⟩, we may easily prove that.13
$$\begin{array}{rcl} \langle \langle \tilde{{u}}_{x},\delta {\phi }_{m}^{C}\rangle \rangle & =& \frac{\partial } {\partial x}\langle \langle \tilde{u},\delta {\phi }_{m}^{C}\rangle \rangle -\Bigg\langle\Bigg\langle\tilde{u}, \frac{\partial \delta {\phi }_{m}^{C}} {\partial x} \Bigg\rangle\Bigg\rangle \\ & &{ \left.-\,\tilde{u}\delta {\phi }_{m}^{C}\right \vert }_{ z=\zeta }\frac{\partial \zeta } {\partial x}{\left.-\tilde{u}\delta {\phi }_{m}^{C}\right \vert }_{ z=-H}\frac{\partial H} {\partial x}, \\ \langle \langle \tilde{{v}}_{x},\delta {\phi }_{m}^{C}\rangle \rangle & =& \frac{\partial } {\partial y}\langle \langle \tilde{v},\delta {\phi }_{m}^{C}\rangle \rangle -\Bigg\langle\Bigg\langle\tilde{v}, \frac{\partial \delta {\phi }_{m}^{C}} {\partial y} \Bigg\rangle\Bigg\rangle \\ & &{ \left.-\,\tilde{v}\delta {\phi }_{m}^{C}\right \vert }_{ z=\zeta }\frac{\partial \zeta } {\partial y}{\left.-\tilde{v}\delta {\phi }_{m}^{C}\right \vert }_{ z=-H}\frac{\partial H} {\partial y}, \\ \langle \langle \tilde{{w}}_{z},\delta {\phi }_{m}^{C}\rangle \rangle & =& -\,\langle \langle w,\delta {\phi }_{{ m}_{z}}^{C}\rangle \rangle {\left.\,+\,w\delta {\phi }_{ m}^{C}\right \vert }_{ z=\zeta }{\left.\,-\,\,w\delta {\phi }_{m}^{C}\right \vert }_{ z=-H}.\end{array}$$
(19.76)
With these expressions the weighted continuity statement (19.74) takes the form
$$\begin{array}{rcl} & & \frac{\partial } {\partial x}\langle \langle \tilde{u},\delta {\phi }_{m}^{C}\rangle \rangle + \frac{\partial } {\partial y}\langle \langle \tilde{v},\delta {\phi }_{m}^{C}\rangle \rangle \\ & & -\,\Bigg{\langle}\Bigg{\langle}\tilde{u}, \frac{\partial \delta {\phi }_{m}^{C}} {\partial x} \Bigg{\rangle}\Bigg{\rangle} -\Bigg{\langle}\Bigg{\langle}\tilde{v}, \frac{\partial \delta {\phi }_{m}^{C}} {\partial y} \Bigg{\rangle}\Bigg{\rangle} -\Bigg{\langle}\Bigg{\langle}w,\delta {\phi }_{{m}_{z}}^{C}\Bigg{\rangle}\Bigg{\rangle} \\ & & -\,\delta {\phi }_{m}^{C}(\zeta )\left ({\left [u{\zeta }_{ x} + v{\zeta }_{y} - w\right ]}_{z=\zeta } -\bar{ u}{\zeta }_{x} -\bar{ v}{\zeta }_{y}\right ) \\ & & -\,\delta {\phi }_{m}^{C}(-H)\left ({\left [u{H}_{ x} + v{H}_{y} - w\right ]}_{z=-H} -\bar{ u}{H}_{x} -\bar{ v}{H}_{y}\right ) \\ & =& \left (\frac{{H}_{x}} {H} \bar{u} + \frac{{H}_{y}} {H} \bar{v}\right )\langle \langle 1,\delta {\phi }_{m}^{C}\rangle \rangle. \end{array}$$
(19.77)
This equation is written down in full in order to demonstrate incorporation of the boundary conditions. The term in brackets in the third line equals − ζ ∕ ∂t and that in the fourth line vanishes. After this substitution we may ignore the two terms involving ζ ∕ ∂t because the rigid-lid assumption is made. Finally, the non-linear term \(\bar{u}{\zeta }_{x},\bar{v}{\zeta }_{y}\) may be omitted because the non-linearities have consistently been dropped in earlier equations. Thus, (19.77) reduces to
$$\begin{array}{rcl} & & \frac{\partial } {\partial x}\langle \langle \tilde{u},\delta {\phi }_{m}^{C}\rangle \rangle + \frac{\partial } {\partial y}\langle \langle \tilde{v},\delta {\phi }_{m}^{C}\rangle \rangle -\left \langle \left \langle \tilde{u}, \frac{\partial \delta {\phi }_{m}^{C}} {\partial x} \right \rangle \right \rangle \\ & & -\,\left \langle \left \langle \tilde{v}, \frac{\partial \delta {\phi }_{m}^{C}} {\partial y} \right \rangle \right \rangle -\langle \langle w,\delta {\phi }_{{m}_{z}}^{C}\rangle \rangle \\ & =& \left (\bar{u}{H}_{x} +\bar{ v}{H}_{y}\right )\left [ \frac{1} {H}\langle \langle 1,\delta {\phi }_{m}^{C}\rangle \rangle - \delta {\phi }_{ m}^{C}(-H)\right ].\end{array}$$
(19.78)
Substituting the expansion (19.52) and making use of the formulae (19.67) at appropriate places yields the equation
$$\begin{array}{rcl} & & {A}_{\mathit{mn}}^{C}\left [\frac{\partial {U}_{n}} {\partial x} + \frac{\partial {V }_{n}} {\partial y} \right ] - {K}_{\mathit{mn}}\left [\frac{{H}_{x}} {H} {U}_{n} + \frac{{H}_{y}} {H} {V }_{m}\right ] - {L}_{\mathit{mn}}\frac{{W}_{n}} {H} \\ & & \quad = \left (\bar{u}\frac{{H}_{x}} {H} +\bar{ v}\frac{{H}_{y}} {H} \right )\left [{D}_{m}^{C} - \delta {\phi }_{ m}^{C}(\zeta = 0)\right ],\quad (m = 1,2,3,\ldots,N),\qquad \end{array}$$
(19.79)
in which
$$\begin{array}{rcl}{ A}_{\mathit{mn}}^{C}& =& \langle \delta {\phi }_{ m}^{C},{\phi }_{ n}\rangle, \\ {D}_{m}^{C}& =& \langle \delta {\phi }_{ m}^{C},1\rangle, \\ & & \\ {K}_{\mathit{mn}}& =& \left \langle (\xi - 1)\frac{\mathrm{d}{\phi }_{n}} {\mathrm{d}\xi },\delta {\phi }_{m}^{C}\right \rangle - \delta {\phi }_{ m}^{C}(0)\delta {\phi }_{ n}^{C}(0), \\ {L}_{\mathit{mn}}& =& \left \langle \Xi, \frac{\mathrm{d}\delta {\phi }_{m}^{C}} {\mathrm{d}\xi } \right \rangle. \end{array}$$
(19.80)
This completes the derivation of the baroclinic equations. They are: (19.68), (19.74) and (19.79) and form 4N partial differential equations for the baroclinic variables U n , V n , W n , P n ; needless to say that the barotropic quantities \(\bar{u},\bar{v},\zeta \) are regarded as being prescribed or governed by (19.45) or (19.57) in the main text.

Footnotes

  1. 1.
    For the derivation of (19.1), see Sect. 11.3 and for a first attempt of interpretation Sect. 9.2 of Chap. 9 in Volume I. An elegant derivation follows from the conservation law of barotropic potential vorticity, which requires (see (5.74)–(5.79) in Chap. 5 in Volume I of this book series)
    $$\begin{array}{rcl} \frac{\mathrm{d}{\Pi }_{bt}} {\mathrm{d}t} = \frac{\mathrm{d}} {\mathrm{d}t}\left (\frac{\zeta + f} {H} \right ) = 0,& & \\ \end{array}$$
    in which ζ = ω z  = ∂v ∕ ∂x − ∂u ∕ ∂y = E[Ψ]. Therefore,
    $$\begin{array}{rcl} \frac{1} {H} \frac{\partial {\omega }_{z}} {\partial t} + \frac{\partial } {\partial x}\left (\frac{{\omega }_{z} + f} {H} \right )u + \frac{\partial } {\partial y}\left (\frac{{\omega }_{z} + f} {H} \right )v = 0,& & \\ \end{array}$$
    which, after linearization, is equivalent to (19.1).
  2. 2.

    For a biographical sketch see Fig. 19.1.

  3. 3.

    There is a vast literature on TWs. A reference text may be LeBlond and Mysak [27] who treat primarily waves in the open ocean. A review, perhaps more adequate to the present topic is by Stocker and Hutter [42] and contains a large number of references pertaining to the propagation of TWs in closed or semi-closed basins. Relevant works are also by Stocker and Hutter [404344] and Johnson [202122], Willmott and Johnson [50], Johnson and Kaoullas [23] and others.

  4. 4.

    Mysak et al. [33, p. 52] list six arguments in support of the TW-model and only the above mentioned discrepancy in the phase relation against it.

  5. 5.

    With \(f = 1{0}^{-4}\,\mathrm{{s}}^{-1},\;g\prime = 0.02\,\mathrm{{ms}}^{-2},\;{D}_{1} = 10\,\mathrm{m},\;{D}_{2} = 270\,\mathrm{m}\) and \({\psi }_{0} = U \cdot L \cdot ({D}_{1} + {D}_{2}) = 0.03 \cdot 1{0}^{4} \times 270\,\mathrm{{m}}^{3}\,\mathrm{{s}}^{-1}\), where U is a velocity scale (approximately 3 cm s − 1 for Lake of Lugano) and L = 104 m, one obtains \({\zeta }_{0} = 40\,\mathrm{m}\).

  6. 6.

    With f = 10 − 4 s − 1,  g′ = 0. 02 ms − 2,  ψ0 = 7 ×104 m3 s − 1 and D 2 = 270 m, (19.21) yields ζ0 = 1 m. Alternatively, using τ0 = ρair c d U w 2 with ρair = 1. 29 kg m − 3,  c d  = 1. 85 ×10 − 3 (an average value for lakes during summer, see Simons [39], p. 92), and U w  = 4 ms − 1, we find τ0 = 0. 038 Nm − 2 and hence according to (19.21), ζ0 = 1. 5 m and according to (19.20), U = τ0 ∕ (ρ1 fD 1) = 3. 8 cm s − 1. Both values are typical observations in Lake Zurich and Lake of Lugano, see Table 19.3.

  7. 7.

    This solution in an elliptic basin with parabolic bottom is constructed in  Chap. 20, Sect. 20.3.3.

  8. 8.

    These equations actually imply a statement regarding \(\mathcal{L}\prime ({D}_{1}\vec{{u}}_{1} + {D}_{2}{h}_{2}\vec{{u}}_{2}) \cdot \hat{\vec{ n}}\) rather than the mass transport itself. However, if \(\mathcal{L}g = 0\) along \(\partial \mathcal{D}\) for all time, then necessarily g = 0 as well.

  9. 9.

    Recall that \((\hat{\vec{k}} \times \nabla \psi ) \cdot \nabla {h}^{-1} = 0\) for all times implies that \({\mathcal{L}}^{-1}\{(\hat{\vec{k}} \times \nabla \psi ) \cdot \nabla {h}^{-1}\} = 0.\)

  10. 10.
    We assume the reader to be familiar with the method of the Principle of Weighted Residuals, see e.g. Finlayson [8]. The principle or method of weighted residuals (MWR) is based on the following mathematical equivalence: Let \(f(\vec{x})\) be a function or functional whose value vanishes for all \(\vec{x} \in \mathcal{D}\subset {\mathbb{R}}^{N}\):
    $$\begin{array}{rcl} f(\vec{x}) = 0\quad \mbox{ for all }\vec{x} \in \mathcal{D}\subset {\mathbb{R}}^{N}.& & \end{array}$$
    (19.53)
    Let, moreover, \(\delta {\phi }_{\alpha }(\vec{x})\) be an arbitrary bounded function from a set {δϕα, α = 1, 2, }. Obviously, (19.53) implies
    $$\begin{array}{rcl} {\int \nolimits \nolimits }_{\mathcal{D}}\delta {\phi }_{\alpha }(\vec{x})f(\vec{x})\mathrm{d}\vec{x} = 0\qquad (\alpha = 1, 2,\ldots ).& & \end{array}$$
    (19.54)
    If (19.54) holds for any complete set of δϕα, then (19.54) also implies (19.53). This equivalence statement lies at the heart of the MWR.
  11. 11.

    An estimate for [P ∕ ρ ∗ ] is obtained as follows: Under hydrostatic conditions the last of (19.51) suggests that [P ∕ ρ ∗ ] ∼ (Δρ ∕ ρ)g[D], where Δρ ∕ ρ is the density anomaly and [D] a typical metalimnion thickness: Thus, with Δρ ∕ ρ ∼ 10 − 3 and [D] ≤ 10 m this yields [P ∕ ρ ∗ ] ≤ 10 − 1m2 s − 2, implying \(\mathbb{B} \leq 1{0}^{-1}\).

  12. 12.

    This argument can even be made more forceful by recognizing that according to (19.58)3 an estimate for H 1 is 10 m (n = 1) so that λ1 = 10 − 3 ×104 ∕ 102 = 10 − 1. Consequently, one baroclinic–barotropic coupling term is about a factor of 100 larger than the corresponding barotropic–baroclinic term.

  13. 13.

    Here, we use the definition ⟨⟨f, g⟩⟩ =  ∫ − H ζfgdz.

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Copyright information

© Springer-Verlag Berlin Heidelberg 2011

Authors and Affiliations

  • Kolumban Hutter
    • 1
    Email author
  • Yongqi Wang
    • 2
  • Irina P. Chubarenko
    • 3
  1. 1.c/o Versuchsanstalt für Wasserbau Hydrologie und Glaziologie ETH-ZentrumETH ZürichZürichSwitzerland
  2. 2.Department of Mechanical EngineeringDarmstadt University of TechnologyDarmstadtGermany
  3. 3.P.P. Shirshov Institute of OceanologyRussian Academy of SciencesKaliningradRussia

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