Physics of Lakes pp 355398  Cite as
Topographic Waves in Enclosed Basins: Fundamentals and Observations
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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 gyrationwaves, see Fig. 19.1.
Keywords
Wind Stress Gravity Wave Potential Vorticity Topographic Wave Baroclinic Effect19.1 Review of Early Work
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 semianalytic seminumerical approach Stocker and Hutter [41, 42, 43] found solutions to the TWequation in a rectangular basin with symmetric bathymetric chart and could identify three typical wave modes of the TWequation: (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 [45, 46].^{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 twolayer 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 TWmode from the solution of the barotropic TWequation 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 TWsignals 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 timeseries 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 25m 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 4day 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 lowpass filtered current velocities from the 32m 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).
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 4day period, the ucomponent leads the vcomponent by a phase angle of 90^{ ∘ } which corresponds to a cyclonic rotation; at the nearinertial period the phase angle of the ucomponent lags that of the vcomponent by approximately 90^{ ∘ }, indicating an anticyclonic oscillation.
Further scrutiny of the data shows that (1) the 4day 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 onshore positions are primarily alongshore while those at offshore stations may have appreciable amplitudes for both, the East and Northcomponents.
Eigenfrequencies (periods) of the first three TWmodes 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  ∞ 
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 temperaturetime series could be analyzed (for a detailed description see [19, 32, 33]). They disclose a moderate longperiodic signal with a period of app. 74 h.
A statistical crosscorrelation analysis of the temperature time series indicated for all stationpairtime series – except stations 3/8 – high coherence and strongly suggested a counter clockwise rotation of the wave signal around the basin.
This was the state of conflicting interpretation of the same observations in the mid1980s 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 TWequation 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^{ ∘ }Cisotherm–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 midlake position is predominantly counterclockwise, but (2) it is predominantly clockwise at some nearshore 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 TWbehaviour. 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 ‘midlake’ (nearshore) mooring stations, suggesting a fundamental TWmode.
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.
 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 TWmodels.
 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.
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.
On the other hand, inferences from simple (analytically accessible) models and more realistic (only numerically exploitable) models which attempt to explain basinwide TWbehaviour are conflicting. Hence, interpretation of basinwide 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 TWequation 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 bayfilling TWs often have eigenfrequencies which are very close to each other, whereas their eigenfunctions 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 TwoLayer Model
Having discussed the purely barotropic topographic Rossby waves already, we proceed directly to their description in a stratified fluid and commence with a twolayer 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 oneway, 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.
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 TwoLayerEquations
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 RigidLid Approximation
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 righthand 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 lefthand side, with the baroclinic processes. The first term on the righthand 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.
When \(\nabla H =\vec{ 0}\) the third and fourth term on the lefthand side vanish, and the equation describes classical internal Kelvin and Poincaré waves. Thus, (19.8) and (19.9) exhibit in general a twoway coupling, a baroclinic–barotropic coupling and a barotropic–baroclinic coupling the strengths of which must be estimated by a scaleanalysis.
19.3.2.2 LowFrequency 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).
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. 2^{a}  ≤ 28^{a}  74^{b}  
Zurich  28^{a}  ≤ 45^{a}  100^{b}  
Geneva  72^{c}  ≤ 78^{d}  72–96^{b} 
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 righthand side of (19.8) may safely be ignored. In this case, the TWequation (19.8) uncouples from (19.9). Since also boundary conditions will be shown to be free of this baroclinic–barotropic coupling, solutions to the TWproblem 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
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}  ζ_{0}  

[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 } 
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 oneway coupling means that traces of the topographically induced motion can be observed by measuring baroclinic quantities such as temperaturetime 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 temperaturetime series.
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.
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 thermoclineelevation amplitude ζ_{0} is smaller than D _{1} in all three cases, a fact which gives some confidence in the scaling procedure.
19.3.4 Boundary Conditions
The boundary condition (19.29) applies, whether the simplifying assumptions D _{1} ≪ D _{2} and ω^{2} ≪ 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).
We conclude: the geometric optics approximation is only consistent in the lowfrequency 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 twolayer 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 twolayer model of Sect. 19.3 can be carried over to a lake with continuous stratification. The oceanographic literature is rich in studies of lowfrequency processes in a stratified medium. A general result, common to all studies, is that increasing the stratification increases the frequencies of the considered longperiod 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.
19.4.2 Spectral Decomposition of the Baroclinic Fields
Matrix elements for the expansion of the field variables in terms of the flatbottom 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{d^{2} Ξ _{ m }} {dξ^{2}} ⟩  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} ^{2}δ_{ 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  δ_{1m }  
N _{ m }  ⟨1, ψ_{ m }⟩ − ψ_{ m }(0)  − \frac{dΞ _{ m }} {dξ} \big{  }_{ξ = 0}  δ_{1m } + ( − 1)^{ m }\sqrt{2m − 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.
 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 thatby virtue of the boundary condition in (19.58)_{2}.$$\langle 1,{\phi }_{m}\rangle =\langle 1, \frac{\mathrm{d}{\Xi }_{m}} {\mathrm{d}\xi } \rangle ={ \int \nolimits \nolimits }_{0}^{1}\frac{\mathrm{d}{Z}_{m}} {\mathrm{d}\xi } \:\mathrm{d}\xi = {Z}_{m}(1)  {Z}_{m}(0) = 0,$$
*
 3.
 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 yieldsFinally, the adiabatic equation and the vertical momentum equation then suggest that$$\{\delta {\phi }_{m}^{M}\} =\{ \delta {\phi }_{ m}^{C}\} =\{ \frac{\mathrm{d}{\Xi }_{m}} {\mathrm{d}\xi } \}.$$$$\{\delta {\phi }_{m}^{V }\} =\{ {\Xi }_{ m}\}$$
Summarizing the governing equations, we obtain for an expansion in buoyancy modes
19.4.3 Scale Analysis
The internal wave problem ((19.63)_{4, 5, 6, 7}) is then solved in a second step. Structurally, this is analogous to the twolayer 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 onesided 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 TWequation. 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

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 TWmodes in elongated basins. As shown by Trösch, whose finite element bay mode solutions contrasted with Mysak et al.’s elliptical wholebasin 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 twolayer equations for a small depth epilimnion and a deep hypolimnion (Gratton scaling) showed that the barotropic–baroclinic scaling is onesided 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 twodimensional 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.
Footnotes
 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)in which ζ = ω_{ z } = ∂v ∕ ∂x − ∂u ∕ ∂y = E[Ψ]. Therefore,$$\begin{array}{rcl} \frac{\mathrm{d}{\Pi }_{bt}} {\mathrm{d}t} = \frac{\mathrm{d}} {\mathrm{d}t}\left (\frac{\zeta + f} {H} \right ) = 0,& & \\ \end{array}$$which, after linearization, is equivalent to (19.1).$$\begin{array}{rcl} \frac{1} {H} \frac{\partial {\omega }_{z}} {\partial t} + \frac{\partial } {\partial x}\left (\frac{{\omega }_{z} + f} {H} \right )u + \frac{\partial } {\partial y}\left (\frac{{\omega }_{z} + f} {H} \right )v = 0,& & \\ \end{array}$$
 2.
For a biographical sketch see Fig. 19.1.
 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 semiclosed basins. Relevant works are also by Stocker and Hutter [40, 43, 44] and Johnson [20, 21, 22], Willmott and Johnson [50], Johnson and Kaoullas [23] and others.
 4.
Mysak et al. [33, p. 52] list six arguments in support of the TWmodel and only the above mentioned discrepancy in the phase relation against it.
 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 = 10^{4} m, one obtains \({\zeta }_{0} = 40\,\mathrm{m}\).
 6.
With f = 10^{ − 4} s^{ − 1}, g′ = 0. 02 ms^{ − 2}, ψ_{0} = 7 ×10^{4} m^{3} s^{ − 1} and D _{2} = 270 m, (19.21) yields ζ_{0} = 1 m. Alternatively, using τ_{0} = ρ_{air} c _{ d } U _{ w } ^{2} with ρ_{air} = 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 τ_{0} = 0. 038 Nm^{ − 2} and hence according to (19.21), ζ_{0} = 1. 5 m and according to (19.20), U = τ_{0} ∕ (ρ_{1} fD _{1}) = 3. 8 cm s^{ − 1}. Both values are typical observations in Lake Zurich and Lake of Lugano, see Table 19.3.
 7.
This solution in an elliptic basin with parabolic bottom is constructed in Chap. 20, Sect. 20.3.3.
 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.
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.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 \(\vec{x} \in \mathcal{D}\subset {\mathbb{R}}^{N}\):Let, moreover, \(\delta {\phi }_{\alpha }(\vec{x})\) be an arbitrary bounded function from a set {δϕ_{α}, α = 1, 2, …}. Obviously, (19.53) implies$$\begin{array}{rcl} f(\vec{x}) = 0\quad \mbox{ for all }\vec{x} \in \mathcal{D}\subset {\mathbb{R}}^{N}.& & \end{array}$$(19.53)If (19.54) holds for any complete set of δϕ_{α}, then (19.54) also implies (19.53). This equivalence statement lies at the heart of the MWR.$$\begin{array}{rcl} {\int \nolimits \nolimits }_{\mathcal{D}}\delta {\phi }_{\alpha }(\vec{x})f(\vec{x})\mathrm{d}\vec{x} = 0\qquad (\alpha = 1, 2,\ldots ).& & \end{array}$$(19.54)
 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^{ − 1}m^{2} s^{ − 2}, implying \(\mathbb{B} \leq 1{0}^{1}\).
 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 λ_{1} = 10^{ − 3} ×10^{4} ∕ 10^{2} = 10^{ − 1}. Consequently, one baroclinic–barotropic coupling term is about a factor of 100 larger than the corresponding barotropic–baroclinic term.
 13.
Here, we use the definition ⟨⟨f, g⟩⟩ = ∫_{ − H } ^{ζ} fgdz.
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