Standing surface waves or seiches are inherent hydrodynamic features of enclosed water bodies. Their two-dimensional structure is important for estimating flood risk, coastal erosion, and bottom sediment transport, and for understanding shoreline habitats and lake ecology in general. In this work, we present analysis of two-dimensional seiche characteristics in Flathead Lake, Montana, USA, a large intermountain lake known to have high seiche amplitudes. To examine spatial characteristics of different seiche modes, we used the original procedure of determining the seiche frequencies from the primitive equation model output with subsequent derivation of the spatial seiche structure at fixed frequencies akin to the tidal harmonic analysis. The proposed procedure revealed specific seiche oscillation features in Flathead Lake, including maximum surface level amplitudes of the first fundamental mode in straights around the largest island; several higher modes appearing locally in the vicinity of the river inflow; the “Helmholtz” open harbor mode, with the period approximately twice that of the longest seiche mode, generated by a large shallow bay connected to the main lake basin; and several rotating seiche modes potentially affecting the lake-wide circulation. We discuss lake management problems related to the spatial seiche distribution, such as shoreline erosion, floods, and transport of sediments and invasive species in Flathead Lake.

Since

Flathead Lake: bathymetry map and location of pressure and water level measurement sites (red circles), referred to in the text.

Flathead Lake is a natural glacial lake located in the Rocky Mountains of the
western US (47

Winds on Flathead Lake are thermally driven in the summer months due to
differential heating and cooling on the lake surface and the adjacent
mountain ranges. This thermal regime is characterized by distinct westerly
thermal winds draining from mountains out on to the lake in the evening, and
northerly winds driven by air masses that drain north to south in the valley
early in the morning. These winds typically have a maximum speed of
around 5 m s

The Princeton Ocean Model

Harmonic analysis relies on a prescribed set of frequencies persisting in the
analyzed signal. Therefore, it has become a state-of-the-art method in the
analysis of tidal motions, where the prevailing frequencies are determined by
external forcing and are fixed independently from the characteristics of
the water body. Harmonic analysis is rarely, if ever, applied to seiches,
as their harmonics cannot be known a priori. Being defined by the
basin morphometry, the contribution of different harmonics into the total
variance of surface oscillations varies spatially over the lake. Therefore,
neither single point data nor the spectrum averaged over the lake area is
representative of the entire basin. In order to uncover the frequencies set
of seiche oscillations relevant to the whole lake, the following procedure
was developed and applied. In order to retain only the oscillatory motions in
the surface elevation time series, the mean values were removed from velocity
and surface elevation records. A spectrum was taken of the free-surface
oscillations at every grid point of the model domain. Then, a
“maximum spectrum estimation” was constructed by superimposing the
significant spectral peaks among all grid points. The significance level of
the spectral peaks was defined as the upper 99 % significance level for red
noise signal containing the same variance as the spectrum estimation

For analysis of spatial structure of vector velocity fields, the distribution
of the rotary coefficients,

In the following analysis, we use data on underwater pressure oscillations
collected during 2011–2012 at five measurement sites around the lake (points
EO, SM, WB, NR, and SB in Fig.

The model was driven by the initial surface slope without any additional
external forcing, so that the resulting spectrum of the surface level
oscillations has a line shape in contrast to continuous spectra of natural
water level oscillations. The seiche frequencies and periods were therefore
easily identifiable from the line spectral peaks (Fig. 2); 16 frequencies
were identified, with the weakest peak containing about 1 % of the spectral
energy of the strongest one. The sum of spectral energies residing at the
16 frequencies contained more than 99 % of the total variance (Fig.

The synthetic maximum spectrum of modeled lake surface oscillations. Significant peaks are marked with circles with corresponding seiche periods in min.

An additional support to the results of the harmonic analysis was provided by
the comparison of the time series restored from the 16 Fourier components
against the original modeled time series; the restored time series captured
all major features of the original variability (Fig.

Periods of free oscillation modes in Flathead Lake.

Lake surface oscillations at a single point in the middle of the lake produced by the model (blue solid line) and restored from the 12 harmonics (modes) of the Fourier analysis (red dashed line).

For further analysis, we conditionally divided the seiche modes in three groups, based on the spatial distribution of the elevation amplitudes and phases: six “lake-wide modes” (Fig. 4), six “strong-bay modes” (Fig. 5), and four short-period seiche modes with the smallest amplitudes. The latter group is characterized by very small energy of oscillations and is therefore excluded from the detailed analysis below. For completeness, the spatial characteristics of the four least energetic short-period modes are provided in Fig. S1 (Supplement I).

Relative amplitudes of the six lake-wide free oscillation modes. Zero-amplitude nodal lines are shown in red.

The amount and distribution of the zero phase of the lake-wide modes with
periods of 63.0, 32.4, 21.6, and 14.2 min (red lines in Fig.

The same as Fig.

The slowest oscillation mode has a period of 117 min, which exceeds the longest possible period of a standing wave in an enclosed
basin significantly. The oscillations contain about 6 % of the energy of the largest
“first longitudinal” mode making it the fifth strongest oscillation mode. The
oscillation has the greatest amplitudes concentrated in Polson Bay
(Fig.

The shape of the lake surface corresponding to the maximum slope due to the slowest open-channel Helmholtz mode (see also animated surface oscillations in Supplement II).

Six local modes do not produce any appreciable water level oscillation in the
open part of Flathead Lake (Fig.

The distribution of the seiche current velocities is characterized by local
spots of strong current intensification, confined to the shallow nearshore
areas or narrow straights connecting different parts of the lake
(Figs.

Normalized amplitudes of the velocity oscillations for the six modes with strong intensification of seiche currents in the nearshore areas (red color). Note the logarithmic color scale.

Currents produced by four seiche modes have rotational features revealed by
the spatial distribution of the rotary coefficients (Fig.

Rotary coefficients for the four free oscillation modes with rotational character. See Fig. S3 in Supplement I for spatial distribution of rotary coefficients for other modes.

The modal structure of seiches derived from the modeling results shows good
qualitative agreement with water level records from Flathead Lake
(Fig.

Several important discrepancies between the modeled and observed spectra
provide an insight into the features not captured by the model, but
potentially significant for the hydrodynamics of Flathead Lake. The high-frequency modes with periods of 10–15 min are generally stronger in the
observations that in the model. Observations at measurement sites WB and NR
(Fig.

Spectra of measured (blue) vs. modeled (red) free-surface oscillations at four different locations in Flathead Lake. The inlet shows the positions of the corresponding measurement points.

Seiches in Flathead Lake have been measured in past studies

Fourier analysis is an established method for studying tidal oscillations,
but is usually considered inapplicable to seiche analysis because seiche
frequencies are basin-conditioned and are not known a priori. We demonstrated
that isolation of the seiche frequencies by constructing a maximum spectrum
over the basin under investigation is an effective way to isolate the
eigenfrequencies from the output of primitive equation models, at the expense
of estimation of the spectral density at each model grid point. The latter
suggests 10

Our method allowed identification of several specific features of the
basin-scale oscillations in Flathead Lake, indistinguishable by the simple
channel-like approximation (Eq.

The essence of seiche analysis presented above consists of establishing the
two-dimensional picture of seiche oscillations and designation of areas with
maximum water level and current speed amplitudes. Among the applied aspects
of these results are lakeshore management issues such as shoreline
erosion and estimation of the flood risks. The north shore and Polson Bay are
two areas of the lake that this modeling effort has shown to have both high
amplitude and high current velocities associated with seiche motions. Both of
these areas have experienced very high levels of shoreline retreat
(1–14 m yr

Seiches have also caused significant flooding to shoreline development activities that have built too close to the lakeshore and at too low elevations. We found through our modeling exercise that the low-frequency Helmholtz mode had high amplitudes in Polson Bay, Big Arm Bay, and areas of the north shore. These areas would be particularly in danger of potential flooding, especially in Polson Bay, given the shorter frequencies and possibility of developing resonance conditions. High-velocity amplitudes were also found along Wildhorse Island and Polson Bay. The high-frequency modes concentrated at the north shore could affect sediment transport distribution from the river inflow as well as the drift of aquatic invasive species. Results from this study will be valuable to local planning departments and developers alike to limit future problems associated with seiche-induced flooding (Fig. S4, Supplement I) by providing maps of spatially distributed zones of expected seiche impacts.

In addition, the maps of rotational coefficients for expected dominant modes
of oscillation (Fig.

As discussed above, the method applied in the present study provides an
effective way of gaining information on the precise seiche temporal
characteristics and, more importantly, on the two-dimensional lateral
distribution of the seiche amplitudes and currents. The latter are difficult
to reveal from direct field observations constrained to irregular point
measurements at the lake surface; however they are crucial for understanding the
seiche contribution to the transport of suspended matter and lake-wide
mixing. Moreover, knowledge on relative distribution of seiche intensity
along the lake shores is of key importance for shoreline management. With
regard to estimation of seiche effects on the littoral zone, our model
effectively complements the observation data on the nearshore water level
variability, as well as providing guidelines for the design of water level
monitoring. Our results do not include information on the absolute magnitudes
of water level oscillations and currents. The latter can vary within a wide
range, depending on wind forcing and wind–seiche resonance, or be produced by
other disturbances, such as earthquakes

Wind, dam operations, and seiche oscillations in lakes can play a significant role in shoreline erosion by exposing fragile and otherwise protected backshore environments to the action of wind waves. An alteration of hydrologic connection with marsh shorelines through seiche motions ultimately impacts the mosaic of riparian habitat. Therefore, understanding food web dynamics in the lake, management of lake level regulation with dams, and the early detection of aquatic invasive species and shoreline development planning can all benefit from a more quantitative understanding of seiche motions, based on a combination of our model results and the monitoring of water level/currents.

Comparison of pressure records from transducers from different locations
around the lake show distinct seiche oscillations (Fig.

The research was supported by the German Science Foundation (DFG projects KI 853/5-1, KI 853/7-1) and by the Leibniz Association (project SAW-2011-IGB-2). M. S. Lorang was partially supported by the FLBS Limnology Professorship endowment. Edited by: H. H. G. Savenije