Hazard assessment of the Tidal Inlet landslide and potential subsequent tsunami, Glacier Bay National Park, Alaska

Abstract

An unstable rock slump, estimated at 5 to 10 × 10⁶ m³, lies perched above the northern shore of Tidal Inlet in Glacier Bay National Park, Alaska. This landslide mass has the potential to rapidly move into Tidal Inlet and generate large, long-period-impulse tsunami waves. Field and photographic examination revealed that the landslide moved between 1892 and 1919 after the retreat of the Little Ice Age glaciers from Tidal Inlet in 1890. Global positioning system measurements over a 2-year period show that the perched mass is presently moving at 3–4 cm annually indicating the landslide remains unstable. Numerical simulations of landslide-generated waves suggest that in the western arm of Glacier Bay, wave amplitudes would be greatest near the mouth of Tidal Inlet and slightly decrease with water depth according to Green’s law. As a function of time, wave amplitude would be greatest within approximately 40 min of the landslide entering water, with significant wave activity continuing for potentially several hours.

Appendices

Appendix

Resonance of Tidal Inlet: Analytic Approach

The geometry of Tidal Inlet is ideally suited to allow a determination of the modes of natural resonance using analytic expressions. For example, Rogers and Mei (1978) used a rectangular bay geometry with a width 2a and length L (Fig. 9). The resonant modes are determined using the Boussinesq equations at a constant water depth (h) that include the effects on nonlinearity and dispersion:

$$ \begin{array}{*{20}l} {{\eta _{{\text{t}}} + \nabla u + \nabla {\left( {\eta u} \right)} = 0} \hfill} \\ {{u_{{\text{t}}} + \nabla \eta + \frac{1} {2}\nabla u^{2} + \frac{1} {3}\mu ^{2} \nabla \eta _{{{\text{tt}}}} = 0} \hfill} \\ \end{array} $$

(7)

where η is the water surface displacement, u is the depth-averaged horizontal velocity field, and the small parameter μ is defined by $\mu ^{2} = {\omega ^{2} h} \mathord{\left/ {\vphantom {{\omega ^{2} h} g}} \right. \kern-\nulldelimiterspace} g$ where ω is frequency and h is water depth. The variables in Eq. 7 have been nondimensionalized with respect to characteristic length and time scales.

Fig. 9

Geometry and coordinate system for a narrow bay

The resonant wave numbers (k _r) are given by

$$ {\left( {k_{{\text{r}}} l} \right)} \cong {\left( {m + \frac{1} {2}} \right)}\pi + {\left[ {{\left( {\frac{1} {{k_{{\text{r}}} l}}} \right)}\operatorname{Im} Z} \right]}_{{k_{{\text{r}}} l = {\left( {m + \tfrac{1} {2}} \right)}\pi }} m = 0,1,2, \ldots K $$

(8)

where l is the nondimensional length of the bay. The entrance impedance of the bay (Z) is defined by Rogers and Mei (1978) as

$$ Z = k^{2} \delta {\left[ {1 + \frac{{2i}} {\pi }\ln {\left( {k\delta \frac{{2\gamma }} {{\pi e}}} \right)}} \right]} $$

(9)

where 2δ is the nondimensional bay width and γ is Euler’s constant. The aspect ratio of Tidal Inlet is approximately $\frac{{2\delta }} {l} = \frac{1} {7}$ such that the first two resonant modes are (k _r l)₁ = 1.272 and (k _r l)₂ = 4.050. These modes are slightly less than the 1/4 and 3/4 wavelength resonant modes predicted by a simple “quarter-wavelength resonator” (Raichlen and Lee 1992). At resonant antinodes, there is no additional amplification of waves from resonance within the bay (i.e., unit response).

Outside the bay and far from the entrance, nonlinear effects can be ignored. Rogers and Mei (1978) indicated that most bay and harbor resonant problems can be decoupled so that the nonlinear theory is used within the bay and the linear theory is used in the ocean. The radiated wave from the bay, exclusive of incident and reflected waves outside the bay, is given by the following expression (Rogers and Mei 1978):

$$ \eta _{{{\text{rad}}}} = iTk\delta \sin {\left( {kl} \right)}H^{{{\left( 1 \right)}}}_{0} {\left( {kr} \right)} $$

(10)

where r ² = x ² + y ⁰ >> δ, x > 0 (Fig. 9),$H^{{{\left( 1 \right)}}}_{0} {\left( {kr} \right)}$is the Bessel function of the third kind (Hankel function), and

$$ T = A{\left[ {\cos {\left( {kl} \right)} - {\left( {{iZ} \mathord{\left/ {\vphantom {{iZ} k}} \right. \kern-\nulldelimiterspace} k} \right)}\sin {\left( {kl} \right)}} \right]}^{{ - 1}} $$

(11)

where A is the nondimensional amplitude of the incident wave. The wave field from Eq. 11 is shown in Fig. 10.

Fig. 10

Modeled radiant component of wave field in Glacier Bay outside the Tidal Inlet from resonance involving monochromatic input. Axes in nondimensional units

When considering the effects of resonance on transient waves, Kowalik (2001) also noted that resonant amplification also depends on the duration of the wave train. Short wave trains, relative to bay length, will not last long enough to set up resonance. Resonance in elongated coastal inlets may be more of a concern with longer-period waves from large seismogenic tsunamis (Carrier and Shaw 1970; Henry and Murty 1995).