2 Strongly nonlinear two-layer model

The two-layer model in three dimensions is strongly nonlinear, where internal waves of mode one are modelled. The wavelength is assumed to be much longer than the width of the pycnocline in a realistic fluid, such as in a layered stratification in the field. The model has been developed and tested with successful comparison with laboratory experiments on transient interfacial wave formation at a geometry in the lower layer (Grue Reference Grue2015a ). It was further extended to model the dead water problem at subcritical speed (Grue Reference Grue2015b ). The interfacial method is a generalization of a fully nonlinear method developed for a single-layer fluid, see Clamond & Grue (Reference Clamond and Grue2001, § 6), Grue (Reference Grue2002, § 6), Fructus, Clamond & Grue (Reference Fructus, Clamond, Grue and Kristiansen2005) and Fructus & Grue (Reference Fructus and Grue2007).

This section specifies the mathematical problem and solution for the case of a two-layer fluid. The upper layer has density $\unicode[STIX]{x1D70C}_{0}$ , thickness $h_{0}$ at rest and is referred to as fluid 0. This is overlying a lower fluid 1 of density $\unicode[STIX]{x1D70C}_{1}$ and depth $h_{1}$ . Incompressible and irrotational motion in each of the layers is assumed. The motion is governed by the Laplacian potentials $\unicode[STIX]{x1D719}_{0}$ and $\unicode[STIX]{x1D719}_{1}$ in the two layers, where index 0 refers to layer 0 and index 1 to layer 1. Let $\boldsymbol{x}=(x_{1},x_{2})$ denote horizontal coordinates in the plane that coincides with the centre of the pycnocline at rest, $y$ the vertical coordinate and $t$ time. The ship is moving at velocity $\boldsymbol{U}=U\boldsymbol{i}$ along the upper boundary of the upper fluid, where $\boldsymbol{i}=\unicode[STIX]{x1D735}x_{1}$ . The position of the ship hull is mathematically described by $y=h_{0}+\unicode[STIX]{x1D6FD}(\boldsymbol{x},t)$ , where $\unicode[STIX]{x1D6FD}(\boldsymbol{x},t)$ , a prescribed function of $\boldsymbol{x}$ and $t$ , determines the hull shape and is specified in (3.1) below. A definition sketch is found in figure 1. Following Grue (Reference Grue2015b ) we denote by $F$ the upper boundary of the upper fluid 0. The kinematic boundary condition along the ship hull is given by $\unicode[STIX]{x2202}\unicode[STIX]{x1D6FD}/\unicode[STIX]{x2202}t+W_{F}=0$ , where the normal velocity $W_{F}=\boldsymbol{U}\boldsymbol{\cdot }\unicode[STIX]{x1D735}_{H}\unicode[STIX]{x1D6FD}$ is connected to the fluid motion by

Here, $\unicode[STIX]{x1D735}_{H}=(\unicode[STIX]{x2202}/\unicode[STIX]{x2202}x_{1},\unicode[STIX]{x2202}/\unicode[STIX]{x2202}x_{2})$ denotes the horizontal gradient. The normal vector points into fluid 0. The potential evaluated at the position of the ship hull is introduced by $\unicode[STIX]{x1D719}_{0F}(\boldsymbol{x},t)=\unicode[STIX]{x1D719}(\boldsymbol{x},y=h_{0}+\unicode[STIX]{x1D6FD}(\boldsymbol{x},t),t)$ . The Froude number of the surface waves based on the ship length is $U/\sqrt{gl_{0}}\ll 1$ (where $g$ is the acceleration of gravity) and means that the rigid-lid condition can be applied along $F$ , outside the ship model, where $y=h_{0}$ . The relative roles of the internal and surface waves can be evaluated by calculating the wave resistance. This is further discussed in § 3.1, finding that the surface wave amplitude due to the observed ship is less than 4 % of the internal wave amplitude, where also the dead water resistance exceeds that of the surface waves.

Figure 1.

Sketch of the two-fluid model. (a) Centre plane section, side view. (b) Mid ship section, frontal view. (c) Flow variables.

The Froude number in relation to the internal wave mode is defined by $Fr=U/c_{0}$ , where $c_{0}$ denotes the interfacial (internal) long-wave speed of the two-fluid system, given by $c_{0}^{2}=g^{\prime }h_{0}/(\unicode[STIX]{x1D70C}_{0}/\unicode[STIX]{x1D70C}_{1}+h_{0}/h_{1})$ , where $g^{\prime }=g(\unicode[STIX]{x1D70C}_{1}-\unicode[STIX]{x1D70C}_{0})/\unicode[STIX]{x1D70C}_{1}$ denotes reduced gravity. Let the interface be denoted by $I$ and its motion described by the elevation $y=\unicode[STIX]{x1D702}(x_{1},x_{2},t)$ , where $y=0$ determines the interface at rest. Values of the potentials along  $I$ are introduced by $\unicode[STIX]{x1D719}_{0I}(\boldsymbol{x},t)=\unicode[STIX]{x1D719}_{0}(\boldsymbol{x},y=\unicode[STIX]{x1D702},t)$ and $\unicode[STIX]{x1D719}_{1I}(\boldsymbol{x},t)=\unicode[STIX]{x1D719}_{1}(\boldsymbol{x},y=\unicode[STIX]{x1D702},t)$ at $I$ . Scaled normal velocities along $I$ are introduced by

for the upper and lower fluid respectively, where $n$ points into the upper fluid 0 (see figure 1).

Solution of the Laplacian potentials $\unicode[STIX]{x1D719}_{0}$ and $\unicode[STIX]{x1D719}_{1}$ is expressed by a set of integral equations, see Grue (Reference Grue2015a ,Reference Grue b ). The velocities $W_{I}$ , $V_{I}$ along $I$ and potential $\unicode[STIX]{x1D719}_{0F}$ along $F$ are obtained using the method of successive approximations (e.g. Mikhlin Reference Mikhlin1957):

The leading linear approximation of the set of equations is obtained by the triple $(W_{I}^{(1)},V_{I}^{(1)},\unicode[STIX]{x1D719}_{0F}^{(1)})$ . The quadratic approximation, obtained by $(W_{I}^{(1)}+W_{I}^{(2)},V_{I}^{(1)}+V_{I}^{(2)},\unicode[STIX]{x1D719}_{0F}^{(1)}+\unicode[STIX]{x1D719}_{0F}^{(2)})$ , includes the leading coupling between the variables $\unicode[STIX]{x1D719}_{0}$ and $\unicode[STIX]{x2202}\unicode[STIX]{x1D719}_{0}/\unicode[STIX]{x2202}n$ , and $\unicode[STIX]{x1D719}_{1}$ and $\unicode[STIX]{x2202}\unicode[STIX]{x1D719}_{1}/\unicode[STIX]{x2202}n$ , as well as the excursions $\unicode[STIX]{x1D702}$ along $I$ and $\unicode[STIX]{x1D6FD}$ along $F$ , where $\unicode[STIX]{x1D6FD}$ is prescribed. The cubic approximation, obtained by $(W_{I}^{(1)}+W_{I}^{(2)}+W_{I}^{(3)},V_{I}^{(1)}+V_{I}^{(2)}+V_{I}^{(3)},\unicode[STIX]{x1D719}_{0F}^{(1)}+\unicode[STIX]{x1D719}_{0F}^{(2)}+\unicode[STIX]{x1D719}_{0F}^{(3)})$ , includes cubic couplings between the potentials, normal velocities and excursions along $I$ and the body geometry. The quadratic and cubic contributions have been implemented and tested in the two-dimensional case, where reference solutions for solitary waves of very large amplitude exist. Further, interfacial wave generation by a moving geometry in two dimensions has been tested. We refer to Grue (Reference Grue2015a , figures 2–5, table 1), where it is documented that the expansions of the variables given in (2.3) above converge very rapidly for interfacial elevations up to the thinner layer depth ( $h_{0}$ ). Thus, for $\unicode[STIX]{x1D702}/h_{0}\sim 1$ the examples provided in Grue (Reference Grue2015a ) document that $|W_{I}^{(3)}|/|W_{I}^{(1)}+W_{I}^{(2)}+W_{I}^{(3)}|\simeq |W_{I}^{(3)}|/|W_{I}|\ll 1$ , $|V_{I}^{(3)}|/|V_{I}^{(1)}+V_{I}^{(2)}+V_{I}^{(3)}|\simeq |V_{I}^{(3)}|/|V_{I}|\ll 1$ . This means that $W_{I}^{(1)}+W_{I}^{(2)}$ and $V_{I}^{(1)}+V_{I}^{(2)}$ represent the full nonlinearity in the calculations for elevations up to $\unicode[STIX]{x1D702}/h_{0}\sim 1$ . This result contributes considerably to the computational speed and requirements of RAM. The evaluations of $W_{I}^{(1)}+W_{I}^{(2)}$ , $V_{I}^{(1)}+V_{I}^{(2)}$ imply an order of magnitude fewer products compared with $W_{I}^{(1)}+W_{I}^{(2)}+W_{I}^{(3)}$ , $V_{I}^{(1)}+V_{I}^{(2)}+V_{I}^{(3)}$ . As a result, computations can be performed for relatively large horizontal domains, where the opposite is true for evaluations including the cubic terms, where the correction terms provide contributions that are approximately zero.

Explicit expressions of the linear and quadratic contributions are obtained by use of the Fourier transform:

Expressions for $W_{I}^{(3)}$ and $V_{I}^{(3)}$ are given in appendix D. Here, ${\mathcal{F}}$ denotes Fourier transform, ${\mathcal{F}}^{-1}$ inverse transform, $\boldsymbol{k}=(k_{1},k_{2})$ wavenumber vector in Fourier space and $k=|\boldsymbol{k}|$ . Further, $T_{0}=\tanh (kh_{0})$ , $T_{1}=\tanh (kh_{1})$ and $C_{0}=\cosh (kh_{0})$ . In (2.5) and (2.9), $\unicode[STIX]{x1D719}_{0F}=\unicode[STIX]{x1D719}_{0F}^{(1)}+\unicode[STIX]{x1D719}_{0F}^{(2)}$ . In the strongly nonlinear calculations presented below, the quadratic approximation is used. Linear calculations are obtained by using the triple $(W_{I}^{(1)},V_{I}^{(1)},\unicode[STIX]{x1D719}_{0F}^{(1)})$ .

The interfacial motion is expressed by the difference and sum potentials along the interface:

where $\unicode[STIX]{x1D707}=\unicode[STIX]{x1D70C}_{0}/\unicode[STIX]{x1D70C}_{1}$ denotes the density ratio. It should be noted that the jump function in (2.10a ) accounts for interactions between the density jump and ambient pressure fields. The interfacial elevation and jump in potential along $I$ are integrated forward in time using the kinematic and dynamic boundary conditions at the interface, giving

where $g^{\prime }$ is defined below (2.1). The right-hand side of (2.11b ) is

and is evaluated in the nonlinear calculations. In the linear calculations, $\mathscr{N}\mathscr{L}_{2}$ is put to zero. Successive approximations of the potential sum $\unicode[STIX]{x1D719}_{0I}+\unicode[STIX]{x1D719}_{1I}=\unicode[STIX]{x1D6F7}=\unicode[STIX]{x1D6F7}^{(1)}+\unicode[STIX]{x1D6F7}^{(2)}+\cdots$ are obtained by applying continuity of the normal velocity along the interface, using the expansions for $V_{I}$ and $W_{I}$ . The potentials $\unicode[STIX]{x1D719}_{0I}$ and $\unicode[STIX]{x1D719}_{1I}$ are then obtained. An RK4 scheme is used for time integration of the Fourier transformed versions of (2.11). The numerical method and implementation are described in Grue (Reference Grue2015a ,Reference Grue b ).

3 Wave wake and resistance

The ship geometry is modelled by

where $l_{0}$ denotes the ship length, $w_{0}$ the width, $d_{0}$ the draught and $x_{1,0}(t)$ the prescribed position of the ship where $U={\dot{x}}_{1,0}$ (note that $\unicode[STIX]{x1D6FD}\leqslant 0$ ). We perform computations relevant to a set of field observations of the wave wake generated by a cargo ship moving along a Canadian fjord; see the description in § 4.1 below. In the observations, an upper mixed layer of depth 5 m is separated from a deep lower salty layer by a 5 m thick pycnocline localized at an average depth of 7.5 m. Applying the two-fluid model, we shall use $h_{0}=7.5~\text{m}$ as the depth of the upper layer in the two-layer fluid. This is also the reference length for the computations. Wave propagation characteristics comparing two- and three-layer fluids, where in the latter the finite thickness of the pycnocline is accounted for, are described in appendix B, with results illustrated in figure 13, finding only minor differences between the wave phase speeds and group velocities of the two- and three-layer fluids for $k_{1}h_{0}<0.7$ (where $k_{1}$ is the wavenumber along the $x_{1}$ -direction). The asymptotic wave wake patterns of the two- and three-layer fluids are compared in figure 14(b) for the background density conditions in the Canadian fjord. We note that a two-layer representation of internal waves moving along a pycnocline of small or moderately finite thickness is further justified by the wave tank experiments by Grue et al. (Reference Grue, Jensen, Rusås and Sveen1999) and the three-layer calculations by Fructus & Grue (Reference Fructus and Grue2004).

Figure 2.

(a) The elevations $\unicode[STIX]{x1D702}_{NL}$ (—-) and $\unicode[STIX]{x1D702}_{LIN}$ (- - -) at the centre plane section ( $x_{2}=0$ ).(b) The elevations $\unicode[STIX]{x1D702}_{NL}$ (—-) and $\unicode[STIX]{x1D702}_{LIN}$ (- - -) at the lateral section at $(x_{1}-Ut)/h_{0}=9.6$ ; $\unicode[STIX]{x1D702}_{NL}$ ( $\cdots \,$ ) and $\unicode[STIX]{x1D702}_{LIN}$ (- - -) at $(x_{1}-Ut)/h_{0}=-20.3$ . Section of the ship geometry at $(x_{1}-Ut)/h_{0}=0$ ; $Fr=6.6$ , $(l_{0},w_{0},d_{0})=(29.4h_{0},4h_{0},1.2h_{0})$ ; same computations as in figure 3(a,b).

The observed ship has the following dimensions: length 221 m, width 23 m and draught 9 m. The corresponding non-dimensional length and width are $l/h_{0}=29.4$ and $w/h_{0}=3$ respectively. The draught of 9 m corresponds to a non-dimensional draught of $d_{0}/h_{0}=1.2$ . In the computations, the degree of nonlinearity is investigated by varying this parameter in the range $d_{0}/h_{0}\sim 0.1{-}1.2$ . A ship draught of $d_{0}/h_{0}=0.1$ gives a wave wake well within the linear regime, while $d_{0}/h_{0}=1.2$ gives a strongly nonlinear interaction between the body and the interface, corresponding to the ship in the observation. The volume of the numerical ship, with dimensions $(l_{0},w_{0},d_{0})=(29.4h_{0},4h_{0},1.2h_{0})$ , becomes $V_{0}=43\,000~\text{m}^{3}$ (with $h_{0}=7.5~\text{m}$ ), corresponding to the volume of the ship in the observation. Non-vertical ship walls at the waterline are assumed with the present mathematical and numerical formulation. In order to keep the sectional area of the ship, and thereby the volume, the width of the ship at the waterline is somewhat greater than the vertically averaged width. In the supercritical flow conditions, the interface is pushed down below the ship. Calculations of the steady motion may thus be performed for ship draughts that slightly exceed the interfacial level at rest where the nonlinear and linear interfaces below the ship are shown in figure 2. This is not true in the initial ramp-up phase in the calculations with $d_{0}/h_{0}$ slightly exceeding 1. The interface rapidly converges to the supercritical position, however. Alternatively, the draught may be gradually increased up to its final value. This has not been necessary. In the present formulation the local draught of the ship geometry is assumed to be a single-valued function of the horizontal coordinate; however, the body coordinates may be generalized to be locally ‘overhanging’, following the generalizations presented in Clamond & Grue (Reference Clamond and Grue2001, appendix B.3) in two dimensions. Steady trim and pitch effects can be accounted for iteratively; however, they are quite small for a free-surface Froude number of $U/\sqrt{gl_{0}}=0.14$ , such as for the present observed ship.

Figure 3.

(a) Nonlinear elevation ( $\unicode[STIX]{x1D702}_{NL}$ ), (b) linear elevation ( $\unicode[STIX]{x1D702}_{LIN}$ ), (c) difference ( $\unicode[STIX]{x1D702}_{NL}-\unicode[STIX]{x1D702}_{LIN}$ ) of ship wake; $Fr=6.6$ , $tc_{0}/h_{0}=100$ . (d) Nonlinear elevation $\unicode[STIX]{x1D702}_{NL}$ far downstream, at $tc_{0}/h_{0}=135$ , and (e) $\unicode[STIX]{x1D702}_{NL}$ at $tc_{0}/h_{0}=195$ . Ship dimensions $(l_{0},w_{0},d_{0})=(29.4h_{0},4h_{0},1.2h_{0})$ . Centre position of ship model at $(x_{1}-Ut=0,x_{2}=0)$ .

Ship wakes are calculated for Froude numbers $Fr=U/c_{0}$ in the range 4–12, where $c_{0}$ is defined below (2.1). A numerical wave tank of (length, width) of $(L_{1}=700h_{0},L_{2}=260h_{0})$ is discretized with $(N_{1},N_{2})=(864,600)$ nodes along the respective directions. This means that $(\unicode[STIX]{x0394}x_{1}/h_{0},\unicode[STIX]{x0394}x_{2}/h_{0})=(0.81,0.43)$ . The nonlinear ( $\unicode[STIX]{x1D702}_{NL}$ ) and linear ( $\unicode[STIX]{x1D702}_{LIN}$ ) elevations in the wave wake are obtained for the ship model with draught $d_{0}/h_{0}=1.2$ moving along the $x_{1}$ -direction at $Fr=6.6$ . The running time is $t=100h_{0}/c_{0}$ ; see figures 3(a,b). The depth ratio is $h_{1}/h_{0}=18$ , and $\unicode[STIX]{x0394}\unicode[STIX]{x1D70C}/\unicode[STIX]{x1D70C}\ll 1$ . The results illustrate that the difference $\unicode[STIX]{x1D702}_{NL}-\unicode[STIX]{x1D702}_{LIN}$ is comparable to the nonlinear elevation; see figures 3(c), 4(a)–(d), for $Fr\sim 4{-}12$ . Computations of the far downstream wave wake, for longer running times of $tc_{0}/h_{0}=135$ and $tc_{0}/h_{0}=195$ , are performed using wider and shorter wave tanks, and discretizations of respectively $(L_{1},L_{2})=(500h_{0},250h_{0})$ , $(N_{1},N_{2})=(720,720)$ (figure 3 d) and $(L_{1},L_{2})=(400h_{0},400h_{0})$ , $(N_{1},N_{2})=(500,1040)$ (figure 3 e). In these calculations the ship model is stopped after having traversed the distance $L_{1}$ .

Figure 4.

(a) Nonlinear elevation ( $\unicode[STIX]{x1D702}_{NL}$ ) and (b) difference $\unicode[STIX]{x1D702}_{NL}-\unicode[STIX]{x1D702}_{LIN}$ of wave wake, for $Fr=4$ , $tc_{0}/h_{0}=140$ , for ship model of $(l_{0},w_{0},d_{0})=(29.4h_{0},4h_{0},0.8h_{0})$ . (c,d) The same for $Fr=12$ , $tc_{0}/h_{0}=80$ and ship model of $(l_{0},w_{0},d_{0})=(29.4h_{0},4h_{0},1.2h_{0})$ .

Wave wake elevations of numerical ships with draughts in the range $d_{0}/h_{0}\sim 0.1{-}1.2$ and Froude numbers of $Fr=4$ , 6.6 and 12 are obtained. The snapshots in figure 5 show the laterally moving wave train, where $\unicode[STIX]{x1D702}$ is a function of the lateral coordinate $x_{2}$ , at far downstream positions $x_{1}-Ut$ . An overprediction of the wave heights by up to 50 % is a main inferiority of the linear predictions. This occurs when the ship draught is comparable to the upper layer depth $(d_{0}/h_{0}\sim 1)$ . In figure 5(f), for $Fr=6.6$ and draught $d_{0}/h_{0}=1.2$ , the wave height is up to $0.2h_{0}$ in the nonlinear prediction and $0.3h_{0}$ in the linear prediction. The overprediction by the linear method is 39 % in figure 5(i), for $Fr=12$ , $d_{0}/h_{0}=1.2$ , and 34 % in figure 5(c), for $Fr=4$ , $d_{0}/h_{0}=0.9$ .

Figure 5.

Nonlinear (solid line) and linear (dashed line) lateral elevation of wave wake at fixed downstream positions for increasing ship draught and Froude number: (a)–(c) $(x_{1}-Ut)/h_{0}=92.6$ , $Fr=4$ , ship draught $d_{0}/h_{0}=0.3,0.7,0.9$ ; (d)–(f) $(x_{1}-Ut)/h_{0}=133.3$ , $Fr=6.6$ , ship draught $d_{0}/h_{0}=0.4,0.8,1.2$ ; (g)–(i) $(x_{1}-Ut)/h_{0}=66.6$ , $Fr=12$ , ship draught $d_{0}/h_{0}=0.1,0.6,1.2$ . The ship model has length $l_{0}=29.4h_{0}$ and width $w_{0}=4h_{0}$ .

What is even more important is the representation of the leading part of the diverging wave train, where the nonlinear leading trough is pronounced, while its linear counterpart totally misrepresents the wave and also moves too slowly. The appreciable leading nonlinear trough including its forerunning slope is very important in the interpretation of what is identified as the leading diverging wave in the field observation described below. This is found to move at supercritical speed; see the discussion in § 5.2.

The negative excursions along the first four troughs are obtained for a ship draught of $d_{0}/h_{0}=0.8$ and $Fr=4$ , and $d_{0}/h_{0}=1.2$ and $Fr=6.6$ and 12. The depressions are obtained as functions of the lateral coordinate; see figure 6. Following studies on ship-generated surface wave wakes, the wave height of the diverging waves exhibits an algebraic decay in the lateral coordinate according to $x_{2}^{-\unicode[STIX]{x1D6FC}}$ , where the exponent $\unicode[STIX]{x1D6FC}$ is in the range between $0.2$ and $1.06$ ; see Doctors & Zilman (Reference Doctors and Zilman2004). The trough amplitudes, $\unicode[STIX]{x1D702}_{min}$ , in the present numerical experiments are similarly compared with

where $B$ and $\unicode[STIX]{x1D6FC}$ are fitted. The exponent $\unicode[STIX]{x1D6FC}$ is in the range 0.46–0.61 for the leading trough, and in the range 0.41–0.52 for the second trough, both resulting from the nonlinear calculations. The decay is weaker for the third and fourth troughs, with $\unicode[STIX]{x1D6FC}\sim 0.30{-}0.40$ and ${\sim}0.16{-}0.38$ respectively. Values of the coefficient $B$ are obtained as well. The results in table 1 may be used to obtain trough amplitudes at any lateral $x_{2}$ and downstream position $x_{1}$ . Linearly calculated depressions along the wave troughs are inferior, as seen in figure 6 and table 1.

Figure 6.

Nonlinear (left plots) and linear (right plots) elevation, $\unicode[STIX]{x1D702}_{min}$ , along troughs 1 to 4 in ship wake, versus lateral coordinate, $x_{2}$ : (a,b) $Fr=4$ , ship dimensions $(l_{0},w_{0},d_{0})=(29.4h_{0},4h_{0},0.8h_{0})$ ; (c,d) $Fr=6.6$ , ship dimensions $(l_{0},w_{0},d_{0})=(29.4h_{0},4h_{0},1.2h_{0})$ ; (e,f) $Fr=12$ , ship dimensions $(l_{0},w_{0},d_{0})=(29.4h_{0},4h_{0},1.2h_{0})$ ; trough 1 (*), trough 2 (●), trough 3 (▫), trough 4 (○). Solid and dotted lines: fits of $\unicode[STIX]{x1D702}_{min}/h_{0}=-B(x_{2}/h_{0})^{-\unicode[STIX]{x1D6FC}}$ .

Root mean square (r.m.s.) values of the difference between the nonlinear and linear wave wake are compared with r.m.s. values of $\unicode[STIX]{x1D702}_{NL}$ ; i.e. we divide $[\int _{\boldsymbol{x}}(\unicode[STIX]{x1D702}_{NL}-\unicode[STIX]{x1D702}_{LIN})^{2}\,\text{d}x_{1}\,\text{d}x_{2}]^{1/2}$ by $[\int _{\boldsymbol{x}}(\unicode[STIX]{x1D702}_{NL})^{2}\,\text{d}x_{1}\,\text{d}x_{2}]^{1/2}$ , where integration is carried out over the entire computational domain, after each run. The r.m.s. ratio, a function of the ship draught ratio ( $d_{0}/h_{0}$ ) and Froude number, is of order unity when $d_{0}/h_{0}\sim 1$ and $Fr=4$ and $Fr=6.6$ (figure 7). The relative difference reduces with increasing Froude number. Linear calculations are in general good only for very small draughts of the ship.

Figure 7.

The r.m.s. value of $\unicode[STIX]{x1D702}_{NL}-\unicode[STIX]{x1D702}_{LIN}$ divided by the r.m.s. value of $\unicode[STIX]{x1D702}_{NL}$ versus the ship draught $d_{0}/h_{0}$ for $Fr=4$ (●), $Fr=6.6$ (*) and  $Fr=12$ (♢).

Table 1.

Values $(\unicode[STIX]{x1D6FC},B)$ of fitted curves, $\unicode[STIX]{x1D702}_{min}/h_{0}=-B(x_{2}/h_{0})^{-\unicode[STIX]{x1D6FC}}$ , of troughs 1–4, obtained by nonlinear (NL) and linear (LIN) predictions. Ship model with $(l_{0},w_{0},d_{0})=(29.4h_{0},4h_{0},1.2h_{0})$ for $Fr=6.6$ and 12, and $(l_{0},w_{0},d_{0})=(29.4h_{0},4h_{0},0.8h_{0})$ for $Fr=4$ .

4 Field observations

A two-week-long field experiment was carried out in June 2013 to observe the propagation of nonlinear internal waves and their reflection off a steep cliff in the Saguenay Fjord in Eastern Canada (figure 9). The experiment was motivated by the preliminary observations of internal wave reflection made by Bourgault, Janes & Galbraith (Reference Bourgault, Janes and Galbraith2011) in this environment. One piece of evidence of the generation and propagation of an internal wave train caused by the passage of a cargo ship was observed during the experiment, and the details of this event are presented here. More information about the general oceanography of this fjord can be found in Bourgault et al. (Reference Bourgault, Janes and Galbraith2011), Bourgault, Galbraith & Winkler (Reference Bourgault, Galbraith and Winkler2012) and Belzile, Galbraith & Bourgault (Reference Belzile, Galbraith and Bourgault2016) and references therein.

Figure 9.

Map of the Saguenay Fjord and the bathymetry (in m). The symbol ‘C’ indicates the location of the time-lapse camera and its field of view (solid lines), and the circle indicates the position of the mooring. The curved solid line in the middle of the fjord is the Arctic ship track on 13 June obtained from the Automatic Identification System database. The red cross next to the mooring shows the position where a CTD cast was obtained at 20:30 UTC and used for comparison with the thermistor chain data (see figure 10 a,b for details).

In situ measurements were collected from a mooring equipped with a downward-looking 300 kHz Workhorse Sentinel acoustic Doppler current profiler (ADCP) manufactured by Teledyne RD Instruments. The ADCP was mounted on a gimbal installed inside a doughnut-like surface buoy in order to minimize pitch and roll caused by surface waves. Both pitch and roll stayed within $1^{\circ }$ for the duration of the internal ship wake event reported here. The ADCP pinging rate was 1.72 Hz and it recorded 10 s ensembles in 1 m vertical bin sizes. The compass calibration yielded an uncertainty of $\pm 7.8^{\circ }$ . The mooring was anchored in 80 m depth at $48^{\circ }~12.642^{\prime }~\text{N}$ , $69^{\circ }~53.719^{\prime }~\text{W}$ (figure 9). A GPS was installed on the buoy to measure its displacements.

The mooring was also equipped with eight temperature recorders (TR-1060 by RBR) and three temperature–depth recorders (TDR-2050) evenly distributed between 3.5 m and 38.5 m (3.5 m separation between each recorder). The three temperature–depth recorders were located at the top (3.5 m), middle (21.0 m) and end (38.5 m) of the temperature chain. The ADCP also recorded surface temperature at 0.5 m depth. Each RBR thermistor recorded at 1 Hz, while the ADCP recorded temperature in 10 s ensembles. Temperature–salinity profiles were also routinely carried out around the bay with a Seabird 19plus CTD profiler deployed from a small Zodiac.

Shore-based time-lapse photography also captured sea-surface patterns caused by internal waves, fronts and eddies in a way similar to that presented in Bourgault et al. (Reference Bourgault, Janes and Galbraith2011) and Richards et al. (Reference Richards, Bourgault, Galbraith, Hay and Kelley2013). We used a Canon EOS 40D with 10.1 megapixels located at $48^{\circ }~12.552^{\prime }~\text{N}$ , $69^{\circ }~54.492^{\prime }~\text{W}$ and altitude $H=50.5~\text{m}$ (figure 9). The camera recorded one image per minute. The images were calibrated to remove lens distortion and each image was stabilized against a reference image to remove small camera movements between successive images that may be caused by wind gusts. The calibrated and stabilized images were then georectified following Pawlowicz (Reference Pawlowicz2003), see also Bourgault (Reference Bourgault2008), using a series of 12 ground control points seen on site (boulders, capes, wharf, etc.). The r.m.s. difference between the ground control points and the georectified pixels is 10 m. The georectified image resolution is highly anisotropic with a much higher resolution in the horizontal field of view than in the vertical field of view due to the high obliquity of the images. The image resolution orthogonal to the line of sight and at the distance the ship passed, roughly 1 km away from the camera, is approximately 1 m. At the same distance, the resolution along the line of sight is approximately 20 m and steadily increases towards the camera.

The background density structure is typical for this subarctic fjord, with a thin brackish surface layer overlying a thick salty bottom layer. These two layers are separated by a 5 m thick pycnocline located between 5 and 10 m and characterized by a density jump of $\unicode[STIX]{x0394}\unicode[STIX]{x1D70C}=15~\text{kg}~\text{m}^{-3}$ ; see figure 10(a). The background conditions in the fjord, such as local velocity shear (figure 10 b), are further elaborated in appendix A.

Figure 10.

Background conditions at the mooring site prior to the arrival of the internal wake. (a) The density profile at 19:05 UTC inferred from the thermistor chain and the T–S relationship (thick black). For comparison a CTD cast was obtained at 20:30 UTC next to the mooring (see figure 9 for the position of that cast) and is shown here (thin grey) along with the density inferred from the thermistors at the same moment (thick grey). (b) The eastward (solid) and northward (dashed) velocity profiles averaged over 15 min between 18:50 and 19:05. The uncertainty in this 15 min average is $\pm 0.01~\text{m}~\text{s}^{-1}$ . (c) Sketch of the theoretical three-layer fluid. (d) Two-layer fluid.

5 Observation and theory compared