2.1. Experimental set-up

The experiments were carried out in a rectangular tank of size $L \times W \times H = 107\ \textrm {cm} \times 80\ \textrm {cm} \times 40\ \textrm {cm}$, filled with water (kinematic viscosity $\nu _{\ast }=1.0 \times 10^{-2}\ \textrm {cm}^2\ \textrm {s}^{-1}$) to a depth of 38 cm (see figure 1). The water was linearly stratified with salt and the buoyancy frequency was chosen as $N= 0.91$ rad s$^{-1}$. The tank dimensions were large enough to not have significant effects on the generated internal waves and induced mean flow. Furthermore, the tank was covered with a transparent plate to avoid parasitic flows due to surrounding air motion.

Figure 1.

Schematic view of the experimental set-up. The rectangular tank is of size $L\times W \times H$, where $L$ is measured from the wave generator which lies against the left-hand wall. Green lines correspond to the intersection of PIV laser sheets with the generator. Isophase planes of the generated internal wave beam are tilted downward at an angle $\theta$ from the horizontal. The group velocity $c_g$ is in the along-beam ($\xi$) direction while the phase velocity $c_\varphi$ is in the cross-beam ($\eta$) direction.

Internal waves were forced by a moving-plate generator (Mercier et al. Reference Mercier, Martinand, Mathur, Gostiaux, Peacock and Dauxois2010), horizontally centred on the left-hand wall of the tank at a height between 22.2 and 33.7 cm from the bottom. The origin of the coordinate system $\textrm {O}_\ast x_{\ast }y_{\ast }z_{\ast }$ is at the centre of the wave generator, at a height of 27.9 cm from the bottom, $y_{\ast }$ is the upward vertical coordinate and $x_{\ast }$ and $z_{\ast }$ are the along-tank and transverse horizontal coordinates, respectively (figure 1). The wave generator comprised 18 plates of 6.3 mm in thickness and 13.9 cm in width that moved horizontally, mimicking an upward-propagating transverse wave

(2.1)\begin{equation} x_{\ast}=a\sin (\omega t - 2{\rm \pi} y_{\ast} /\lambda_{\ast}) \end{equation}

with amplitude $a = 1$ cm and wavelength $\lambda _{\ast } = 3.83$ cm. This forcing generated an internal wave beam of frequency $\omega$ that propagated downward along the tank at an angle $\theta$ to the horizontal specified by the dispersion relation $\omega = N \sin \theta$ (see figure 1). The experiments focused on three driving frequencies, $\omega = 0.24, 0.45$ and 0.64 rad s$^{-1}$, corresponding to the propagation angles $\theta = 15^\circ$, $30^\circ$ and $45^\circ$, respectively.

To get a representative view of the three-dimensional velocity field, two separate measurements, in a vertical and a horizontal plane, were performed using particle image velocimetry (PIV). The tank was seeded with 10 $\mathrm {\mu }$m silver-coated hollow glass spheres with density of 1.4 kg m$^{-3}$, illuminated by a vertical/horizontal laser sheet produced by a 532 nm 2 W continuous laser and passing through the centre of the generator ($y_{\ast }= z_{\ast }=0$), while a camera was filming from the side/top towards the $-z_{\ast }$ direction/$-y_{\ast }$ direction. Two frames per second were recorded with a resolution of 4.51 and 5.66 pixels mm$^{-1}$ for the side and top views, imaging a region of $650\ \textrm {mm} \times 380\ \textrm {mm}$ and $650\ \textrm {mm} \times 490\ \textrm {mm}$, respectively. Final windows of the cross-correlation algorithm (Fincham & Delerce Reference Fincham and Delerce2000) had a size of $21 \times 21$ pixels with a 10-pixel grid spacing, corresponding to a mesh size of 2.2 mm/1.8 mm.

2.2. Long-time mean-flow evolution

We present detailed experimental observations only for $\omega =0.45$ rad s$^{-1}$, corresponding to the wave beam propagation angle $\theta = 30^\circ$ to the horizontal. The responses for the other two driving frequencies we considered are qualitatively similar to those discussed here. Figure 2 shows horizontal velocity fields in the $x_{\ast }$ direction obtained from PIV measurements in the central vertical plane ($z_{\ast } = 0$) and in the horizontal plane through the centre of the wave generator ($y_{\ast } = 0$), at three different times normalized with the wave beam period $T_0 = 2{\rm \pi} /\omega =14$ s. Specifically, the three rows of flow images in figure 2 correspond to $t/T_0=20$, 40 and 100. The left-hand image in each row shows the wave beam response in the vertical plane, while the middle and right-hand images depict the induced mean flow in the vertical and horizontal planes, respectively. The mean-flow responses were obtained by averaging the velocity in the $x_{\ast }$ direction over $10T_0$.

Figure 2.

The PIV measurements of horizontal velocity fields in the $x_{\ast }$ direction at three times, $t/T_0$, normalized with the wave beam period $T_0 = 2{\rm \pi} /\omega = 14$ s: (a) 20; (b) 40; (c) 100. The left-hand images show the wave beam responses in the vertical plane $z_{\ast } = 0$, while the middle and right-hand images depict the induced mean flow in the vertical plane $z_{\ast } = 0$ and the horizontal plane $y_{\ast } = 0$, respectively. These planes are indicated by dashed lines. The thick black line marks the location of the wave generator. The zero-velocity line around $x=10$ cm in the right-hand images is an experimental artifact due to laser sheet reflection.

Figure 2 reveals that the wave beam response has essentially reached steady state by $t/T_0 = 20$ and changes very little thereafter. This is in sharp contrast to the induced mean flow, which at early times is confined in the region of the generated wave beam (figure 2a), but later starts to spread in the $x_{\ast }$ direction (figure 2b), and eventually reaches a quasi-steady state that stretches along the $x_{\ast }$ direction far away from the beam (figure 2c). It should be noted that the mean flow is purely horizontal (i.e. negligible in the $y_{\ast }$ direction) and also features a component in the $z_{\ast }$ direction (not shown here) as required by incompressibility. At late times, however, when the mean flow is highly elongated in the $x_{\ast }$ direction relative to the $z_{\ast }$ direction (figure 2c), the transverse ($z_{\ast }$) component is much smaller than the along-tank ($x_{\ast }$) component.

Previous experiments using a similar set-up (Bordes Reference Bordes2012; Bordes et al. Reference Bordes, Venaille, Joubaud, Odier and Dauxois2012) concentrated on the beam–mean-flow interaction while the induced mean flow still resides in the vicinity of the wave beam (figure 2a). This early stage is characterized by rapid growth (linear in time) of the mean PV associated with the horizontal mean flow, which also affects the underlying internal wave beam as indicated by the cross-beam bending of the beam profile in figure 2 (left-hand images). The focus here, however, is on the long-time mean-flow development (figure 2b,c).

3.1. Onset of streaming

We begin by reviewing the model of Fan et al. (Reference Fan, Kataoka and Akylas2018). Their analysis focused on the ‘distinguished limit’ where the nonlinear beam–mean-flow coupling is as important as the effects of the along-beam dispersion, transverse dispersion and viscous attenuation of the beam envelope. For the benefit of readers interested in a detailed discussion of the scalings appropriate to this flow regime, we adopt the same non-dimensionalization (with $1/N$ as the time scale, where $N$ is the constant buoyancy frequency, and $\lambda _{\ast }/2{\rm \pi}$ as the length scale, where $\lambda _{\ast }$ is the beam carrier wavelength set by the wave generator) and use the same notation as Fan et al. (Reference Fan, Kataoka and Akylas2018). Here, as our interest centres on the experiments in § 2, background rotation is ignored ($\,f=0$).

In terms of $\varepsilon \ll 1$ defined in (3.1), which controls the assumed weak transverse variations, the beam envelope and the induced mean flow depend on the ‘slow’ spatial variables

(3.2a–c)\begin{equation} X=\varepsilon^{2}\xi,\quad Y=\varepsilon \eta,\quad Z=\varepsilon z, \end{equation}

where $\xi$, $\eta$ and $z$ are the along-beam, cross-beam and transverse coordinates, respectively (figure 3). Referring to the experimental set-up (figure 1), the coordinate system $\mathrm {O}xyz$ in figure 3 is the non-dimensional counterpart of $\textrm {O}_\ast x_{\ast }y_{\ast }z_{\ast }$, and the $(\xi ,\eta )$ coordinates are rotated by $\theta$ relative to the horizontal and vertical $(x,y)$ coordinates. Furthermore, the appropriate ‘slow’ time is

(3.3)\begin{equation} T=\varepsilon^{2} t, \end{equation}

indicating that the initial phase of the mean-flow development lasts $O(1/\varepsilon ^{2})$ beam periods.

Figure 3.

Schematic of theoretical model for the generated wave beam. The coordinate system $\mathrm {O}xyz$ is the non-dimensional counterpart of $\textrm {O}_\ast x_{\ast }y_{\ast }z_{\ast }$ (figure 1), $y$ is the upward vertical and $x$ and $z$ are the along-tank and transverse horizontal coordinates, respectively. The along- and cross-beam coordinates $\xi$ and $\eta$ (also marked in figure 1) are rotated by $\theta$ relative to $(x,y)$. The beam has a nearly monochromatic profile. Dotted lines indicate lines of constant phase of the sinusoidal carrier whose wavelength (set by the wave generator) has been normalized to $2{\rm \pi}$ in the asymptotic analysis. The envelope scale is $O(\varepsilon ^{-2})$ along the $\xi$ direction and $O(\varepsilon ^{-1})$ along the $\eta$ and $z$ directions, in keeping with the scalings (3.2a–c).

At this stage, the disturbance comprises an $O(\varepsilon )$ primary harmonic, representing a weakly nonlinear nearly monochromatic beam, and an $O(\varepsilon ^{2})$ mean flow. Specifically, the velocity components ${\boldsymbol u}=(u,v,w)$ in the $(\xi ,\eta ,z)$ coordinate system, the density perturbation $\rho$ and the pressure perturbation $p$ take the form

(3.4a)\begin{gather} {\boldsymbol u}=\varepsilon \{ {(U,\varepsilon^{2}V,\varepsilon W)\,\mbox{e}^{{\mathrm{i}}\varphi }+\mbox{c.c.}} \}+ \varepsilon^{2}(\bar{{U}},\bar{{V}},\bar{{W}})+\cdots, \end{gather}

(3.4b)\begin{gather}\rho =\varepsilon \{ {R\,\mbox{e}^{{\mathrm{i}}\varphi }+\mbox{c.c.}} \}+\varepsilon^{4}\bar{{R}}+\cdots,\quad p=\varepsilon \{{P\,\mbox{e}^{{\mathrm{i}}\varphi}+\mbox{c.c.}}\}+\varepsilon^{3}\bar{{P}}+\cdots, \end{gather}

where $\varphi =\eta -\omega t$ and $\omega =\sin \theta$. (These perturbation expansions are somewhat different from (2.11) in Fan et al. (Reference Fan, Kataoka and Akylas2018) owing to our assumption of no rotation, $f=0$; details of the analysis for $0\le f \ll 1$ including the scaling of the mean flow above can be found in Fan (Reference Fan2017).) The coupled beam–mean-flow dynamics is described via the various amplitudes in (3.4), which depend on $X$, $Y$, $Z$ and $T$. The equations governing these amplitudes are obtained by imposing incompressibility, mass conservation and momentum balance. Specifically, upon substituting (3.4) into these fundamental equations and collecting terms $\propto \exp ( {\mathrm {i}}\varphi )$, the beam amplitudes can be expressed in terms of the along-beam velocity amplitude $U$:

(3.5a–d)\begin{equation} V={\mathrm{i}} U_{X} +\cot \theta U_{ZZ},\quad W=-{\mathrm{i}}\cot \theta U_{Z},\quad R=-{\mathrm{i}} U,\quad P=\cos \theta U. \end{equation}

Thus, in view of (3.2a–c) and (3.4a), the beam response obeys incompressibility correct to $O(\varepsilon ^3)$. Furthermore, to the same order of approximation, $U$ satisfies the evolution equation

(3.6)\begin{equation} U_{T} +{\mathrm{i}}\bar{V}U+\cos \theta \left({U_{X} -\frac{{\mathrm{i}}}{2}\cot \theta U_{ZZ} } \right)+ \frac{\beta }{2}U=0. \end{equation}

The second term in (3.6) is the feedback of the induced mean flow to the beam dynamics, while the third and fourth terms, respectively, account for the effects of (along-beam and transverse) dispersion and viscous attenuation of the beam envelope. This balance of dissipation with nonlinear coupling and dispersion assumes that the inverse Reynolds number $\nu =4{\rm \pi} ^2 \nu _{\ast }/N {\lambda _{\ast }}^2$ scales like

(3.7)\begin{equation} \nu =\beta \varepsilon^{2}, \end{equation}

where $\beta =O(1)$.

Turning next to the induced mean flow, from mass conservation it is deduced that the vertical velocity is $O(\varepsilon ^{4})$. Thus, to leading order, the mean flow is purely horizontal:

(3.8)\begin{equation} \bar{U}=\bar{V}\cot \theta. \end{equation}

Furthermore, as along-beam variations are weaker than cross-beam and transverse variations in view of (3.2a–c), incompressibility to leading order requires

(3.9)\begin{equation} \bar{V}_{Y} +\bar{W}_{Z} =0. \end{equation}

Finally, to complete the description of the beam–mean-flow interaction, we find it convenient to utilize the transport equation

(3.10)\begin{equation} q_{t} +{\boldsymbol u}\boldsymbol{\cdot} \boldsymbol{\nabla} q=\nu ({\nabla^{2}\zeta -\boldsymbol{\nabla} \rho \boldsymbol{\cdot} \nabla^{2}(\boldsymbol{\nabla} \times {\boldsymbol u})}) \end{equation}

for the PV

(3.11)\begin{equation} q=\zeta -\boldsymbol{\nabla} \rho \boldsymbol{\cdot} (\boldsymbol{\nabla} \times {\boldsymbol u}), \end{equation}

where $\zeta =(\boldsymbol {\nabla } \times {\boldsymbol u})\boldsymbol {\cdot } {\boldsymbol j}$ is the vertical vorticity and ${\boldsymbol j}$ denotes the vertical ($y$) unit vector. From (3.11), upon making use of (3.4)–(3.5a–d), it follows that $q=\varepsilon ^{3}\bar {Q}+\cdots ,$ where

(3.12)\begin{equation} \bar{Q}=\varOmega -2(UU^{\ast })_{Z} \end{equation}

is the mean PV amplitude and

(3.13)\begin{equation} \varOmega =\frac{1}{\sin \theta }\bar{V}_{Z} -\sin \theta \bar{W}_{Y} \end{equation}

is the amplitude of the mean vertical vorticity, $\zeta =\varepsilon ^{3}\varOmega +\cdots$. Thus, taking into account (3.4), (3.7), (3.12) and (3.13), the balance of mean terms in the PV transport equation (3.10), correct to $O(\varepsilon ^{5})$, yields

(3.14)\begin{equation} \varOmega_{T} =2\left({\frac{\partial }{\partial T}+\beta } \right)(UU^{\ast })_{Z}. \end{equation}

This evolution equation along with (3.6), (3.9) and (3.13), which govern the beam–mean-flow interaction for $T=O(1)$, are the end results of Fan et al. (Reference Fan, Kataoka and Akylas2018).

It should be noted that the vertical vorticity associated with the induced mean flow derives from two separate sources, one of inviscid and the other of viscous origin, corresponding to the two contributions to the forcing term on the right-hand side of (3.14). The focus here is on the mean flow due to the viscous source of vertical vorticity: this mean flow component, referred to as streaming, is forced resonantly and eventually overwhelms its inviscid counterpart. Specifically, for $T=O(1)$, the beam amplitude $U$, governed by (3.6), reaches a quasi-steady state. Thus, in response to the viscous forcing term in (3.14), $\varOmega$ grows linearly in $T$,

(3.15)\begin{equation} \varOmega \sim 2\beta T(UU^{\ast })_{Z}, \end{equation}

and so do $\bar {V}$ and $\bar {W}$. As a result, the convective derivative term on the left-hand side of the PV transport equation (3.10), which does not partake in the $O(\varepsilon ^5)$ dominant balance leading to (3.14), eventually comes into play.

3.2. Saturation of streaming

As noted above, the dominant balance of $O(\varepsilon ^5)$ terms in (3.10) that provides the mean vorticity equation (3.14) eventually is invalidated owing to the resonant growth of streaming. Specifically, this breakdown occurs at $T=O(\varepsilon ^{-1/2})$, when $\bar {V}$ and $\bar {W}$ are of $O(\varepsilon ^{-1/2})$ so the convective derivative term is as important as the time evolution term in (3.10). Thus, to bring out the proper balance of mean terms in (3.10) when $T=O(\varepsilon ^{-1/2})$, we introduce the rescaled mean-flow variables

(3.16a,b)\begin{equation} (\bar{V}, \bar{W}) \to \varepsilon^{-1/2} (\bar{V}, \bar{W}),\quad \varOmega \to \varepsilon^{-1/2} \varOmega, \end{equation}

which depend on $X$, $Y$, $Z$ defined in (3.2a–c) and the rescaled slow time

(3.17)\begin{equation} \hat{T}= \varepsilon^{1/2} T. \end{equation}

Upon implementing these scalings in the PV transport equation (3.10), equation (3.14) for the mean vertical vorticity is replaced by

(3.18)\begin{equation} \left({\frac{\partial }{\partial \hat{T}}} +\bar{V} {\frac{\partial }{\partial Y}} + \bar{W} {\frac{\partial }{\partial Z}} \right) \varOmega =2 \beta(UU^{\ast})_{Z} . \end{equation}

In this evolution equation, the advection by the induced mean flow counterbalances the viscous forcing of $\varOmega$. Thus, the resonant growth (3.15) predicted at early times ($\hat {T} \ll 1$) is expected to saturate for $\hat {T}=O(1)$ and, as a result, the mean flow would level off and start being transported away from the beam region. Qualitatively, this scenario is consistent with the experimental observations presented in § 2.2: the growth of streaming in the beam vicinity is essentially complete after 20 beam periods (figure 2a), and by 40 periods (figure 2b) the effects of advection are clearly visible. Detailed comparison between theory and experiments is made in § 4.

3.3. Far-field flow regime

As the mean vertical vorticity continues to be transported away from the beam, the effect of the forcing term on the right-hand side of (3.18) gradually diminishes. Thus, for $\hat {T} \gg 1$ the mean flow far from the beam reaches a quasi-steady state due to the balance of horizontal advection with viscous diffusion in the PV transport equation (3.10). In this far-field regime, as revealed by the experimental observations (figure 2c), the mean flow is highly elongated in the horizontal ($x$) direction. This suggests replacing the beam-oriented variables $X=\varepsilon ^{2} \xi$ and $Y=\varepsilon \eta$ in (3.2a–c) with $x$ and $y$ (see figure 3), scaled such that $x$ variations are weaker than vertical ($y$) and transverse ($z$) variations. Additionally, the far-field flow scalings should bring out the anticipated balance of horizontal advection with viscous diffusion of the mean PV.

The scaled spatial variables appropriate to the far field are

(3.19a–c)\begin{equation} \tilde{X}=\varepsilon^{5/2} x, \quad \tilde{Y}=\varepsilon y, \quad Z=\varepsilon z. \end{equation}

It should be noted that $\tilde {X}$ is ‘slower’ than $\tilde {Y}$ and $Z$ in keeping with the observed far-field characteristics of the mean flow (figure 2c), while $Z$ remains as in (3.2a–c) to permit matching with the near field as $\tilde {X} \to 0$. Furthermore, as $\tilde {X}$ is scaled differently from $Z$, it becomes necessary to rescale the transverse ($z$) mean-flow velocity,

(3.20)\begin{equation} \bar{W} \to \varepsilon^{3/2} \bar{W}, \end{equation}

so as to satisfy incompressibility

(3.21)\begin{equation} \frac{1}{\sin \theta }\bar{V}_{\tilde{X}} +\bar{W}_{Z} =0. \end{equation}

In view of the rescaling (3.20), in the far field the $z$-velocity component $\varepsilon ^3 \bar {W}$ is smaller than the $x$-velocity component $\varepsilon ^{3/2} \bar {V}/\sin \theta$, which as noted in § 2.2 is also consistent with our experimental observations. Thus, the amplitude of the mean vertical vorticity to leading order is given by

(3.22)\begin{equation} \varOmega=\frac{\bar{V}_{Z}}{\sin \theta}. \end{equation}

Finally, returning to (3.10), in view of (3.19a–c) and (3.20), the dominant balance of mean PV terms in the far field yields

(3.23)\begin{equation} \left({\frac{\partial }{\partial \tilde{T}}+\frac{\bar{V}}{\sin \theta } \frac{\partial }{\partial \tilde{X}}+\bar{W}\frac{\partial }{\partial Z}-\beta \left({\frac{\partial^{2}}{\partial \tilde{Y}^{2}}+\frac{\partial^{2}}{\partial Z^{2}}} \right)}\right)\varOmega =0, \end{equation}

where

(3.24)\begin{equation} \tilde{T}= \varepsilon^{3/2} \hat{T} = \varepsilon^2 T= \varepsilon^4 t \end{equation}

is the ‘very slow’ time appropriate to the mean-flow dynamics at this late stage. As expected, the evolution equation (3.23) accounts for the horizontal advection and viscous diffusion of $\varOmega$, which are the main controlling factors in the far field.

3.4. Combined mean-flow equation

To investigate quantitatively the three stages of mean-flow dynamics identified above, it would be necessary to solve the evolution equations (3.14), (3.18) and (3.23), subject to appropriate initial and matching conditions. Considering that these problems would have to be tackled numerically, we find it more convenient instead to work with a single mean-flow evolution equation that combines the dominant effects in the three flow regimes. This extended equation reduces asymptotically to the simpler evolution equations (3.14), (3.18) and (3.23), respectively, under the appropriate flow scalings for $T$, $\hat {T}$ and $\tilde {T}=O(1)$.

Specifically, since the mean flow is horizontal to leading order, we introduce a streamfunction $\varPsi (\chi ,\tilde {Y},Z,T)$, where $(\chi ,\tilde {Y},Z)=\varepsilon (x,y,z)$ and $T=\varepsilon ^2 t$, such that the mean-flow velocity components in (3.4a) are given by

(3.25a,b)\begin{equation} \bar{V}= \sin\theta \varPsi_{Z},\quad \bar{W}=-\varPsi_{\chi}, \end{equation}

with $\bar {U}= \cos \theta \varPsi _{Z}$ according to (3.8). Thus,

(3.26)\begin{equation} \frac{1}{\sin \theta }\bar{V}_{\chi} +\bar{W}_{Z} =0, \end{equation}

which ensures that the incompressibility equations (3.9) and (3.21) are satisfied automatically in the corresponding flow regimes. Furthermore, in view of (3.25a,b), the mean vertical vorticity takes the form

(3.27)\begin{equation} \varOmega =\frac{1}{\sin \theta }\bar{V}_{Z} -\bar{W}_{\chi}=\varPsi_{\chi\chi} +\varPsi_{ZZ}, \end{equation}

which reduces asymptotically to expressions (3.13) and (3.22) under the corresponding flow scalings. Finally, the evolution equations (3.14), (3.18) and (3.23) can be combined to the following transport equation for $\varOmega$:

(3.28)\begin{equation} \varOmega_{T} +\varepsilon J(\varOmega,\varPsi) -\varepsilon^2 \beta \left({\frac{\partial^{2}}{\partial \tilde{Y}^{2}}+\frac{\partial^{2}}{\partial Z^{2}}} \right) \varOmega=2\left({\frac{\partial }{\partial T}+\beta } \right)(UU^{\ast })_{Z}, \end{equation}

where $J(a,b)\equiv a_{\chi } b_{Z} -a_{Z} b_{\chi }$.

Setting $\varepsilon =0$ in (3.28) recovers the evolution equation (3.14) derived in Fan et al. (Reference Fan, Kataoka and Akylas2018) that is valid at the early stage of mean-flow generation, $T=O(1)$, when the production of mean PV due to the underlying modulated beam is the dominant effect. At the later times $\hat {T}=O(1)$ and $\tilde {T}=O(1)$, the $O(\varepsilon )$ and $O(\varepsilon ^2)$ terms in (3.28), which account for the effects of advection and viscous diffusion of mean PV, respectively, come into play as discussed above. The extended mean-flow equation (3.28) along with (3.25a,b), (3.27) and the beam evolution equation (3.6) comprise the theoretical model to be used below for quantitative comparison against our experimental observations and numerical Navier–Stokes simulations.