3.2.1 Total energy

Unlike rotating inertial waves, where no potential energy can be defined, equipartition of energy between kinetic and potential subparts can be defined for internal waves. The time variation of the total energy can thus be safely obtained from PIV that gives accurate measurements of the velocity components: $\langle e_{tot}\rangle _{T_{0}}=\int _{{\mathcal{V}}}\,\text{d}V\langle \boldsymbol{u}^{2}\rangle _{T_{0}}/{\mathcal{V}}$ .

The velocity field has been shown to be bidimensional (i.e. only two components invariant in the $y$ -direction) except in the boundary layers, of typical width $d=\sqrt{\unicode[STIX]{x1D708}/\unicode[STIX]{x1D714}_{0}}\sim 1~\text{mm}$ (Brouzet et al. Reference Brouzet, Sibgatullin, Scolan, Ermanyuk and Dauxois2016b ). So their total volume, $2d{\mathcal{A}}+d{\mathcal{L}}W$ (where ${\mathcal{L}}$ is the perimeter of the vertical trapezoidal section of the domain, ${\mathcal{A}}$ its area and $W$ the width of the experimental tank) represents only $1$  % of the total volume of the tank. The measured value of the total energy will thus be computed as $\langle e_{tot}\rangle _{T_{0}}=\int _{{\mathcal{A}}}\,\text{d}x\,\text{d}z\langle \boldsymbol{u}^{2}\rangle _{T_{0}}/{\mathcal{A}}$ , where $\boldsymbol{u}$ is measured in the mid vertical plane only.

Because of some experimental issues near boundaries, the measured area is a bit smaller than the total area of the trapezoidal domain. However since this difference is small we assume that the sampled area is representative enough so that our measure of the average energy density is correct.

3.2.2 Injected power

As the generator is everywhere almost vertical, we can safely approximate (3.8) by

(3.10) $$\begin{eqnarray}p_{inj}\simeq \frac{1}{\unicode[STIX]{x1D70C}_{r}{\mathcal{A}}}\int _{z}Pu\,\text{d}z.\end{eqnarray}$$

The determination of the injected power requires therefore the simultaneous measurement of the horizontal velocity field and the pressure near the wavemaker on the left boundary of the tank ( $x=0$ ). This kind of generator forces a purely propagating mode-1 with an amplitude $u_{0}$ slightly lower than $a_{0}\unicode[STIX]{x1D714}_{0}$ depending on the efficiency of the wavemaker (see Mercier et al. Reference Mercier, Martinand, Mathur, Gostiaux, Peacock and Dauxois2010),

(3.11) $$\begin{eqnarray}\displaystyle u^{M1}(x,z,t)=u_{0}\cos (mz)\cos (\unicode[STIX]{x1D714}_{0}t-\ell x). & & \displaystyle\end{eqnarray}$$

Using the incompressibility equation and assuming that the flow is two-dimensional, we can straightforwardly get the vertical component,

(3.12) $$\begin{eqnarray}\displaystyle w^{M1}(x,z,t)=-u_{0}\frac{\ell }{m}\sin (mz)\sin (\unicode[STIX]{x1D714}_{0}t-\ell x), & & \displaystyle\end{eqnarray}$$

induced in the fluid by the generator. The associated pressure field can be deduced by projection of the linear inviscid version of (3.1) on the $x$ -axis, followed by a straightforward integration

(3.13) $$\begin{eqnarray}P^{M1}(x,z,t)=\unicode[STIX]{x1D70C}_{r}u_{0}\frac{\unicode[STIX]{x1D714}_{0}}{\ell }\cos (mz)\cos (\unicode[STIX]{x1D714}_{0}t-\ell x),\end{eqnarray}$$

in which the constant of integration is set to zero since $P$ accounts only for the pressure variation with respect to the hydrostatic equilibrium pressure. The time-averaged theoretically injected power can then be computed as

(3.14) $$\begin{eqnarray}\langle p_{inj}^{M1}\rangle _{T_{0}}=u_{0}^{2}N\frac{H^{2}}{4\unicode[STIX]{x03C0}}\sqrt{1-\left(\frac{\unicode[STIX]{x1D714}_{0}}{N}\right)^{2}}\frac{1}{{\mathcal{A}}}.\end{eqnarray}$$

This theoretical development is valid for a pure propagating mode-1. However, such approximation may be questioned in the presence of the slope, since the size of the closed domain has been chosen to get strong reflected beams, which are required for the formation of the attractor. Some feedback on the forced wave field from its reflection on the slope is then expected.

To go beyond, it is possible to use the PIV measurements to compute both the pressure and horizontal velocity appearing in (3.10). Indeed, if the PIV technique gives directly the horizontal velocity field, the pressure field can be obtained from $u$ and $w$ as follows. Since the width of the attractor $\unicode[STIX]{x1D706}$ , which is the typical length of variation of the velocity field in the transverse direction of the branch, is much larger than $d$ , we use the linear and inviscid approximation of (3.1) and (3.2) to get the experimental pressure gradient, which is then integrated using the Matlab^© code intgrad2, available at www.mathworks.com/matlabcentral/fileexchange/. The unknown constant of integration necessary for this last step is not important since mass conservation implies $\int _{{\mathcal{S}}_{g}}u\,\text{d}y\,\text{d}z=0$ , leading to a vanishing contribution to the injected power. Thus we arbitrarily choose $\langle P\rangle _{{\mathcal{A}}}=0$ .

An example of the computed pressure field is presented in figure 3(a). The pressure field is different from the pure mode-1 pattern (3.13): this will have some consequences on the estimation of the injected power. Because of the integration, the pressure field emphasizes the large scale motion contrary to the density gradient or velocity fields that show the attractor (see for example figure 2).

Figure 3.

(a) Pressure field $P$ , computed via an integration of the experimental velocity fields $u$ and $w$ . The unimportant constant of integration is arbitrarily chosen to have a zero average pressure field. The pattern is computed approximately $50T_{0}$ after the generator was started. (b) Evolution in time of the injected power computed at $x_{0}=5~\text{cm}$ . The oscillating black curve corresponds to the instantaneous measurement, while the red dashed curve is the associated local mean. The generator is switched on at $t=0$ while it is turned off at $t=60T_{0}$ . The thick blue line corresponds to the theoretical injected power (3.14) expected for a pure mode-1, i.e. neglecting the reflection on the slope. Experimental parameters are those given in the caption of figure 2.

Figure 3(b) shows the time evolution of the injected power $p_{inj}$ (solid black line) for a typical experiment in the linear regime. For visualization purposes, this flux of energy is measured at $x_{0}\simeq 5~\text{cm}$ , as will be discussed in § 4.1; $p_{inj}$ is oscillating with a mean value (dashed line) that first increases until reaching its stationary value. As soon as the generator is turned off, it decreases towards zero. We note that the measured value is above $\langle p_{inj}^{M1}\rangle _{T_{0}}$ (thick blue line), i.e. the theoretical injected power (3.14) expected for a pure mode-1 neglecting the finite size of the domain due to the reflection on the slope.

3.2.3 Disentangling the different contributions for the dissipative terms

As already mentioned, the two-dimensional PIV measurements give access to the velocity field in the middle plane $y=0$ and can be extended to the volume except in the boundary layers. In the remainder of the paper, this is named the bulk part. Since the thickness of the boundary layers is very small, the velocity gradient may be strong in these regions and the associated dissipative term cannot be neglected. It is therefore convenient to distinguish several different contributions as follows:

(3.15) $$\begin{eqnarray}\displaystyle p_{diss} & = & \displaystyle \frac{1}{{\mathcal{V}}}\int _{{\mathcal{V}}}\,\text{d}V\unicode[STIX]{x1D708}|\unicode[STIX]{x1D735}\boldsymbol{u}|^{2}\nonumber\\ \displaystyle & = & \displaystyle \underbrace{\frac{1}{{\mathcal{V}}}\int _{{\mathcal{V}}_{bulk}}\,\text{d}V\underbrace{\unicode[STIX]{x1D708}|\unicode[STIX]{x1D735}\boldsymbol{u}|^{2}}_{\unicode[STIX]{x1D716}_{bulk}}}_{p_{bulk}}+\underbrace{\frac{1}{{\mathcal{V}}}\int _{{\mathcal{V}}_{BL,\bot }}\,\text{d}V\underbrace{\unicode[STIX]{x1D708}|\unicode[STIX]{x1D735}\boldsymbol{u}|^{2}}_{\unicode[STIX]{x1D716}_{\bot }}}_{p_{\bot }}+\underbrace{\frac{1}{{\mathcal{V}}}\int _{{\mathcal{V}}_{BL,\Vert }}\,\text{d}V\underbrace{\unicode[STIX]{x1D708}|\unicode[STIX]{x1D735}\boldsymbol{u}|^{2}}_{\unicode[STIX]{x1D716}_{\Vert }}}_{p_{\Vert }}.\qquad\end{eqnarray}$$

This definition disentangles the bulk dissipation (denoted with the index bulk and that we can expect to measure accurately), from the contributions of the boundary layers close to the walls. Among the last ones, we distinguish the dissipation in the boundary layers along the longitudinal walls (parallel to the measurement plane and therefore perpendicular to $y$ , denoted with the index $\Vert$ ), from the dissipation in the boundary layers perpendicular to the attractor plane (denoted with the index $\bot$ ). Let us discuss these three different terms successively.

In the bulk, the transverse velocity $v$ is equal to zero and the velocity components $u$ and $w$ do not depend on $y$ . The contribution of the bulk can thus be computed rather straightforwardly from the measurements of the velocity field shown in figure 2 using the formula

(3.16) $$\begin{eqnarray}\unicode[STIX]{x1D716}_{bulk}=\unicode[STIX]{x1D708}|\unicode[STIX]{x1D735}\boldsymbol{u}|^{2}=\unicode[STIX]{x1D708}(\unicode[STIX]{x2202}_{x}u)^{2}+\unicode[STIX]{x1D708}(\unicode[STIX]{x2202}_{z}u)^{2}+\unicode[STIX]{x1D708}(\unicode[STIX]{x2202}_{x}w)^{2}+\unicode[STIX]{x1D708}(\unicode[STIX]{x2202}_{z}w)^{2}.\end{eqnarray}$$

Figure 4(a) presents the spatial dependence of this quantity in the mid-plane averaged over 10 $T_{0}$ . As expected, its amplitude is particularly important along the first branch of the attractor, that is located just after the reflection on the slope where the focusing takes place, and then decreases along the attractor. The mean of this quantity on the whole domain leads to $p_{bulk}$ , the energy dissipation rate due to the bulk.

It is much more difficult to directly estimate the dissipation in the boundary layers. Indeed, since their thicknesses are very small compared to the size of the tank (approximately $1$  %), it is very hard to measure the velocity in the boundary layers as accurately as in the bulk. A method to evaluate the velocity using the velocity measured in the mid-plane ( $y=0$ ) will then be presented for the two types of boundary layer, parallel to the lateral walls (§ 3.2.4) and perpendicular to the attractor plane (§ 3.2.5). In both cases, theoretical developments recently reported by Beckebanze et al. (Reference Beckebanze, Brouzet, Sibgatullin and Maas2018) will be used and adapted. For the sake of brevity and clarity, only the main ingredients of their results will be recalled. Notice that linear equations will be used in the two following sections as the Reynolds number is small in viscous boundary layers. Brouzet et al. (Reference Brouzet, Sibgatullin, Scolan, Ermanyuk and Dauxois2016b ) showed that the experimental velocity field in the parallel boundary layers is compatible with the theoretical description.

Figure 4.

(a) Dissipation rate field within the bulk of the fluid, $\langle \unicode[STIX]{x1D716}_{bulk}(x,z)\rangle _{T_{0}}$ , measured in the mid-plane. (b) Dissipation rate field in the longitudinal boundary layers, $\langle \unicode[STIX]{x1D716}_{\Vert }\rangle _{T_{0}}$ , averaged along the transverse direction. (a,b) The field has been averaged over 10 forcing periods $T_{0}$ around $t=50T_{0}$ and the theoretically expected attractor for this geometry is depicted with the dashed line. Experimental parameters are those given in the caption of figure 2.

3.2.4 The dissipation in the longitudinal boundary layers $\unicode[STIX]{x1D716}_{\Vert }$

Close to both longitudinal vertical walls, the largest contribution to the viscous term $\unicode[STIX]{x1D708}\unicode[STIX]{x1D6FB}^{2}\boldsymbol{u}$ comes from the gradients in the $y$ -direction. Indeed, experimental measurements show that variations in the $xz$ -plane are typically on the centimetre length scale, that is significantly larger than the millimetre length scale that appears in the viscous boundary layer. The evolution of the velocity (3.1) and buoyancy fields (3.2) can thus be simplified as

(3.17) $$\begin{eqnarray}\displaystyle & \displaystyle \unicode[STIX]{x2202}_{t}u=-\frac{\unicode[STIX]{x2202}_{x}P}{\unicode[STIX]{x1D70C}_{r}}+\unicode[STIX]{x1D708}\unicode[STIX]{x2202}_{yy}u, & \displaystyle\end{eqnarray}$$

(3.18) $$\begin{eqnarray}\displaystyle & \displaystyle \unicode[STIX]{x2202}_{t}w=-\frac{\unicode[STIX]{x2202}_{z}P}{\unicode[STIX]{x1D70C}_{r}}+b+\unicode[STIX]{x1D708}\unicode[STIX]{x2202}_{yy}w, & \displaystyle\end{eqnarray}$$

(3.19) $$\begin{eqnarray}\displaystyle & \displaystyle \unicode[STIX]{x2202}_{t}b=-wN^{2}. & \displaystyle\end{eqnarray}$$

The transverse velocity $v$ does not vanish any more but its contribution to the dissipation rate remains very small (Beckebanze et al. Reference Beckebanze, Brouzet, Sibgatullin and Maas2018). Let us consider a monochromatic internal wave $a=\text{Re}[\text{}\underline{a}\,\text{e}^{-\text{i}\unicode[STIX]{x1D714}_{0}t}]$ at the frequency $\unicode[STIX]{x1D714}_{0}$ in which $a$ stands for $u$ , $w$ , $b$ or $P$ . The case of polychromatic signals will be discussed in § 4.2. Introducing the angle $\unicode[STIX]{x1D703}$ defined through the dispersion relation $\unicode[STIX]{x1D714}_{0}^{2}=N^{2}\sin ^{2}\unicode[STIX]{x1D703}$ , (3.19) can be rewritten as $\text{}\underline{b}=-\text{i}\unicode[STIX]{x1D714}_{0}\text{}\underline{w}/\text{sin}^{2}\unicode[STIX]{x1D703}$ . Equations (3.17) and (3.18) can then be simplified as

(3.20) $$\begin{eqnarray}\displaystyle & \displaystyle -\text{i}\unicode[STIX]{x1D714}_{0}\text{}\underline{u}=-\unicode[STIX]{x2202}_{x}\text{}\underline{P}/\unicode[STIX]{x1D70C}_{r}+\unicode[STIX]{x1D708}\unicode[STIX]{x2202}_{yy}\text{}\underline{u}, & \displaystyle\end{eqnarray}$$

(3.21) $$\begin{eqnarray}\displaystyle & \displaystyle \text{i}\unicode[STIX]{x1D714}_{0}\cot ^{2}\unicode[STIX]{x1D703}\text{}\underline{w}=-\unicode[STIX]{x2202}_{z}\text{}\underline{P}/\unicode[STIX]{x1D70C}_{r}+\unicode[STIX]{x1D708}\unicode[STIX]{x2202}_{yy}\text{}\underline{w}, & \displaystyle\end{eqnarray}$$

that have to be solved with the no-slip boundary conditions $\text{}\underline{u}(y=\pm W/2)=\text{}\underline{w}(y=\pm W/2)=0$ , while $\text{}\underline{u}(y=0)$ and $\text{}\underline{w}(y=0)$ can be measured at the centre of the tank with the PIV measurements described above. One finally gets

(3.22) $$\begin{eqnarray}\displaystyle & \displaystyle \text{}\underline{u}(y)=\left(1-\frac{\cosh [\text{i}^{1/2}y/d]}{\cosh [\text{i}^{1/2}W/(2d)]}\right)\text{}\underline{u}(y=0), & \displaystyle\end{eqnarray}$$

(3.23) $$\begin{eqnarray}\displaystyle & \displaystyle \text{}\underline{w}(y)=\left(1-\frac{\cosh [\text{i}^{-1/2}\cot \unicode[STIX]{x1D703}y/d]}{\cosh [\text{i}^{-1/2}\cot \unicode[STIX]{x1D703}W/(2d)]}\right)\text{}\underline{w}(y=0), & \displaystyle\end{eqnarray}$$

in which we use $d=\sqrt{\unicode[STIX]{x1D708}/\unicode[STIX]{x1D714}_{0}}$ , the thickness of the boundary layers.

The dissipation within the longitudinal boundary layers is thus obtained by performing successively the following operations: a time Fourier transform of the wave field in the mid-plane $\boldsymbol{u}(x,y=0,z,t)$ , a band-pass filtering in the Fourier space around the forcing frequency $\unicode[STIX]{x1D714}_{0}$ , multiplication by the theoretically derived expressions (3.22) and (3.23) with respect to the $y$ -coordinate to get $\unicode[STIX]{x2202}_{y}\text{}\underline{\boldsymbol{u}}(x,y,z)$ . The real part of the inverse Fourier transform is then taken to generate the signal $\unicode[STIX]{x2202}_{y}\boldsymbol{u}(x,y,z)$ . The value of $\unicode[STIX]{x1D716}_{\Vert }=\unicode[STIX]{x1D708}[(\unicode[STIX]{x2202}_{y}u)^{2}+(\unicode[STIX]{x2202}_{y}w)^{2}]$ at any point of the domain is then computed and the integration over the whole domain leads to the value of the energy dissipation rate $p_{\Vert }=\int \unicode[STIX]{x1D716}_{\Vert }\,\text{d}V/{\mathcal{V}}$ due to the parallel boundary layers. The $y$ -averaged value $\langle \unicode[STIX]{x1D716}_{\Vert }\rangle _{y,T_{0}}=(1/W)\int _{-W/2}^{+W/2}\,\text{d}y\langle \unicode[STIX]{x1D716}_{\Vert }\rangle _{T_{0}}$ is displayed in figure 4(b) for all points of the domain. Here again, the dissipation is larger along the first branch of the attractor and then decreases along the skeleton of the attractor. By comparing figure 4(a,b), we realize that this boundary layer dissipation is much larger than the bulk dissipation. This difference is a factor of $10$ when averaged over the entire domain. Indeed the $y$ -averaged dissipation of the bulk is of the order of $\unicode[STIX]{x1D708}~(U/\unicode[STIX]{x1D706})^{2}$ , while the $y$ -averaged longitudinal boundary layer dissipation is of the order $\unicode[STIX]{x1D708}~(2d/W)~(U/d)^{2}$ . We can then estimate the ratio of the two types of dissipation $\langle \unicode[STIX]{x1D716}_{bulk}\rangle _{y,T_{0}}/\langle \unicode[STIX]{x1D716}_{\Vert }\rangle _{y,T_{0}}\sim Wd/\unicode[STIX]{x1D706}^{2}\sim 10\,\%$ , a value in good agreement with the measured ratio.

3.2.5 The dissipation in the transversal boundary layers $\unicode[STIX]{x1D716}_{\bot }$

The second dissipation term in the boundary layers, $\unicode[STIX]{x1D716}_{\bot }$ , corresponds to the boundary layers perpendicular to the attractor plane, and, more precisely, where a reflection of the internal wave beam occurs. For the present experimental set-up, three walls have to be considered while, due to the free-slip boundary condition, the reflection at the top surface is not a source of dissipation. Note that no linear coupling between internal and surface waves can occur due to the low frequency which leads to a surface wavelength much larger than the size of the set-up. We have therefore $p_{\bot }=p_{\bot }^{[1]}+p_{\bot }^{[3]}+p_{\bot }^{[4]}$ where $j=1$ stands for the boundary layer along the slope, $j=3$ close to the generator and $j=4$ near the bottom of the tank. We will develop the method to estimate the dissipation term associated with a reflection on a wall inclined at an arbitrarily angle $\unicode[STIX]{x1D719}$ , that will stand for $\unicode[STIX]{x1D6FC}$ , $\unicode[STIX]{x03C0}$ or $-\unicode[STIX]{x03C0}/2$ respectively for $j=1,3,4$ .

Considering the reflection on a boundary in the inviscid case, the velocity field corresponds to the sum of the incident beam $\boldsymbol{u}^{[I]}$ and the inviscid reflected beam $\boldsymbol{u}^{[R]}$ . It has to fulfil the no-penetration condition and is therefore parallel to the boundary along the wall (free-slip condition). As depicted in figure 5, we define in the real case the viscous contribution $\boldsymbol{u}_{\unicode[STIX]{x1D708}}=u_{\unicode[STIX]{x1D708}}\boldsymbol{e}_{x}+w_{\unicode[STIX]{x1D708}}\boldsymbol{e}_{z}$ ensuring that the velocity field vanishes along the wall (no-slip condition), yielding the complete velocity field $\boldsymbol{u}=\boldsymbol{u}^{[I]}+\boldsymbol{u}^{[R]}+\boldsymbol{u}_{\unicode[STIX]{x1D708}}$ . Thus $\boldsymbol{u}_{\unicode[STIX]{x1D708}}$ , that has to compensate the inviscid free-slip velocity field, is mainly on the $\boldsymbol{e}_{x^{\prime }}$ direction and may therefore be written as $\boldsymbol{u}_{\unicode[STIX]{x1D708}}=u_{\unicode[STIX]{x1D708}}^{\prime }\boldsymbol{e}_{x^{\prime }}$ in the tilted coordinate system $(x^{\prime },z^{\prime })$ shown in figure 5, with $\lim _{z^{\prime }\rightarrow -\infty }u_{\unicode[STIX]{x1D708}}^{\prime }=0$ . As shown in Beckebanze et al. (Reference Beckebanze, Brouzet, Sibgatullin and Maas2018), this is valid at leading order in $d/\unicode[STIX]{x1D706}$ – equal to $1\,\%$ in our experiment.

Figure 5.

Schematic of a reflection: the incident beam with velocity $\boldsymbol{u}^{[I]}$ is bounced on a boundary inclined by an angle $\unicode[STIX]{x1D719}$ with respect to the vertical, leading to the reflected beam with velocity $\boldsymbol{u}^{[R]}$ . The no-slip condition induces a viscous velocity field $\boldsymbol{u}_{\unicode[STIX]{x1D708}}$ that counter-balances at the boundary the inviscid velocity field $\boldsymbol{u}^{[I]}+\boldsymbol{u}^{[R]}$ . Its exponential decrease as a function of the distance from the boundary is schematically represented. Note the introduction of the wall-oriented coordinate system in which $x^{\prime }$ is the distance along the wall and $z^{\prime }$ is the distance normal to the wall.

In a boundary layer, the dominant part of the viscous term is given by the direction perpendicular to the wall $(z^{\prime })$ , and we have $\unicode[STIX]{x1D6FB}^{2}\boldsymbol{u}\approx \unicode[STIX]{x1D6FB}^{2}\boldsymbol{u}_{\unicode[STIX]{x1D708}}\approx \unicode[STIX]{x1D708}\unicode[STIX]{x2202}_{z^{\prime }z^{\prime }}\boldsymbol{u}_{\unicode[STIX]{x1D708}}$ , as $d\ll \unicode[STIX]{x1D706}$ . The associated dissipation is thus $\langle p_{\bot }^{[j]}\rangle _{T_{0}}=\langle \int _{j}\,\text{d}x^{\prime }\,\text{d}z^{\prime }[\unicode[STIX]{x2202}_{z^{\prime }}u_{\unicode[STIX]{x1D708}}^{\prime }(x^{\prime })]^{2}\rangle _{T_{0}}/{\mathcal{A}}$ . The viscous velocity, pressure and buoyancy field $(u_{\unicode[STIX]{x1D708}}^{\prime },p_{\unicode[STIX]{x1D708}},b_{\unicode[STIX]{x1D708}})$ are solutions of the system

(3.24) $$\begin{eqnarray}\displaystyle & \displaystyle \unicode[STIX]{x2202}_{t}u_{\unicode[STIX]{x1D708}}^{\prime }=-\unicode[STIX]{x2202}_{x^{\prime }}p_{\unicode[STIX]{x1D708}}-b_{\unicode[STIX]{x1D708}}\cos \unicode[STIX]{x1D719}+\unicode[STIX]{x1D708}\unicode[STIX]{x2202}_{z^{\prime }z^{\prime }}u_{\unicode[STIX]{x1D708}}^{\prime }, & \displaystyle\end{eqnarray}$$

(3.25) $$\begin{eqnarray}\displaystyle & \displaystyle 0=-\unicode[STIX]{x2202}_{z^{\prime }}p_{\unicode[STIX]{x1D708}}+b_{\unicode[STIX]{x1D708}}\sin \unicode[STIX]{x1D719}, & \displaystyle\end{eqnarray}$$

(3.26) $$\begin{eqnarray}\displaystyle & \displaystyle \unicode[STIX]{x2202}_{t}b_{\unicode[STIX]{x1D708}}=N^{2}u_{\unicode[STIX]{x1D708}}^{\prime }\cos \unicode[STIX]{x1D719}, & \displaystyle\end{eqnarray}$$

where we used $w_{\unicode[STIX]{x1D708}}=-u_{\unicode[STIX]{x1D708}}^{\prime }\cos \unicode[STIX]{x1D719}$ (since $\boldsymbol{u}_{\unicode[STIX]{x1D708}}$ is parallel to $\boldsymbol{e}_{x^{\prime }}$ ). Equation (3.25) leads to $p_{\unicode[STIX]{x1D708}}\sim d\,b_{\unicode[STIX]{x1D708}}$ , thus the term $\unicode[STIX]{x2202}_{x^{\prime }}p_{\unicode[STIX]{x1D708}}$ in (3.24) can be neglected at zeroth order in $d/\unicode[STIX]{x1D706}$ .

As in the previous subsection, taking the ansatz $a=\text{Re}[\text{}\underline{a}\,\text{e}^{-\text{i}\unicode[STIX]{x1D714}_{0}t}]$ , (3.26) leads to $\text{}\underline{b}_{\unicode[STIX]{x1D708}}=\text{i}\unicode[STIX]{x1D714}_{0}\text{}\underline{u}_{\unicode[STIX]{x1D708}}^{\prime }\cos \unicode[STIX]{x1D719}/\text{sin}^{2}\unicode[STIX]{x1D703}$ . Using (3.24), we finally get

(3.27) $$\begin{eqnarray}\displaystyle \text{i}\text{}\underline{u}_{\unicode[STIX]{x1D708}}^{\prime }=(d/\unicode[STIX]{x1D707})^{2}\unicode[STIX]{x2202}_{z^{\prime }z^{\prime }}\text{}\underline{u}_{\unicode[STIX]{x1D708}}^{\prime }, & & \displaystyle\end{eqnarray}$$

by introducing the geometrical factor $\unicode[STIX]{x1D707}=(\cos ^{2}\unicode[STIX]{x1D719}/\sin ^{2}\unicode[STIX]{x1D703}-1)^{1/2}$ and $d$ the thickness of the boundary layer which is again equal to $\sqrt{\unicode[STIX]{x1D708}/\unicode[STIX]{x1D714}_{0}}$ . As this equation has to be solved with the boundary conditions $\lim _{z^{\prime }\rightarrow -\infty }\text{}\underline{u}_{\unicode[STIX]{x1D708}}^{\prime }=0$ and $\text{}\underline{u}_{\unicode[STIX]{x1D708}}^{\prime }(x^{\prime },z^{\prime }=0)=-\text{}\underline{u}_{0}(x^{\prime })$ , where $u_{0}^{\prime }(x^{\prime })$ is the inviscid along-wall velocity field, the solution reads

(3.28) $$\begin{eqnarray}\text{}\underline{u}_{\unicode[STIX]{x1D708}}^{\prime }(x^{\prime },z^{\prime })=\left\{\begin{array}{@{}ll@{}}\displaystyle -\text{}\underline{u}_{0}^{\prime }(x^{\prime })\exp \left(\frac{1+\text{i}}{\sqrt{2}}\unicode[STIX]{x1D707}_{j}z^{\prime }/d\right),\quad & \text{for }\unicode[STIX]{x1D719}=\unicode[STIX]{x1D6FC},0\;(j=1,3),\\ \displaystyle -\text{}\underline{u}_{0}^{\prime }(x^{\prime })\exp \left(\frac{1-\text{i}}{\sqrt{2}}z^{\prime }/d\right),\quad & \text{for }\unicode[STIX]{x1D719}=\unicode[STIX]{x03C0}/2\;(j=4).\end{array}\right.\end{eqnarray}$$

After some calculations, we get

(3.29) $$\begin{eqnarray}\langle p_{\bot }^{[j]}\rangle _{T_{0}}=\frac{\unicode[STIX]{x1D708}\sqrt{2}|\unicode[STIX]{x1D707}_{j}|}{4{\mathcal{A}}d}\int \text{d}x^{\prime }|\text{}\underline{u}_{0}^{\prime }(x^{\prime })|^{2}.\end{eqnarray}$$

The amplitude of the inviscid along-wall velocity field $u_{0}^{\prime }(x^{\prime })$ is required to proceed and compute the integral (3.29). As $u_{\unicode[STIX]{x1D708}}^{\prime }$ vanishes outside from the boundary layer, the PIV measurements give a good estimate of the inviscid velocity field. Furthermore, rather than measuring $u_{0}^{\prime }(x^{\prime })$ directly on the PIV field by zooming on the region as close as possible to the wall, we use the relation $\boldsymbol{u}=\boldsymbol{u}^{[I]}+\boldsymbol{u}^{[R]}$ , where $\boldsymbol{u}^{[I]}$ and $\boldsymbol{u}^{[R]}$ do not depend strongly on $z^{\prime }$ along the rays which allows us to measure them where we want in the region next to the wall. To get $\boldsymbol{u}^{[I]}$ and $\boldsymbol{u}^{[R]}$ , we use the Hilbert transform (Mercier, Garnier & Dauxois Reference Mercier, Garnier and Dauxois2008) that allows us to distinguish the different branches of the attractor by restricting ourselves to internal waves propagating in a given direction. Because of experimental errors, $\boldsymbol{u}^{[I]}+\boldsymbol{u}^{[R]}$ is not perfectly parallel to the wall but this is solved by computing its projection

(3.30) $$\begin{eqnarray}u_{0}^{\prime }=(\boldsymbol{u}^{[I]}+\boldsymbol{u}^{[R]})\boldsymbol{\cdot }\boldsymbol{e}_{x^{\prime }}=(u^{[I]}+u^{[R]})\sin \unicode[STIX]{x1D719}-(w^{[I]}+w^{[R]})\cos \unicode[STIX]{x1D719}.\end{eqnarray}$$

For visualization purposes, $\boldsymbol{u}^{[I]}$ and $\boldsymbol{u}^{[R]}$ are experimentally evaluated at a small but finite distance from the boundary, $z^{\prime }=100~\text{mm}$ . Because of this, we have to shift both functions so that their maxima coincide as is the case on the wall $z^{\prime }=0$ . Figure 6 shows an example of the incident and reflected beams, in the case of the bottom reflection. From this profile, we can compute $\langle p_{\bot }\rangle _{T_{0}}=\langle p_{\bot }^{[1]}\rangle _{T_{0}}+\langle p_{\bot }^{[3]}\rangle _{T_{0}}+\langle p_{\bot }^{[4]}\rangle _{T_{0}}$ using (3.29). As we will see in next section, this sum is of the same order of magnitude of the dissipation in the bulk and is ten times smaller than the dissipation taking place in the parallel boundary layers. This result is expected since the volume of the parallel boundary layers is much larger due to the small width of the tank. Varying $W$ clearly changes the importance of the parallel boundary layers compared to the perpendicular ones, as will be shown in § 4.3.

We will now apply these methods to measure the different terms in the energy budget for the attractors in several regimes.

Figure 6.

Incident (a,c) and reflected (b,d) velocity amplitudes (a,b) along the direction of the bottom boundary, obtained at $t=50T_{0}$ thanks to Hilbert filtering. Panels (c,d) correspond to the horizontal cut at the altitude $z=100~\text{mm}$ shown in (a,b) by the red horizontal lines. Experimental parameters are those given in the caption of figure 2.