# Computation of Density Perturbation and Energy Flux of Internal Waves from Experimental Data

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## Abstract

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## 1. Introduction

## 2. Methods

#### 2.1. The Mxzt Method

#### 2.2. The MxtUp Method

#### 2.3. Vertical Energy Flux

_{x}is the number of periods in the x-interval.

## 3. Results

#### 3.1. Evaluation of the Methods on Analytic Fields

#### 3.1.1. Analytic IW Fields

_{ideal}), periodic in time (t) and space (x, z) and containing only waves with upward group velocity, i.e., downward phase velocity:

_{damped}), closer to experimental observations. The amplitude of the internal waves in laboratory experiments decreases upwards, away from the generation source. To represent this vertical damping analytically, we multiply the Brunt–Väisälä frequency anomaly in (26) by a decreasing exponential

_{damped}field is therefore not periodic in z, in contrast to the IW

_{ideal}field.

_{ideal}field in (26) an IW field containing only waves with downward group velocity (referred to as IW

_{down}):

^{−1}, $\text{}{n}_{z}=2$; three horizontal wavelengths: ${k}_{0}=21.42{\text{}\mathrm{m}}^{-1},\text{}{n}_{x}=3$; 10 time periods: ${\mathsf{\omega}}_{0}=0.44\text{}\mathrm{rad}.{\mathrm{s}}^{-1},\text{}{n}_{t}=10$ as in the experiment of Section 3.2). The density stratification is linear, so that the Brunt–Väisälä frequency N is constant (N = 0.85 s

^{−1}) and larger than the angular frequency ${\omega}_{0}$ of the internal wave. The amplitude A is set to 0.02 s

^{−2}, and the vertical damping coefficient a is set to 0.62 ${\mathrm{m}}^{-1}$. The vertical damping coefficient a is chosen to correspond to the mean vertical decrease observed in the experiment (τ = −20%, see below in Section 3.2.2).

#### 3.1.2. Validation and Comparison of Reconstructed Fields

_{damped}signal. The non-periodic signal is seen as a periodic discontinuous signal by the FFT, and the discontinuity distorts the spectrum by adding false overtones that are aliased into the spectrum. The figure also shows the spectrum of $\delta {N}_{mix}^{2}$ (dashed-dotted lines with circle markers). The real part has an additional peak at $-{m}_{0}$, which is the signature of the downward propagating beam (IW

_{down}).

_{ideal}), the nrms difference between δρ from the Mxzt method and the analytical δρ is $3\times {10}^{-13}\%$. With the MxtUp method, the nrms difference for δρ is $4\times {10}^{-13}\%$. As expected, therefore, in the ideal case, both methods give very small errors. The nrms difference between the reconstructed δN

^{2}and the analytical δN

^{2}is the same for both methods: $nrmse\left(\delta {N}_{Mxzt,MxtUp}^{2}\right)=0.1\%$. This error comes from differentiating $\delta \rho $ along z by finite differences when using (7) to calculate δN

^{2}. The same error is obtained when differentiating $\delta {\rho}_{ideal}$ numerically and comparing to $\delta {N}_{ideal}^{2}$.

_{damped}, $nrmse\left(\delta {\rho}_{Mxzt}\right)$ with the Mxzt method is $6.7\%$, whereas with the MxtUp method $nrmse\left(\delta {\rho}_{MxtUp}\right)$ is less than 2%. As expected, therefore, the non-periodicity of the signal degrades the accuracy of the Mxzt method. The MxtUp method, which doesn’t rely on the z-periodicity hypothesis, is the most accurate in this case. However, the MxtUp method still gives a larger error than in the ideal case ($1.8\%$ instead of ${10}^{-13}\%$). The larger the coefficient $a$, the larger is the error of the resulting density field. For example, for a coefficient a of 1.7 ${\mathrm{m}}^{-1}$, the error of the resulting density field is increased to 4.8%. One reason for this error may be that the ideal polarization relations for an inviscid fluid are used to reconstruct the density field. It would be possible, but more complicated, to use instead the viscous polarization relations, both when constructing the analytical solution and when reconstructing the density field. However, it is not clear that the damping in the experiment is in fact mainly caused by viscosity, and this error is in any case small compared to other error sources in the experiment.

_{ideal}analytical fields (not IW

_{mix}). The reference density anomaly field is thus $\delta {\rho}_{ideal}$. When using the Mxzt method, therefore, we first filter away (in Fourier space) all downward propagating wave components (with upward phase velocity) from the IW

_{mix}field ($\widehat{\tilde{\tilde{\delta {N}^{2}}}}(\omega ,k,m>0$) = 0), in order to reconstruct only the upward propagating wave’s field. The same filtering will be done below, when calculating the vertical energy flux from the experimental data in Section 3.2.2. This filter cannot be applied when using the MxtUp method, since it does not involve Fourier transformation in z.

_{mix}, $nrmse\left(\delta {\rho}_{Mxzt}\right)$ with the Mxzt method is $3\times {10}^{-13}\%$, wheras with the MxtUp method, $nrmse\left(\delta {\rho}_{MxtUp}\right)$ is around 20%. Thus, as expected, the presence of downward propagating waves in the original signal degrades the accuracy of the MxtUp method. The error in the density field obtained with MxtUp is of the same magnitude as B/A (20%), the ratio of the amplitude of the downward propagating waves to the amplitude of the upward propagating waves. The Mxzt method is thus the most accurate in this case.

_{down}, for example, by multiplying it by a cosine function of x, the error induced by MxtUp on the reconstructed fields is no longer homogeneous along x. The rmse of the $\delta {\rho}_{MxtUp}$ and $\delta {N}_{MxtUp}^{2}$ fields are maximal where the amplitude of the downward propagating waves is maximal. This change has no impact on the resulting Mxzt fields, as the downward propagation is filtered away, and with it the heterogeneity along x.

_{damped}but in the opposite direction), boundary errors appear on the resulting Mxzt field. The associated nrmse is therefore slightly increased: $nrmse\left(\delta {N}_{Mxzt}^{2}\right)=$0.4% and $nrmse\left(\delta {\rho}_{Mxzt}\right)=$0.6% for a = 0.62 ${\mathrm{m}}^{-1}$.

#### 3.1.3. Vertical Energy Flux

_{x}is the number of periods in the x-interval. The numerical energy flux is calculated by using the reconstructed fields w and $\delta p$ in (25). The reconstructed fields w and $\delta p$ are obtained from either (13) or (15) (for the Mxzt method), or (18) and (20) (for the MxtUp method).

_{ideal}, plain curves) and the damped IW field (IW

_{damped}, dashed curves). The value given by (32) is shown by the red line. For IW

_{ideal}, the nrms difference between the analytical value and the energy flux from both the Mxzt and MxtUp method is $0.26\%$. This is of the same magnitude as the error caused by numerical differentiation. The results from both methods agree: the green and black plain lines are indistinguishable on Figure 5a, and the nrms difference between the energy flux from Mxzt and MxtUp is less than ${10}^{-13}\%$.

_{damped}, the profiles of the vertical energy flux are very different with the two methods. With Mxzt (dashed black curve) the profile presents Gibb’s oscillations related to the non-periodicity of the IW field in z. The assumption that the fields are periodic in z forces the energy flux to be periodic, which means that it is overestimated at the upper boundary and underestimated at the lower boundary. The method MxtUp, on the other hand, accurately captures the vertical damping of the radiated energy away from the generation source.

_{mix}. The analytic value obtained from (32) is shown separately for the upward and downward components (red lines). The values obtained with Mxzt are shown by the black lines. The nrms difference between the analytical value of the upward energy flux and the upward energy flux from the Mxzt method (plain black line) is $0.26\%$. The Mxzt method also gives a very accurate solution for the downward energy flux (dashed black line), the nrms difference to the analytical value being only $0.26\%$. The dotted line shows the flux obtained with the Mxzt method if the modes with $m<0$ (downward group velocity) are not filtered away from the Fourier spectrum of the mixed field. It is equal to the difference between the solutions for the upward and downward energy flux.

_{mix}. The number of oscillations is proportional to the number of vertical wavelengths inside the domain. The nrms difference between the analytical profile of the upward vertical energy flux (red plain line) and the upward energy flux from the MxtUp method is 28%, which is in the same range as B/A (20%). Averaging over z, and comparing the mean of the MxtUp EF to the analytical value of the upward EF, the nrmse falls to 3.73%. This average overestimate is in the same range as the ratio of the downward flux EF to the upward EF: $E{F}_{down}/E{F}_{up}={\left(B/A\right)}^{2}=4\%$.

_{down}, for example, by multiplying it by a cosine function, IW

_{down}is no longer perfectly periodic along x. Thus, the FFT along x introduces artificial harmonics of k, which affect the MxtUp method. The most significant impact is a slightly larger overestimation of the MxtUp EF average. This change has no impact on the Mxzt energy flux, as the downward propagating component is filtered away.

_{down}, for example, by multiplying it by ${e}^{az}$ (a vertical damping similar to IW

_{damped}but in the opposite direction), Gibb’s oscillations appear in the Mxzt EF (both upward and downward). The nrmse on the upward EF profile is thus slightly increased: $nrmse\left(E{F}_{Mxzt}\left(I{W}_{Up}\right)\right)=1.05\%$ for a = 0.62 ${\mathrm{m}}^{-1}$. However, the mean of the upward EF stays very close to the analytical value: $nrmse\left({\langle E{F}_{Mxzt}\left(I{W}_{Up}\right)\rangle}_{z}\right)=$ 0.28% for $a=$ 0.62 ${\mathrm{m}}^{-1}$.

_{ideal}), both methods are very accurate and the errors of the reconstructed fields are close to zero. For the damped wave field (IW

_{damped}), the signal is not periodic in z, and the periodic extension of the input signal is discontinuous at the upper and lower boundaries. Thus, the FFT along z introduces artificial higher harmonics of m (Figure 2). The false overtones corrupt the Mxzt method, creating Gibb’s oscillations. These oscillations result in errors near the upper and lower boundaries, both for the fields (Figure 3) and for the energy flux (Figure 5). Nevertheless, for a weak vertical damping, the errors stay in an acceptable range. The false overtones also slightly affect the MxtUp method by creating artificial modes with upward phase velocity. However, the induced errors in the calculated energy flux are very small even for a large vertical damping coefficient. The MxtUp method is therefore the most accurate one for the damped field.

#### 3.2. Application to Experimental Data

#### 3.2.1. Presentation of the Laboratory Experiments

_{0}= 20 cm (15 cm for the vertical part plus 5 cm for the half-circle with a radius of 5 cm covering the top). This periodic topography is attached to a long plate, in turn linked to a motor, forcing a sinusoidal back and forth motion at a precisely controlled frequency and amplitude. After the initial density profile has been established and measured, therefore, the topographic obstacle is moved at a given tidal frequency. This allows a better control of the forcing amplitude and frequency than in a set up with a fluid being pushed back and forth, as in the ocean. The equivalence between forcing by the barotropic tide and the oscillating topography for the generation of internal gravity waves has been shown by Gerkema and Zimmerman [20] in the linear case.

#### 3.2.2. Application to Experimental Data

_{Mxzt}& EF

_{MxtUp}). When using the Mxzt method, we first filter away all downward propagating wave components (with upward phase velocity) from the experimental data, as illustrated in Figure 10. This filtering cannot be done with the MxtUp method. This filtering was also not performed when calculating the error of the reconstructed $\delta {N}_{Mxzt}^{2}$ in Figure 7, and when comparing the calculated $\delta {\rho}_{Mxzt}$ to the probe measurements in Figure 9. The resulting profiles of the energy flux are shown in Figure 11.

^{−1}, N = 0.82 s

^{−1}, H = 0.2 m, L = 0.2 m and $U=\omega \xi $, where the tidal amplitude is $\xi =$ 2.3 mm. With these values, we obtain k = 0.29 × 10

^{−6}W/m and C

_{iso}= 22.3 × 10

^{−6}W/m. The energy flux measured in our experiment is approximately 1 × 10

^{−6}W/m per ridge, as seen in Figure 11. It is difficult to compare this to the theoretical value given by Equation (33), since the geometric factor $I\left(S\right)$ is singular at the resonances, and therefore very sensitive to the exact value of S near a resonance. Furthermore, the theoretical value is certainly not singular when there is a finite number of ridges rather than a periodic array, and the ridges in the experiment are rather broad and have rounded tops. Nevertheless, it is interesting to note that the measured energy flux is slightly more than three times larger than the dimensional factor k in Equation (33).

## 4. Discussion and Conclusions

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

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**Figure 1.**Schematic chart of the Mxzt and MxtUp methods for calculating the density anomaly and the upward vertical energy flux from synthetic Schlieren data. * When applied to internal wave fields observed in the laboratory experiments (Section 3.2), a moving mean is applied along the z axis to the experimental data ($\delta {N}^{2}$) in order to reduce the small-scale noise.

**Figure 2.**Spectra of $\delta {N}_{ideal}^{2}$ (solid lines), $\delta {N}_{damped}^{2}$ (dashed lines with star markers) and $\delta {N}_{mix}^{2}$ (dashed-dotted lines with circle markers) as a function of $m$ for $k={k}_{0}$ and $\omega ={\omega}_{0}$. Blue lines: imaginary part; red lines: real part.

**Figure 3.**Vertical profiles of the root mean square (rms) error, averaged over $x$ and $t$, of the reconstructed fields for IW

_{damped}resulting from the Mxzt and MxtUp methods. (

**a**) rms error of $\delta \rho $. (

**b**) rms error of $\delta {N}^{2}$.

**Figure 4.**Vertical profiles of the rms error, averaged over $x$ and $t$, of the reconstructed upward propagating wave fields of IW

_{mix}resulting from the Mxzt and MxtUp methods. (

**a**) rms error of $\delta \rho $. (

**b**) rms error of $\delta {N}^{2}$.

**Figure 5.**(

**a**) The vertical energy flux for IW

_{damped}(dashed curves) and IW

_{ideal}(plain curves) resulting from the MxtUp (green curves) and Mxzt (black curves) methods. The red curve represents the analytic vertical energy flux for IW

_{ideal}, as defined by Equation (32). (

**b**) Vertical energy flux for IW

_{mix}, obtained with MxtUp (green curve) and Mxzt (black curves); analytic flux in red, as defined by Equation (32). Plain curves are upward and dashed curves are downward flux. The black dotted line shows the total flux (upward minus downward) obtained with Mxzt and the combined field IW

_{mix}.

**Figure 6.**Sketch of the experimental set-up. The top panel shows a view from the top whereas the bottom panel shows a view from the side. The rope and weights are used to create a symmetry in the forces moving the plate in both directions and improving the quality of the forcing. The horizontal to vertical ratio is not maintained for the sloping plates on the bottom panel.

**Figure 7.**Application of the two methods to experimental data. (

**a**) Brunt–Väisälä frequency anomaly $\delta {N}^{2}$ obtained from synthetic Schlieren measurements at t = 24 T. The black square represents the conductivity probe position. The dashed line locates the finite-space window ($x,z\text{}\in \left[{X}_{1}:{X}_{2},\text{}{Z}_{1}\text{}:{Z}_{2}\right]$) selected for the Mxzt & MxtUp methods, and used in the lower panels. (

**b**) Root mean square error of the reconstructed $\delta {N}_{MxtUp}^{2}$ field. To calculate the rmse, the $\delta {N}^{2}$ field filtered at the fundamental frequency (denoted $\delta {N}_{{\omega}_{0}}^{2}\text{}$) is used as the reference field. (

**c**) Root mean square error of the reconstructed $\delta {N}_{Mxzt}^{2}$ field (with reference field $\delta {N}_{{\omega}_{0}}^{2}$). The red arrows represent the path of the primary beams.

**Figure 8.**Sketch of the experimental arrangement in the second resonance condition ($B=0.20\text{}m,\text{}\theta =30.7\xb0,\text{}\omega =0.42\text{}\mathrm{rad}/\mathrm{s})$. The arrows represent the path of the primary beams. The downward propagating IW beams are first reflected on the neighbouring wall and then on the tank bottom, midway between two rounded ridges, and finally on the original ridge. The phase is constant along each beam, and changes by $\pi $ at each reflection on a solid surface. This leads to a constructive interference between the IW beams: the downward propagating IW beam generated by the blue ridge (blue arrow) is superimposed on the upward propagating IW beam generated by the red ridge (red arrow).

**Figure 9.**Density measurement filtered at the fundamental frequency ω

_{0}(red curve) and estimation (black and green curves) of the density anomaly from synthetic Schlieren using our two methods. Measurements are taken at a location (black square on Figure 7) in the middle of an upward propagating ray. The red crosses represent the raw probe data.

**Figure 10.**Spectrum of $\delta {N}^{2}$ as a function of $m$ for $k={k}_{0}$ and $\omega ={\omega}_{0}$. The plain line represents the spectrum of the raw data ($|\tilde{\tilde{\widehat{\delta {N}^{2}}}}|$) whereas the dotted line represents the spectrum of the filtered data (without the downward propagation). The red vertical line indicates the value of m

_{0}.

**Figure 11.**Upward (plain curves) and downward (dashed curve, only for the Mxzt method) energy flux resulting from the MxtUp (green curve) and Mxzt (black curve) methods, associated with the fundamental frequency ω

_{0.}The blue line represents a mixed solution for the upward energy flux constructed from the mean slope of the MxtUp solution and the mean of the Mxzt solution.

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**MDPI and ACS Style**

Bordois, L.; Nycander, J.; Paci, A.
Computation of Density Perturbation and Energy Flux of Internal Waves from Experimental Data. *Fluids* **2020**, *5*, 119.
https://doi.org/10.3390/fluids5030119

**AMA Style**

Bordois L, Nycander J, Paci A.
Computation of Density Perturbation and Energy Flux of Internal Waves from Experimental Data. *Fluids*. 2020; 5(3):119.
https://doi.org/10.3390/fluids5030119

**Chicago/Turabian Style**

Bordois, Lucie, Jonas Nycander, and Alexandre Paci.
2020. "Computation of Density Perturbation and Energy Flux of Internal Waves from Experimental Data" *Fluids* 5, no. 3: 119.
https://doi.org/10.3390/fluids5030119