Modeling Internal Wave Propagation Through Stratified Fluids

A few hundred meters below the surface of the Arctic Ocean, there is a layer of Atlantic water which is warmer than the top layer in contact with the sea ice above [10]. If all the heat in this Atlantic layer were to rise to the surface, all Arctic sea ice would melt within five years [11]. Winds across the ocean's surface create downward-moving internal waves which can cause vertical mixing of heat [3]. However, in the top of the Atlantic layer, the water's density changes suddenly every few meters forming a density “staircase,” a barrier to vertical mixing at each step [8]. As Arctic sea ice declines, it will be important to predict how the interaction between internal waves and density staircases might change.

Working with Prof. Nicolas Grisouard, I used numerical simulations to investigate this phenomenon and successfully reproduced the results of a wave tank experiment done in Prof. Peacock’s ENDLab at MIT [4]. These simulations give insight into whether the propagation of internal waves through density staircases could reasonably become significant to vertical heat transport as the Arctic climate continues to change. This work is presented in full detail in Chapter 4 of my thesis.

Background

The Arctic Ocean has a layer of relatively warm water originating from the Atlantic at couple hundred meters depth [10]. Figure 1 shows typical temperature and salinity profiles measured by Ice Tethered Profilers (ITP’s) in the Canadian Basin of the Arctic Ocean [7]. The transition to the Atlantic layer starts around 200 meters depth and contains structures called thermohaline staircases, which can be seen in the insets of Figure 1. Each stair-step is a couple meters thick and well-resolved by the ITP’s which take measurements every ~25 cm [10].

The Atlantic layer contains enough heat to melt all Arctic sea ice if it rose to the surface [11]. However, the Arctic Ocean is very stably stratified with far less wave activity than other oceans. The Atlantic layer currently sits isolated from the colder waters above.

Wind blowing across the ocean’s surface can create internal waves which propagate downward into the ocean. When these waves break, they can cause vertical mixing which could bring warm water closer to the surface [3]. As the sea ice in the Arctic continues to decline, there is more open ocean for winds to blow across.

Figure 1: The temperature profile number 1257 from ITP 1 (left) and the corresponding salinity profile (right) taken on Monday, June 26th, 2006 at 136.1525◦W, 77.2870◦N. Both profiles show insets of the same depth ranges where double-diffusive staircases can be seen. Following Shibley et al. (2017) Figure 3b [7].

Thermohaline Staircases

Many areas of the Arctic Ocean contain vertical density structures at a depth of a few hundred meters [6]. These are sometimes called double diffusive staircases after processes that occur because the rates of diffusion for heat and salinity, which both have an effect on seawater density, are vastly different from one another.

In Figure 2, the left side shows a density profile typical of the Canadian Basin of the Arctic Ocean. The density profile more closely resembles the salinity profile from Figure 1 than the temperature profile because salinity has a greater effect on density in the Arctic Ocean. On the right side of Figure 2, the density staircase is shown in terms of buoyancy frequency or stratification, which is proportional to the rate of change in density with respect to depth [4].

Figure 2: The density anomaly (left) and buoyancy frequency (right) profiles calculated from the temperature and salinity profiles of ITP 1 profile 1257 taken on June 26th, 2006.

Internal Waves and Evanescence

The dispersion relation for internal waves predicts that when the oscillation frequency (ω) of the waves are larger than the fluid’s stratification number (N, units of frequency), the waves are evanescent. This means that they decay rapidly. Figures 3 and 4 are videos of an oscillating cylinder in a wave tank by the GFD Dennou Club. The setup is very similar to the wave tank at the University of Washington. In Figure 3, ω < N and four clear internal wave beams can be seen. In Figure 4, ω > N and the waves do not propagate in a coherent manner. In double diffusive staircases, each step has N = 0 which means all waves are evanescent in those regions. This presents a series of obstacles for propagating internal waves. However, depending on how the wave compares to the spacing between the steps, the wave could counter-intuitively transmit through the barriers [9]. This phenomenon is know as “internal gravity wave tunnelling” and analogous to the tunnelling seen in quantum mechanics [2].

Figure 3: The oscillation frequency (ω = 0.143 Hz) is less than the stratification (N = 0.256 Hz), so internal waves can propagate. Reproduced from the GFD Dennou Club.

Figure 4: The oscillation frequency (ω = 0.263 Hz) is greater than the stratification (N = 0.256 Hz), so internal waves cannot propagate. Reproduced from the GFD Dennou Club. — Figure 4: The oscillation frequency (ω = 0.263 Hz) is greater than the stratification (N = 0.256 Hz), so internal waves are evanescent. Reproduced from the GFD Dennou Club.

Reproducing Wave Tank Results with Numerical Experiments

To study the effects of staircase-like stratification structures on the propagation of internal waves, Professor Peacock and his collaborators performed experiments with the wave tank at ENDLab. The tank was filled with varying concentrations of salt water to produce either a single or double mixed layer profile. An oscillating cylinder created internal waves which propagating down and to the right across the measurement domain [4]. The results were as expected: the waves reflected and transmitted when encountering the mixed layers.

Using Dedalus, a Python framework for solving partial differential equations, I created numerical experiments to recreate the conditions in the wave tank. Figure 5 shows a side-by-side comparison of snapshots from the experimental results of Ghaemsaidi et al. 2016 [4] in the left column and my numerical experiments in the right column. There is a close qualitative match between them, showing the wave coming in from the upper left corner, reflecting and transmitting off the top layer and, for the double layer case, both reflecting and transmitting off the bottom layer as well.

Figure 5: The experimental results from ENDLab for (a) single and (c) double mixed layer. Adapted from Ghaemsaidi et al. 2016 [4]. My direct numerical simulation of the (b) single and (d) double mixed layer conditions of the wave tank.

Transmission Coefficient

In order to determine whether internal wave propagation through density staircases is significant for vertical heat transport up from the warm Atlantic layer, it is necessary to quantify how much of an incoming internal wave gets through the staircases. For this, we use the transmission coefficient which is the ratio between the amplitude of the wave that comes out the bottom of the staircase to the amplitude of the original wave that went in the top of the staircase.

Using the Boussinesq equations, Ghaemsaidi et al. 2016 solved for the transmission coefficient and found it depended on two factors: mL, the ratio between the wavelength and thickness of the layer, and θ, the angle at which the wave hits the top of the staircase [4]. As mL gets bigger, the wave gets smaller in comparison to the layer.

Figure 6 shows a heat map of the transmission coefficient Ghaemsaidi et al. 2016 calculated in 4 cases: either 1 or 2 layers and either assuming the fluid is inviscid or viscous [4]. The 1 layer case is exactly what you might intuitively expect; the transmission coefficient drops off as mL gets larger. But, the effect of wave tunnelling can be seen in the 2 layer case where tendrils of high transmission reach across the plots at certain combinations of mL and θ.

Measuring transmission coefficients in numerical experiments

Ghamesaidi et al. 2016 solved for the transmission coefficient analytically [4]. However, doing so becomes much more difficult in more complicated situations, e.g., including both viscosity and the effect of Earth’s rotation. However, adding in such effects is relatively easy in the Dedalus framework. I adapted the code I used above to measure the transmission coefficient for a variety of cases.

To save on computational resources, I collapsed my simulations to 1 spatial dimension. This also allows me to plot an entire simulation in one image as shown in Figure 7. I simulate waves being produced at the top of the domain which propagate downwards through the stratification structure and get absorbed by a sponge layer at the bottom. I run the simulation until it reaches a steady state and then calculate the transmission coefficient using the wavefield enclosed in the “Analyzed domain” box shown in Figure 7.

To calculate the transmission coefficient, I first remove the upward propagating reflected wave from the analyzed domain using a process called complex demodulation, following Mercier et al. 2008 [5]. After that, I take the average amplitude squared above (below) to get a value for the incident (transmitted) wave, then take the ratio to get the transmission coefficient.

Figure 7: An example simulation with 2 layers where mL = 1, θ = 45°, and omega = 0.71. The background stratification profile is shown on the left. The incoming and transmitted waves can be seen separately as indicated by the arrows. The reflected wave can be seen superimposed on the incoming wave. The wavefield enclosed by the “Analyzed domain” box was used to compute a transmission coefficient of 0.28.

Transmission coefficient for multiple layers

Figure 8 shows the transmission coefficient T for many numerical experiments across a certain range of the ratio between wavelength and layer thickness kL at an incident angle of θ = 45°. For the experiments with 1 layer, T decreases monotonically as the layer becomes larger in relation to the wave and matches very well with the analytical prediction. However, for the experiments with multiple layers, there are peaks in transmission at intermediate values of kL. These peaks in transmission come from wave tunnelling and seem to be moving to lower values of kL as the number of layers increases.

Figure 8: The transmission coefficient for many numerical experiments ran in Dedalus across different values of the ratio between wavelength and layer thickness kL for the cases of 1, 2, 3, 4, and 5 layers all with an incident angle of θ = 45°.

Comparing to previous results

Ghaemsaidi et al. 2016 used MATLAB to calculate the analytical transmission coefficients for the cases of 1 and 2 layers [4]. Figure 9 shows a comparison of their results and what I found from numerical experiments for 1 and 2 layers at θ = 45°. The overall shape is quite similar, with the secondary peak in the 2 layer cases corresponding to the main tendril in Figure 6b.

Figure 9: A comparison between the MATLAB code used by Ghaemsaidi et al. 2016 [4] and my simulations using Dedalus for 1 and 2 mixed layers at θ = 45°.

Future work

The work presented here shows that numerical experiments using the Dedalus framework reproduce the results of previous work done using a laboratory wave tank, numerical experiments, or analytical calculations. This implies that such numerical experiments as I’ve shown here could be used to investigate the interactions between internal waves and staircase-like stratification structures in situations that more closely represent the reality of the Arctic Ocean. This would involve using wavelengths and angles of propagation commonly seen in the ocean, which are much larger and shallower than the values I used [4]. This would also mean using stratification structures with values of buoyancy frequency common to the Arctic and interfaces which are much thinner than the layers.

There are many avenues for improving upon the work presented here. For one, I used only weakly-nonlinear, monochromatic waves. However, the internal wave field in the Arctic Ocean contains many different frequencies and nonlinear processes such as harmonic generation can change which frequencies are present and has been shown to modulate internal wave interactions with staircase structures [12]. Bracamontes-Ramirez and Sutherland 2024 showed how transient internal wave packets do not exhibit peaks in transmission as did the steady state in my experiments [1]. To my knowledge, no study has yet explored how the inclusion of both viscosity and Earth’s rotation affects internal wave’s interactions with staircase structures.

References

[1] Bracamontes-Ramirez, J. and B. Sutherland (2024). Transient internal wave excitation of resonant modes in a density staircase. Physical Review Fluids. 9)6):1–21. DOI: 10.1103/PhysRevFluids.9.064801

[2] Eckart, C. (1961). Internal waves in the ocean. Physics of Fluids, 4(7): 791–799. DOI: 10.1063/1.1706408

[3] Fer, I. (2014). Near-Inertial Mixing in the Central Arctic Ocean. Journal of Physical Oceanography, 44 (8): 2031–2049. DOI: 10.1175/JPO-D-13-0133.1

[4] Ghaemsaidi, S. J., H.V. Dosser, , L. Rainville, and T. Peacock (2016). The impact of multiple layering on internal wave transmission. Journal of Fluid Mechanics, 789: 617–629. DOI: 10.1017/jfm.2015.682

[5] Mercier, M., N. Garnier, and T. Dauxois (2008). Reflection and diffraction of internal waves analyzed with the Hilbert transform. Physics of Fluids, 20 (8) 086601. DOI: 10.1063/1.2963136

[6] Shibley, N. and M.L. Timmermans (2019) The Formation of Double-Diffusive Layers in a Weakly Turbulent Environment. Journal of Geophysical Research: Oceans, 124. DOI: 10.1029/2018JC014625

[7] Shibley, N., M.L.Timmermans, J.R. Carpenter, and J.M. Toole (2017). Spatial variability of the Arctic Ocean’s double-diffusive staircase. Journal of Geophysical Research: Oceans, 122 (2):980-994. DOI: 10.1002/2016JC01241

[8] Sutherland, B. R. (2016). Internal wave transmission through a thermohaline staircase. Physical Review Fluids, 1 (1): 013701. DOI: 10.1103/PhysRevFluids.1.013701

[9] Sutherland, B. R. and Yewchuk, K. (2004). Internal wave tunnelling. Journal of Fluid Mechanics, 511: 125–134. DOI: 10.1017/S0022112004009863

[10] Timmermans, M.-L., Toole, J., Krishfield, R., and Winsor, P. (2008). Ice-Tethered Profiler observations of the double-diffusive staircase in the Canada Basin thermocline. Journal of Geophysical Research, 113: 1–10. DOI: 10.1029/2008jc004829

[11] Turner, J. (2010). The melting of ice in the arctic ocean: The influence of Double-Diffusive transport of heat from below. Journal of Physical Oceanography, 40(1):249–256. DOI: 10.1175/2009JPO4279.1

[12] Wunsch, S. (2018). Nonlinear harmonic generation by internal waves in a density staircase. Physical Review Fluids. 3(11):114803. DOI: 10.1103/PhysRevFluids.3.114803