Internal waves as a proposed mechanism for increasing ambient noise in an increasingly acidic ocean Open Access

The possible effect of increasing acidification on the level of ambient acoustic noise in the ocean has recently received attention both in academic conferences and the popular press.¹ The ability of seawater to absorb sound decreases as the ocean grows more acidic. As global climate change or other processes cause the $pH$ level in the ocean to drop, sound might propagate more freely resulting in an increasingly noisy environment. Hester et al. estimate that the intrinsic sound absorption coefficient of seawater will decrease by almost 40% by mid-century.² Fig. 1 shows the absorption coefficient calculated using the equations in Duda³ and plotted as a function of frequency at different $pH$ levels.

There are reasons to believe why increased acidification might have a less dramatic effect on the ambient noise level than is implied by Fig. 1. Other loss factors, such as attenuation by the seabed, might be so dominant as to render seawater absorption insignificant by comparison. This argument can be understood from the standpoint of acoustic mode theory. Noise sources in the ocean, both natural like wind-driven surface waves and man-made like ships, are predominantly near the sea surface. Under typical summertime conditions, consequently, the noise sources excite only the high-order acoustic modes that span the entire water column. The low-order modes that are trapped by the thermocline are not excited by the shallow noise sources. One can show that it is these trapped low-order modes that are disproportionately affected by ocean absorption; the high-order modes, by contrast, are disproportionately affected by the seabed and so largely immune to changes in acidity. If no means were available for driving energy from the high-order modes excited by the noise sources into the low-order modes, then the ambient noise level would be little affected by an increasingly acidic ocean.

In the present work, the premise explored is that random background ocean internal waves provide the means for coupling energy into the low-order modes that propagate with low loss. A coupled-mode acoustic model for the ambient noise field is developed. A quantity called the characteristic noise range is defined and is shown to be proportional to the ambient noise level at a particular frequency. The model is evaluated using input parameters similar to those observed during the 2001 East China Sea Experiment. It is argued that a noise prediction model for mid-frequencies that neglects the acoustic mode coupling caused by internal waves will under predict the effects of increasing acidification.

To develop the model, expand the time-harmonic $exp(−iωt)$ acoustic pressure from a single noise source in its normal modes

The factor $r−1/2$ accounts for cylindrical spreading with range. Each mode function, $Ψm$⁠, and each lossless part of the wavenumber, $Re(ξm)$⁠, are determined by solving a real eigenvalue problem for the average background sound speed profile $c0(z)$⁠. The associated lossy part of the wavenumber then can be derived from a perturbation expansion⁴ yielding

where $α(z)$ is the volume absorption profile. The depth integration extends over both the water column and the seabed. At low frequencies, absorption in the water column is negligible and in practical applications the integration is often truncated to just the seabed region. To study the effect of increasing ocean acidity on noise levels, the full integration over both regions must be considered.

If the environment were range-independent, there would be no mode coupling. The energy in a particular mode would decrease with range due to cylindrical spreading and absorption but not get transferred into other modes. To within an unimportant constant, the complex mode amplitude $Am$ in Eq. (1) for this special case is simply

where $ξm$ is the full, complex wavenumber for mode $m$⁠. The imaginary part of $ξm$ causes the exponential decrease in amplitude with range. Since there is no coupling, the mode function is simply evaluated at $zs$⁠, the depth of the noise source.

A more realistic model would recognize that the ocean is never truly range-independent. Ocean variability is responsible for the interchange of energy between the acoustic modes. After making a parabolic approximation, the range-dependent mode amplitude $Am$ in this improved model can be shown to satisfy⁵ 

where the mode coupling coefficient,

quantifies the dependence on the range- and depth-varying sound speed perturbation $δc$⁠. In Eq. (5), $k=ω/c0$ is the reference wavenumber and implicitly only the real parts of the individual wavenumbers are retained. In the limiting case as $δc→0$⁠, the solution to Eq. (4) reduces to Eq. (3) as it must.

The model to this point is deterministic: the pressure field for a single noise source is calculated at range $r$ after propagating through an environment with sound speed perturbation $δc$⁠. In reality, the ambient noise is the summation of the fields from many different sources. The ambient noise should be modeled as a random quantity with the randomness introduced both through the sound speed perturbations and the locations of the noise sources. In the present model, the two types of randomness are considered separately, starting with $δc$⁠.

Modal transport theory may be used to calculate the average mode intensity for a single acoustic source after propagating through a random internal wave field. The magnitude-squared of Eq. (1) is taken to get intensity and then ensemble averaged. The details are beyond the scope of this short communication, but are available in the original literature^5,6 and textbooks.⁷ Specifically, modal transport theory has been combined with a shallow-water version of the Garrett–Munk spectral representation for the $δc$ caused by the random internal waves.⁸ Summarizing the main result, let $Γ$ represent a column vector containing the average mode intensities $Γm≡⟨|Am|2⟩$⁠. It can be shown that $Γ$ satisfies the transport equation

The off-diagonal elements in the transport matrix $S$ describes coupling between a pair of acoustic modes as caused by the internal waves. The diagonal elements include terms $2Im(ξm)$ and so describe the decrease of intensity due to absorption. It is these diagonal elements that are affected by changes in acidity.

The solution to Eq. (6) can be written formally using the matrix exponential or, more usefully, with a singular value decomposition

where each $λm$ is a singular value of the transport matrix. The reciprocal, $1/λm$⁠, represents the $e$-folding range for a particular singular value. The smallest singular value characterizes the asymptotic, far-field behavior of the mean intensity.⁹ For present purposes, define $rch≡1/min(λm)$ as the characteristic noise range.

It remains to consider the randomness from having many different, randomly positioned noise sources. The total noise field is the summation from all noise sources. Assume the noise intensities add incoherently. Further assume that the noise sources are uniformly distributed horizontally in coordinates $x$ and $y$⁠. Then, when integrating over the noise sources, $dxdy→rdrdθ=2πrdr$⁠. Note that the factor of $r$ from the Jacobian cancels the factor of $r−1$ for cylindrical spreading when taking the magnitude-squared of Eq. (1). The detailed total noise intensity will depend on all the singular values in Eq. (7), but for simplicity consider only the smallest. Within this approximation,

Equation (8) says that the mean noise intensity at a particular frequency is proportional to the characteristic noise range.

The effect of changing acidity on the model for mean noise intensity is best illustrated by a particular example. Consider a scenario typical of the East China Sea. The region is attractive for present purposes because extensive acoustic and oceanographic data were collected there during a 2001 experiment.¹⁰ The 105 m deep water column has a 40 m thick surface mixed layer above a sharp thermocline. The sound speed contrast between the surface mixed layer and bottom of the water column is 12 m/s. Moderate-strength internal waves are assumed consistent with the ambient noise field observed during the experiment; see Rouseff and Tang⁸ for details.

As noted earlier, both the water column and the seabed (bottom) will contribute to sound absorption. To compare their relative importance and to look at the effect of changing acidity, it is useful to split the integration in Eq. (2) into two parts, $Im(ξm)≡Ib+Iw$⁠, where

For the integration over the bottom, $αb=0.25dB/m/kHz$ consistent with inversion results.¹¹ For the integration over the water column, $αw$ is selected from Fig. 1 at the appropriate frequency and $pH$ level.

Figure 2 shows the two parts of the lossy wavenumber at frequency 3 kHz as a function of the mode index $m$ for various $pH$ levels. Note that $Iw$ is a function of acidity while $Ib$ is not. Index $m=30$ is the approximate transition between the low-order modes that are trapped by the thermocline and the high-order modes that extend up to the sea surface. For the high-order modes, typically $Ib⪢Iw$ meaning the seabed is the dominant loss mechanism and changes in acidity are of little consequence. It is these modes that would be excited by shallow noise sources. For many trapped modes, however, $Iw$ is of comparable magnitude to $Ib$⁠. Indeed, $Iw>Ib$ for the very lowest order modes. For these modes, $Ib$ and hence $Im(ξm)$ are strong functions of acidity. If these modes become strongly populated through internal wave action, the effect on the ambient noise level could be appreciable.

The transport matrix $S$ in Eq. (7) was calculated at 3 kHz using the $Im(ξm)$ from Fig. 2 at the different $pH$ levels. The characteristic noise range, proportional to the noise level via Eq. (8), was then determined. The calculations were repeated at 500 Hz and 1 kHz using $αw$ from Fig. 1 with the results summarized in Table I.

The numerical results suggest strong frequency dependence with acidification becoming more of an issue at higher frequencies. At 500 Hz, a 52.8% drop in the ability of seawater to absorb sound results in only an 8.2% increase in the noise intensity. At 3 kHz, however, the same change in $pH$ results in a 30.2% increase or about one decibel. A mid-frequency noise model that ignores internal wave effects might under predict the noise intensity.

For the frequencies considered in the present shallow water study, wind-driven surface waves are the dominant mechanism for generating ambient noise. Shipping noise becomes more important at lower frequencies. The frequency dependence in Table I implies that shipping noise will be little affected by increased acidification. This facile interpretation, however, should be regarded with some caution. Shipping noise in shallow water is highly variable and site specific.¹² The assumption used in Eq. (8) of uniformly distributed noise sources becomes dubious. A better approach might use known shipping lanes to create a model for the source distribution.¹³ The relative importance of absorption by seawater and by the seabed will be site and season specific; the effects of increased acidity will be enhanced when there is a low-loss seabed or when there is a strong sound speed duct. The transport theory model outlined in the present study for internal wave induced mode coupling is sufficiently general that it might be applied to these site- and season-specific cases.