A Relationship Between Baroclinic Tidal and Wave Horizontal Spectral Energy and Stratification on the Australian Continental Shelf

Technical Summary: Relationship Between Baroclinic Tidal/Wave Energy and Stratification on the Australian Continental Shelf

Problem Statement
Ocean mixing is a critical but poorly defined process in ocean dynamics, heavily influenced by internal tides and waves. Current ocean models often overestimate vertical mixing and lack accurate parameterizations for baroclinic tides. While measuring the energy associated with these internal waves requires expensive, long-term deployment of current meters, temperature data is more readily available. Previous studies have investigated seasonal dependencies of tides using models or short-term observations, but few have utilized long-term time series (over 4 years) to link stratification directly to baroclinic energy across multiple frequency bands. Furthermore, existing literature has not consistently established a predictive relationship between the upper-ocean-to-bottom temperature difference and horizontal baroclinic kinetic energy, particularly across the diurnal, semidiurnal, harmonic, and high-frequency bands.

Methodology
The study analyzed temperature and velocity time series from 28 mooring sites off the Australian coast, sourced from the Integrated Marine Observing System (IMOS). The sites ranged from approximately 12°S to 38°S, with time series durations of at least 4 years.

• Data Processing: Time series were segmented into 10-day intervals with 50% overlap to calculate spectral energy. This duration was selected to balance the need for sufficient data points against the noise introduced by storm events.
• Spectral Analysis: Spectra were computed for East-West and North-South velocities and averaged across depths for each segment. Energy was integrated over five frequency bands:
  1. All frequencies (0.167–12 cpd)
  2. Diurnal (0.5–1.5 cpd)
  3. Semidiurnal (1.5–2.5 cpd)
  4. Harmonics (3–10 cpd)
  5. High frequencies (10–12 cpd)
• Stratification Metric: The upper-ocean (typically ~20 m) to bottom temperature difference ($dT$) was used as a proxy for stratification. Negative differences were excluded as they indicated wave breaking.
• Regression Analysis: The study sought a relationship between the horizontal baroclinic spectral energy ($E_H$) and the temperature difference ($dT$), normalized by the horizontal barotropic spectral energy ($E_{BT}$). Both square root and power law regressions were tested. The power law form $E_H = E_{BT} B dT^A$ was found to provide a better fit, likely due to non-linear effects (friction, advection) not captured by linear internal wave theory.
• Latitude Dependence: Coefficients $A$ and $B$ were analyzed for latitude dependence, specifically investigating the influence of the diurnal critical latitudes (27.6°S for O1 and 30°S for K1).

Key Results

1. Validated Relationships: A clear power-law relationship between the upper-ocean-to-bottom temperature difference and horizontal baroclinic spectral energy was established for 16 of the 28 sites.
2. Validity Constraints: The relationship held only when:
   • The temperature difference exceeded 4°C.
   • The water depth was between 60 m and 200 m.
   • Sites outside these parameters (shallow <60m, deep >200m, or weak stratification <4°C) showed no clear relationship, often exhibiting essentially barotropic velocities.
3. Frequency Band Variations:
   • The relationship was strongest for diurnal, semidiurnal, and all-frequency combined bands.
   • The exponent $A$ was generally less than 0.5 (the linear theory prediction), indicating non-linear effects.
   • The magnitude coefficient $B$ and the exponent $A$ exhibited significant latitude dependence, particularly for tidal frequencies.
4. Latitude Effects:
   • Near and poleward of the diurnal critical latitudes, the semidiurnal coefficients generally increased, reflecting enhanced energy transfers from diurnal to semidiurnal frequencies and harmonics.
   • Latitude-dependent regression formulas ($A = C_1|\psi| + C_2$ and $B = C_3|\psi| + C_4$) were derived. These latitude-dependent coefficients provided a better fit (lower RMS differences) than mean coefficients, particularly for the semidiurnal band.
5. Site Specificity: While site-specific coefficients yielded the most accurate reproduction of observed values, the study demonstrated that generalized relationships using mean or latitude-dependent coefficients are viable for estimation where long-term site-specific data is unavailable.

Significance and Claims
The paper claims that these relationships offer a practical method to estimate horizontal baroclinic energy fields for specific frequency ranges without deploying expensive velocity observations. By utilizing the more accessible temperature difference and barotropic tidal estimates (e.g., from TPXO10v2a), researchers can:

• Estimate the internal tidal and wave energy available for mixing in water depths of 60–200 m.
• Provide parameterizations for sub-grid cell mixing in ocean models, particularly those that do not explicitly resolve tides.
• Address the need for better baroclinic tide parameterizations identified by previous researchers (e.g., Müller et al., 2014).

The authors modestly note that while these relationships do not allow for the exact prediction of coastal internal tides as suggested by Nash et al. (2012), they represent a significant step toward that goal by providing a range of horizontal baroclinic kinetic energy based on stratification and barotropic forcing. The study emphasizes that these relationships are likely limited to local generation on the continental shelf near the shelf-slope break and do not account for all external factors such as currents, eddies, or specific topographic orientations. Future work is suggested to include deeper sites and additional long-term data from other locations to refine depth-dependent relationships and account for other environmental variables.