Barotropic Tidal Elevations

• Barotropic tidal elevations are near-uniform vertical ocean surface motions induced by gravitational and centrifugal forces from the Moon and Sun.
• Advanced methodologies such as linearized shallow-water equations, harmonic analysis, and finite element modeling yield enhanced accuracy in simulating tidal behavior.
• Observational tools including tide gauges, bottom pressure recorders, and satellite altimetry, combined with innovative models, inform coastal risk assessments and global tidal energy budgets.

Barotropic tidal elevations refer to the large-scale, near-uniform vertical motions of the ocean’s free surface that arise as a response to external tidal forcing. These motions manifest as coherent changes in sea level and are a fundamental component of ocean tidal dynamics, distinct from baroclinic tides that involve internal stratification and density-driven structure. Barotropic tides play a central role in global tidal energy budgets, coastal flooding risk, and are crucial in understanding both oceanographic and astrophysical fluid systems. This article summarizes recent advances and models addressing the physical origins, mathematical representation, numerical modeling, observational techniques, stratification effects, and geophysical implications of barotropic tidal elevations.

1. Physical Mechanisms and Theoretical Formulation

Classically, barotropic tidal elevations have been modeled as the ocean’s response to the astronomical tidal potential from the Moon and Sun, decomposed into the gravitational and centrifugal “tractive force” acting mostly tangentially to the Earth’s surface. This leads to the linearized shallow-water equations on a rotating sphere: $$\frac{\partial \eta}{\partial t} + \nabla \cdot (H\mathbf{u}) = 0$$

$$\frac{\partial \mathbf{u}}{\partial t} + f \mathbf{k} \times \mathbf{u} = -g \nabla \eta + F_{\text{fric}} + \cdots$$

where $\eta$ is the free-surface elevation, $H$ the mean depth, $\mathbf{u}$ the horizontal velocity, $f$ the Coriolis parameter, and $F_{\text{fric}}$ frictional terms.

Barotropic elevation $\eta$ is commonly represented as a harmonic sum: $$\eta(t) = \sum_{j=1}^{N} A_j \cos(\omega_j t + \phi_j)$$ with constituent amplitudes $A_j$, frequencies $\omega_j$, and phases $\phi_j$ (Wang et al., 2022).

A new paradigm—the Oceanic Basin Oscillation-Driving Mechanism (OBODTM)—reverses the causal framework. Instead of direct water forcing, it posits that lunar and solar torques deform the solid Earth into an elongated ellipsoid, which rocks ocean basins and effectively “sloshes” water, generating coherent barotropic tides. The algebraic model sums basin-floor displacements with finite propagation delay, producing RMS fits to deep-ocean records an order of magnitude more accurate than traditional hydrodynamic (Laplace-tidal-equation) models (Yang, 28 Jan 2025).

2. Observational Evidence and Quantification

Direct measurement of barotropic tidal elevations relies on tide gauges, bottom pressure recorders, and increasingly, satellite altimetry. In regions like the northwestern Mediterranean, observed barotropic tidal amplitudes are exceptionally small (peak-to-peak semidiurnal elevation $H \simeq 0.1$ m). Bottom-mounted pressure sensors record equivalent pressure oscillations (unattenuated with depth), which, via the local adiabatic lapse rate,

$\Gamma = \frac{\partial T}{\partial p} \simeq 1.68 \times 10^{-8}\, ^\circ\mathrm{C}\,\mathrm{m}^2\,\mathrm{N}^{-1}$

allow conversion of pressure variations to temperature-equivalent signals, yielding detectable $\approx 1.5 \times 10^{-5}$ °C amplitudes at 2500 m depth under near-homogeneous conditions ($N < f$, with $N$ local buoyancy frequency and $f$ inertial frequency) (Haren, 7 Jan 2026).

Satellite altimetry offers near-global surface elevation mapping but is often under-sampled relative to tidal frequencies. Recent advances employ regularized least-squares harmonic analyses incorporating prior gauge information to robustly extract harmonic tidal constants, even when classical Nyquist-based methods fail, maintaining RMS errors below 5% for all dominant tidal constituents (Wang et al., 2022).

3. Mathematical and Numerical Modeling

Harmonic Analysis: The decomposition of tidal elevation into a harmonic sum remains foundational. However, data undersampling is a major challenge. Regularized approaches (ReLSHA) integrate prior spatial information to yield stable, high-fidelity reconstructions of barotropic amplitude and phase at coastal sites under sparse sampling constraints.

Finite Element Modeling: Mixed finite element discretizations of the rotating shallow-water equations yield stable, convergent global tide solutions. Given compatible $H(\text{div})$ velocity spaces ($V_h$) and matching pressure spaces ($W_h$), with discrete systems of the form: $$(1/H\,u_{h,t}, v_h) + (1/\epsilon)(f/H\,u_h^\perp, v_h) - (\beta/\epsilon^2)(\eta_h, \nabla \cdot v_h) + (C/H\,u_h, v_h) = (F, v_h)$$

$(\eta_{h,t}, w_h) + (\nabla \cdot u_h, w_h) = 0$

the method achieves energy conservation or decay, uniqueness of the steady state, and optimal $L^2$-norm convergence for free-surface elevation $\eta_h$ (Cotter et al., 2014).

OBODTM Algebraic Model: For global or basin-scale prediction, the algebraic basin-slosh model expresses barotropic elevation as: $$\eta(t,\mathbf{x}) = \sum_{i=1}^{N} k_i \left[h_i(t - L_i/v) - h(\mathbf{x}, t) \right]$$ where $h_i$ are basin-floor displacements, $L_i$ transit distances, $v$ the barotropic wave speed, and $k_i$ geometry-based weights (Yang, 28 Jan 2025).

4. Influence of Stratification and Internal Structure

Barotropic tides are traditionally regarded as depth-uniform. However, underlying stratification, topography, and basin configuration modulate their amplitude and phase. In two-layer analytical models:

• Stratification modifies the effective gravity-wave speed of the barotropic mode ($k^{BT}$), with explicit sensitivity to reduced gravity $g'$ and upper-layer depth ratio $\gamma$.
• Basin resonance properties and boundary conditions introduce significant variability: small perturbations in stratification can lead to $\sim$10% changes in barotropic amplitude in standing-wave basins, but weaker effects in open domains.
• Secular changes in stratification driven by climate perturb either the amplitude or phase of barotropic tides, as predicted by the linear response formulas (Wetzel et al., 2013).

In rotating astrophysical or planetary analogs, the barotropic response to tidal forcing is quantified by the Love number $k_{l,m}(\omega)$, whose imaginary component encodes energy dissipation and angular-momentum transfer. The presence of a core, internal density structure, and stratification all modulate the frequency-averaged dissipation, with sectoral (|m|=l) and tesseral (|m|<l) harmonics displaying different core-size dependencies (Ogilvie, 2012, Landry, 2017).

5. Separation from Baroclinic Components

Barotropic and baroclinic tidal components are separated via physical and spectral diagnostics:

• Barotropic signals are unattenuated with depth and primarily manifest in pressure (or sea-level) records.
• Baroclinic internal tides and waves, by contrast, dominate temperature variability under stratified conditions.
• In practice, bottom pressure records are band-pass filtered around the tidal frequency band (e.g., semidiurnal M₂), converted to temperature units via the adiabatic lapse rate, and subtracted from measured deep-sea temperature signals. Under near-homogeneous conditions, the barotropic correction explains $\sim$75% of the semidiurnal variance; under strong stratification, the correction is negligible (Haren, 7 Jan 2026).

A concise comparison is given in the following table:

Condition	Dominant Signal in Temperature	Barotropic Correction Significance
$N < f$	Barotropic tide visible	Large (removes most variance)
$N \gg f$	Baroclinic waves dominate	Negligible

6. Applications, Implications, and Future Directions

Barotropic tidal elevation datasets inform a spectrum of applications:

• Coastal flooding and risk: Precise characterization of barotropic swings informs models of extreme events, high-tide flooding, and sea-level rise impacts (Wang et al., 2022).
• Climate-driven secular variation: Long-term monitoring and model sensitivity analyses link changes in ocean stratification to observable trends in tidal amplitude/phase (Wetzel et al., 2013).
• Ocean mixing: While barotropic tides do not directly mix the deep Mediterranean, accurate correction for their contribution is required before assessing sub-millikelvin internal-wave (baroclinic) signatures (Haren, 7 Jan 2026).
• Planetary and stellar interiors: Barotropic tidal elevation theory, via Love numbers and rotational-tidal responses, informs understanding of energy dissipation, angular momentum exchange, and gravitational-wave signals in compact object binaries (Ogilvie, 2012, Landry, 2017).

Continued advances in theoretical models (e.g., OBODTM), improved numerical methods, and synthesis of disparate observational streams are driving refinement of barotropic tidal elevation knowledge, with direct implications for global change detection, operational oceanography, and geophysical fluid theory.

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References:

• "Tidal motions in the deep Mediterranean" (Haren, 7 Jan 2026)
• "Determining Undersampled Coastal Tidal Harmonics using Regularized Least Squares" (Wang et al., 2022)
• "Mixed finite elements for global tide models" (Cotter et al., 2014)
• "An oceanic basin oscillation-driving mechanism for tides" (Yang, 28 Jan 2025)
• "On stratification, barotropic tides, and secular changes in surface tidal elevations: Two-layer analytical model" (Wetzel et al., 2013)
• "Tides in rotating barotropic fluid bodies: the contribution of inertial waves and the role of internal structure" (Ogilvie, 2012)
• "Tidal deformation of a slowly rotating material body: Interior metric and Love numbers" (Landry, 2017)