Harmonic decomposition, discriminant conditions, and structural two-maxima thresholds
Consider a seasonal discharge driven by a single harmonic and filtered through a linear time-invariant (LTI) kernel. The output has the form:
where \(\omega = 2\pi / T\) is the annual angular frequency, \(B\) is the amplitude (set by the kernel gain at \(\omega\)), and \(\psi\) is the phase lag.
Taking the derivative:
This yields exactly two roots per period: \(\omega t - \psi = 0\) (maximum) and \(\omega t - \psi = \pi\) (minimum). No LTI filter can increase the number of extrema beyond the number present in the forcing.
A single-frequency harmonic through any LTI kernel produces exactly one maximum and one minimum per period. Two maxima per year requires either higher harmonics or nonlinearity.
The simplest extension that can produce two maxima adds the first overtone (semiannual harmonic) to the annual fundamental:
Define the amplitude ratio and effective phase difference:
These two parameters fully control the shape of the seasonal waveform (up to overall amplitude and mean).
Substituting \(x = \omega t - \psi_1\) and taking the derivative:
Using the double-angle identity \(\sin(2x + \delta) = \sin 2x\cos\delta + \cos 2x\sin\delta\):
This is a trigonometric equation in \(x\) parameterised by \((r, \delta)\). For a given \(\delta\), there exists a critical amplitude ratio \(r_\mathrm{c}(\delta)\) above which the equation has four roots per period (two maxima, two minima), and below which it has two roots (one maximum, one minimum — unimodal regime).
The system transitions from unimodal to bimodal when \(r\) crosses \(r_\mathrm{c}(\delta)\). This boundary is \(\delta\)-dependent and must be evaluated numerically. For \(\delta \approx 0.9\;\mathrm{rad}\) (the Óbidos value), \(r_\mathrm{c}\) is significantly larger than the observed \(r \approx 0.13\), placing Óbidos firmly in the unimodal regime.
Independent of the specific harmonic model, any smooth seasonal hydrograph \(Q(t)\) can be tested directly:
In \(t \in [0,T)\), count the roots of \(\mathrm{d}Q/\mathrm{d}t = 0\). If \(\geq 4\) roots exist, with \(\geq 2\) satisfying \(\mathrm{d}^2Q/\mathrm{d}t^2 < 0\), the year is classified as "two-maxima."
This discriminant applies to any smooth representation of \(Q(t)\) — harmonic fit, spline, low-pass filtered daily series — and is model-independent.
The linear harmonic model establishes a necessary condition for two-maxima behaviour. In the real Amazon system, additional mechanisms may push the hydrograph toward or away from the bimodal boundary:
Floodplain activation threshold. When discharge exceeds a critical value \(Q_\mathrm{fp}\), water spills onto the floodplain. The effective storage and routing kernel change abruptly. This is a state-dependent nonlinearity: the kernel \(G(\tau)\) becomes \(G(\tau, Q)\).
Backwater effects. Tributary-mainstem interactions at confluences produce stage-dependent damming. Rising and falling limbs of the hydrograph experience different effective routing delays, introducing hysteresis.
Land-use driven kernel drift. Deforestation, urbanisation, and wetland modification alter catchment response time. Over decadal timescales, the routing kernel \(G_i(\tau)\) drifts, potentially moving sub-basins across the \(r_\mathrm{c}\) boundary.
Quantifying these effects requires the underconstrained variables listed in Parameters: UC-1 through UC-4. The falsification programme addresses them through F-3 (kernel drift / hysteresis check).
Following the Harbour vocabulary:
Locks (structural, stable): Discriminants DC-A and DC-B; the single-harmonic impossibility theorem (CL-1); the definition of "two-maxima year" in §1.2.
Keys (interpretive, updatable): Specific fitted values \((r_y, \delta_y)\) at any gauge; climate-mode conditioning indices; routing kernel parameterisations. These are updated when new data arrive without altering the structural framework.