Quantitative attribution of sea-surface-temperature and seasonal-thermocline variability to stochastic atmospheric forcing versus internal oceanic processes (Frankignoul & Hasselmann, 1977)

Frankignoul and Hasselmann (Tellus, 1977) showed that midlatitude sea surface temperature (SST) anomalies behave as the response of the oceanic mixed layer to white-noise atmospheric forcing, with a linear stabilizing feedback rendering the response stationary - the observational confirmation of Hasselmann's stochastic climate mechanism. For the seasonal thermocline, however, they found that local atmospheric forcing through wind-stress curl (Ekman pumping) produced the right spectral shape but roughly two orders of magnitude too little displacement variance, and they left as open questions: the high-wavenumber wind-stress-curl spectra needed for a quantitative input-response comparison; the regional partition between direct atmospheric forcing and internal oceanic processes (baroclinic eddies, advection, mixed-layer deepening, upwelling); the generalization of the one-parameter linear-feedback mixed-layer model to seasonal modulation, multi-component interactions, and higher-order feedbacks; and the correction of atmospheric-model biases in low-frequency and high-wavenumber forcing levels. Later literature resolved parts of it - satellite scatterometry now measures small-scale wind-stress structure, eddying ocean models quantify intrinsic-versus-forced variance, reemergence and multivariate linear-inverse extensions treat seasonal modulation, and stochastic parameterizations address model noise - but no complete, regionally resolved, quantitative attribution budget for observed SST and seasonal-thermocline variance exists, and 2025 work within the same stochastic-model program still documents structural gaps. The question remains open in this form.

Background

Midlatitude sea surface temperature (SST) records exhibit anomalies that persist for several months and carry most of their variance at long periods, and the seasonal thermocline - the upper, seasonally varying part of the ocean's layer of strong vertical temperature gradient, which lies beneath the mixed layer and moves up and down with the annual cycle of wintertime convective mixing and summertime heating - exhibits variability at horizontal scales of tens to a few hundred kilometers. Hasselmann (1976) proposed the stochastic climate model to explain such slow oceanic variability: the atmosphere evolves on a short time scale (days, its 'weather'), while the ocean surface layer responds on much longer time scales (months to years), so the rapidly varying atmospheric forcing acts on the slow ocean variable as random noise. For the heat content of the mixed layer (the wind-stirred, nearly uniform upper 25-100 m of the ocean) forced by random surface heat fluxes, the model reduces to a Langevin-type equation (a first-order stochastic differential equation of the kind used to describe Brownian motion), $dy/dt=v'(t)-\lambda y$, where $y$ is the SST anomaly, $v'$ the anomalous atmospheric forcing with an approximately white (frequency-flat) spectrum at periods longer than the weather band, and $\lambda>0$ a linear stabilizing feedback rate (mainly the damping of SST anomalies by air-sea heat exchange). The predicted response spectrum is red (its variance is concentrated at low frequencies), $G(\omega)=F(0)/(\omega^{2}+\lambda^{2})$, where $F(0)$ is the white-noise level of the input and $\omega$ the angular frequency; without feedback the variance would grow diffusively, $\langle y'^{2}\rangle=2Dt$, with diffusion coefficient $D=\pi F(0)$.

Frankignoul & Hasselmann (1977) tested this picture: Monte-Carlo experiments with a highly idealized barotropic (single-layer, with no vertical density contrasts) atmospheric model coupled to a fixed-depth ('copper-plate') mixed layer reproduced the observed spectral shape and orders of magnitude of midlatitude SST anomalies, and direct comparisons of observed heat-flux and SST spectra at Ocean Weather Ship India (59N, 19W), a long-running stationary measurement station in the North Atlantic, yielded a feedback time of about 4.5 months and $D\approx1.5\,(^{\circ}\mathrm{C})^{2}\mathrm{yr}^{-1}$. Wind-stress curl (the horizontal rotation of the surface wind stress, which drives Ekman pumping - a vertical velocity $\nabla\times\mathbf{S}/(\rho^{w} f)$, where $\mathbf{S}$ is the wind stress, $\rho^{w}$ the density of seawater and $f$ the Coriolis parameter, that displaces isotherms vertically) gave thermocline fluctuations of the correct spectral shape but far too small an amplitude: observed displacement spectra (Wunsch 1972; Bernstein & White 1974) exceeded the simulated ones by roughly two orders of magnitude, and closing the gap would have required wind-stress-curl variance concentrated at wavelengths as short as about 220 km, which the strongly dissipated atmospheric model, spectrally truncated at wavenumber 8 (retaining only atmospheric patterns of wavelengths longer than roughly 1000 km), could not represent. Bernstein & White (1974) suggested instead that the North Pacific thermocline variability is caused by internal baroclinic eddies (eddies generated by baroclinic instability, the release of potential energy stored in the tilted density surfaces of the mean circulation - the ocean's analogue of weather).

The paper concluded that SST anomalies in mid-ocean regions away from intense currents or fronts are consistent with the ocean acting as a first-order Markov integrator of white-noise atmospheric input, decaying with an e-folding time (the time in which an anomaly decays by a factor $e$) of order half a year, but identified four unresolved issues that constitute this problem: (1) whether the observed magnitude of seasonal-thermocline displacements and associated mesoscale (tens to a few hundred kilometer scale) variance can be accounted for by local atmospheric forcing can be decided only by detailed wavenumber-frequency measurements of the wind-stress curl, particularly at high wavenumbers (short wavelengths), which enter a quantitative comparison of the input (wind-stress-curl and surface heat-flux) and response (thermocline-displacement and SST) spectra; (2) the relative regional importance of local atmospheric forcing versus internal oceanic processes (baroclinic eddies, advection, mixed-layer deepening, upwelling) in producing thermocline and SST variability is not settled; (3) the single-parameter linear-feedback mixed-layer representation ignores seasonal modulation, interactions within the ocean, and higher-order anomaly-pattern interactions, and quantifying how these modify the spectral relations and the effective parameters ($\lambda$, $D$) remains to be done; and (4) the atmospheric model used underestimates low-frequency and high-wavenumber forcing levels, so input-response energy budgets from simulated forcing are not definitive.

Problem Statement

The problem, as stated in the concluding discussion of Frankignoul & Hasselmann (1977), is to determine whether the observed variability of the seasonal thermocline in midlatitude oceans - and, as the paper further asks, whether a significant part of the mesoscale eddy energy near the surface (and possibly deeper) - can be explained by local short-time-scale stochastic atmospheric forcing (Ekman pumping driven by the curl of the wind stress), or whether internal oceanic processes dominate, in the paper's own regional framing: local atmospheric forcing 'should be regarded as a serious candidate' in 'certain regions of the ocean', and this attribution 'should be investigated further'. Resolving this attribution requires establishing whether the atmospherically forced response reproduces the observed spectra to within the uncertainties of the measured input and response spectra, so that the variability is accounted for by local atmospheric forcing, or whether a significant residual variance must instead be attributed to internal oceanic processes; establishing it for a single qualifying midlatitude region settles the question for that region, while the analogous determination across the source's remaining 'certain regions of the ocean' is the broader open agenda.

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