Feedbacks and eddy diffusivity in an energy balance model of tropical rainfall shifts

Atmospheric circulations transport both water and energy, linking the spatial patterns of precipitation to those of atmospheric energy transport. Using moist energy frameworks that include latent heat enables diagnosis of the position and intensity of tropical precipitation without explicitly resolving latent heating in precipitation. These frameworks have clarified many tropical phenomena and, importantly, revealed that the latitude of the ITCZ is sensitive to high-latitude anomalies in atmospheric moist energy sources. In steady state, the zonal-mean ITCZ tends to shift toward an anomalous energy source, opposite to the direction of the cross-equatorial energy flux required to balance the forcing. The ITCZ shift can be quantified via the energy flux equator (EFE), defined by the zero of zonal-mean meridional energy transport, and related to imposed energy sources through empirical and Taylor-series-based approaches. Most such frameworks are diagnostic, requiring knowledge of net energy source anomalies, because feedbacks (radiative and surface turbulent) alter the energy balance and required energy transports. Feedbacks can be large: ocean heat transport damps ITCZ shifts, water vapor provides a positive feedback, and clouds can be positive or negative depending on longwave versus shortwave effects. This study uses a diffusive moist energy balance model (MEBM) to investigate how individual radiative feedbacks (Planck, water vapor, lapse rate, albedo) and the meridional structure of eddy diffusivity influence ITCZ/EFE shifts caused by localized forcings. It is further motivated by prior idealized GCM work, including Clark et al. (2018), which isolated the role of water vapor feedback under tropical versus extratropical forcings.

Prior work using moist energy frameworks has related tropical precipitation shifts to anomalous atmospheric energy sources and transport, introducing the EFE as a diagnostic of ITCZ latitude (e.g., Donohoe et al. 2013; Bischoff & Schneider 2014; Adam et al. 2016). Studies show that the ITCZ shifts toward anomalous energy sources and that regional precipitation shifts can be interpreted with energy budgets. Feedbacks complicate prediction: ocean heat transport damps ITCZ shifts; water vapor typically provides a positive feedback akin to its role in global climate sensitivity; and cloud feedbacks can be of either sign, varying across models (e.g., Kang et al. 2008, 2009; Voigt et al. 2014). Clark et al. (2018) systematically examined the water vapor feedback’s role in ITCZ shifts for tropical and extratropical forcings. Additional literature on diffusive energy balance models highlights how spatially varying effective temperature diffusivity and feedbacks shape regional responses and polar amplification, informing expectations for Planck, lapse rate, and water vapor feedback behavior in zonal-mean settings.

Models: A one-dimensional moist energy balance model (MEBM) and a general circulation model (CESM2 aquaplanet) are used. The MEBM represents the steady-state, vertically integrated moist energy budget in sine-latitude x = sinφ: ∂/∂x[D(x) ∂h/∂x] = NEI, where D(x) is a prescribed meridional diffusivity of surface-air moist static energy h, and NEI = S(1 − α) − L is the net energy input (insolation S, surface albedo α, outgoing longwave radiation L). Cloud-radiative effects are omitted. Ice–albedo feedback is a step function of surface temperature. Longwave radiation is computed with RRTMG using prescribed vertical temperature and humidity profiles in single-column radiative transfer (30 levels), with surface air RH = 0.8 and idealized vertical/meridional RH structure that shifts with the EFE. The model is solved with second-order finite differences and multigrid V-cycles on a hierarchy of grids until convergence. Default uniform D = 2.6×10 kg m−2 s−1; experiments also impose diagnosed D(x) from CESM2 and idealized square-wave D1 (tropical maximum) and D2 (extratropical maximum) with the same global mean as the uniform D. Forcings: Following Clark et al. (2018), Gaussian insolation anomalies (energy sinks) are imposed centered at 15°N (tropical) or 60°N (extratropical) with several magnitudes, defined by global-mean −M = S′ and applied as net absorbed shortwave S′(1 − α) (without activating ice–albedo feedback for the forcing definition). Responses are diagnosed via EFE latitude φε, which correlates with ITCZ latitude in GCMs. Forcings are expressed as cross-equatorial transport in the absence of feedbacks by meridionally integrating the net absorbed shortwave anomaly after removing the global mean: Ts(1 − α)(x0) (Eq. 2), evaluated at the basic-state EFE x0 = 0. Sensitivity metric: Linear sensitivity λ = φε / Ts(1 − α) (Eq. 3) is computed using the smallest forcing (M = 5 W m−2) to minimize nonlinearity. Feedback quantification: A linear feedback analysis relates EFE shifts to cross-equatorial transports (Eq. 4): φε = λNF Ts(1 − α) + Σ(ci φε), with λNF the no-feedback sensitivity and ci the cross-equatorial transport per unit EFE shift due to feedback i (Planck PL, water vapor WV, albedo AL, lapse rate LR). Each ci is estimated by partial radiative perturbation: ci = (∂T/∂Xi)(∂Xi/∂φε) (Eq. 5), where ∂T/∂Xi is obtained from offline RRTMG calculations perturbing only Xi (e.g., specific humidity, albedo, lapse rate) about the unforced state, and ∂Xi/∂φε from full forced simulations. Feedback factors fi = λNF ci (Eq. 6) and the total sensitivity approximation λ ≈ λNF/(1 − Σfi) (Eq. 7) are evaluated. Suppressed-feedback MEBM runs hold fixed, from the control, either albedo (no AL), specific humidity and RH distribution (no WV), or vertical lapse rate (no LR); PL is not suppressed prognostically due to instability (net positive feedback). Relative humidity feedback parameterization: Based on CESM2 behavior, the MEBM’s default RH distribution (which only shifts meridionally with φε) is augmented by prescribing linear variations of the 600 hPa RH minima in each hemisphere with EFE latitude, using regressions from CESM2 across forcings. This introduces an EFE-linked cross-equatorial asymmetry in subtropical dry-zone RH. GCM configuration: CESM2 (CAM6 finite-volume, 1°×1°) aquaplanet with a 1-m mixed-layer slab ocean, equinoctial insolation averaged to match the MEBM’s S(x), uniform surface albedo of 0.2725, no sea ice, and cloud–radiative interactions disabled. Diagnostics include EFE–ITCZ relation (for CESM2: φE = 0.784 φITCZ, r2 = 0.996) and energetics from zonal means. Data/code: Simulation outputs available on request; MEBM code available at https://github.com/henrygrantpeterson/mebm.

• EFE sensitivity: The MEBM reproduces the general magnitude and qualitative behavior of EFE shifts in GCMs for both tropical and extratropical forcings. Sensitivity to extratropical forcings is smaller—about one-half to one-third of the tropical sensitivity—consistent with prior cloud-free studies. CESM2 exhibits nonlinearity: a 4.8° EFE shift for a tropical forcing of M = 5 W m−2, while tripling the forcing only doubles the shift. - Feedback contributions and signs: Water vapor (WV) and albedo (AL) provide positive feedbacks, Planck (PL) and lapse rate (LR) negative, on EFE shifts—matching their signs in global-mean climate sensitivity contexts. Removing WV strongly reduces EFE sensitivity for both tropical and extratropical forcings; suppressing LR slightly increases sensitivity (weak negative LR). AL has negligible effect for tropical forcings but is a moderate positive feedback for extratropical forcings. - Planck feedback explains weaker extratropical response: Feedback-factor analysis shows fi magnitudes are larger for extratropical than tropical forcings, with |fPL| roughly twice |fWV| extratropically, producing smaller net EFE shifts. The stronger extratropical PL damping arises because effective temperature diffusivity κ is smaller in colder regions, yielding larger, more localized temperature anomalies and hence more hemispherically asymmetric OLR changes for extratropical forcings. - Near-linearity of feedback decomposition: Offline radiative transfer components (PL, WV, LR, AL) superimpose to match NEI′ well, even for strong forcings (M = 15 W m−2). The total sensitivity is accurately approximated by λ ≈ λNF/(1 − Σfi). For tropical forcings, approximate cancellation between PL (negative) and WV (positive) leaves λ ≈ λNF; for extratropical forcings, stronger PL yields λ significantly smaller than λNF. - Role of water vapor RH asymmetry: In CESM2, ITCZ shifts are accompanied by substantial changes in the RH of subtropical dry zones beyond a simple meridional shift. Parameterizing this in the MEBM (linking 600 hPa RH minima to φε) nearly doubles the EFE sensitivity, exceeding GCM sensitivities. Thus, the WV feedback on EFE shifts is dominated by cross-equatorial asymmetry in subtropical dry-zone RH, with additional contributions from fixed-RH mixing ratio changes and the humid ITCZ’s meridional shift. - Diffusivity structure strongly modulates EFE shifts: Diagnosed D(x) from CESM2 varies by nearly an order of magnitude (midlatitude maxima, minima at equator and poles). Imposing this D(x) in the MEBM substantially weakens EFE sensitivity for both tropical and extratropical forcings—comparable to removing WV for tropical forcings. Idealized tests with square-wave diffusivities show sensitivities are largest when diffusivity is enhanced in the tropics (D1) and smallest when enhanced in extratropics (D2), for a fixed global-mean D. The near-equatorial diffusivity exerts the strongest control, mainly by allowing h′ and its gradients (and thus energy transport anomalies) to spread to the equator, increasing cross-equatorial transport anomalies for a given forcing. - EFE–ITCZ relation: In CESM2 aquaplanet runs, φE = 0.784 φITCZ (r2 = 0.996), consistent with but distinct from the C18 model’s φE = 0.64 φITCZ; analyses therefore focus on EFE shifts rather than converting to ITCZ shifts.

The study set out to understand how radiative feedbacks and the meridional structure of eddy diffusivity govern ITCZ (EFE) shifts in response to localized tropical and extratropical energy forcings. Results show that the smaller EFE shifts for extratropical forcings emerge primarily from a stronger Planck feedback in colder regions, where lower effective temperature diffusivity produces larger, more localized temperature anomalies and hence greater hemispheric asymmetry in clear-sky OLR. This mechanism operates without invoking rotational constraints and clarifies that the reduced extratropical sensitivity is feedback-driven rather than a direct consequence of diffusive transport alone. Water vapor provides the main opposing positive feedback, but unlike global-mean sensitivity where fixed-RH mixing ratio changes dominate, the EFE response is controlled by changes in the cross-equatorial asymmetry of subtropical dry-zone RH. Consequently, capturing RH structure changes is essential for predicting ITCZ shifts. Additionally, the EFE response is highly sensitive to the near-equatorial diffusivity of moist energy: larger tropical diffusivity increases EFE sensitivity by enhancing cross-equatorial transport anomalies and/or flattening the basic-state transport slope at the equator. Together, these findings indicate that both radiative feedbacks and the spatial structure of eddy diffusivity must be quantified to prognostically predict ITCZ/EFE shifts, and that compensation between biases in feedbacks and diffusivity can yield similar net sensitivities across models.

This work demonstrates that: (1) The Planck feedback damps ITCZ/EFE shifts substantially more for extratropical forcings than for tropical ones due to smaller effective temperature diffusivity in colder regions, (2) The water vapor feedback on ITCZ shifts is dominated by cross-equatorial asymmetry in subtropical dry-zone RH, not solely by fixed-RH mixing ratio changes or simple translation of the humid ITCZ, and (3) The EFE response is highly sensitive to the meridional structure of moist energy diffusivity, increasing with larger tropical diffusivity even at fixed global-mean D. The MEBM, corroborated by CESM2 aquaplanet experiments, provides a near-linear feedback framework in which total EFE sensitivity is well approximated by λ ≈ λNF/(1 − Σfi). The results imply that accurately predicting ITCZ/EFE shifts requires joint knowledge of radiative feedback strengths and the effective diffusivity acting on moist energy and temperature anomalies. Future work should incorporate cloud–radiative effects and the seasonal cycle, and further constrain state-dependent diffusivity to improve quantitative predictions and model–observation comparisons.

• No cloud–radiative effects are represented in either the MEBM or the CESM2 runs analyzed here, precluding assessment of cloud feedbacks. - The MEBM omits the seasonal cycle, land, and ocean heat transport, and has no explicit precipitation or ITCZ, relying on EFE as a proxy. - Default MEBM employs a uniform diffusivity; while alternative D(x) profiles are tested, state dependence and process origins of diffusivity are not dynamically simulated. - Linear feedback analysis and partial radiative perturbations assume near-linearity; some nonlinearity is evident at strong forcings and in specific latitude bands. - Suppressing the Planck feedback prognostically is unstable, so PL is only diagnosed offline. - Aquaplanet CESM2 with a 1-m slab ocean limits ocean–atmosphere coupling realism and excludes sea ice (in GCM) while the MEBM includes a simple ice–albedo parameterization.