  The diurnal variability of sea surface temperature (SST) may play an
important role for cloud organization above the tropical ocean, with
implications for precipitation extremes, storminess, and climate sensitivity.
Recent cloud-resolving simulations demonstrate how imposed diurnal SST
oscillations can strongly, and delicately, impact mesoscale convective
organization. In spite of this nuanced interaction, many idealized modeling
studies of tropical convection either assume a constant, homogeneous SST or, in
case of a responsive sea surface, represent the upper ocean by a slab with
fixed thickness. Here we show that slab ocean models with constant heat
capacity fail to capture the wind-dependence of observed diurnal sea surface
warming. To alleviate this shortcoming, we present a simple, yet explicitly
depth-resolved model of upper-ocean temperature dynamics under atmospheric
forcing. Our modular scheme describes turbulent mixing as diffusion with a
wind-dependent diffusivity, in addition to a bulk mixing term and heat fluxes
entering as sources and sinks. Using observational data, we apply Bayesian
inference to calibrate the model. In contrast with a slab model, our model
captures the exponential reduction of the diurnal warming amplitude with
increasing wind speed. Further, our model performs comparably to a more
elaborately parameterized diurnal warm layer model. Formulated as a single
partial differential equation with three key tuning parameters, the model is
suitable as an interactive numerical boundary condition for idealized
atmospheric simulations.
