We present an analytical model reproducing literature‐based numerical simulations of the Marine Atmospheric Boundary Layer (MABL) over a SST front, with wind blowing from the cold to the warm side. Turbulence is parameterised through a varying diffusion coefficient with two critical features: it is parabolic on the vertical and its mean value is decoupled from the MABL height (unlike an Ekman layer model). These two novel features are found essential to recover the internal structure of the MABL from numerical simulations. Different dynamical regimes are obtained and interpreted in terms of non‐dimensional numbers characterising the relative importance of terms driving the momentum equation. A closed‐form expression of the vertically integrated wind divergence in the MABL is then obtained. The resulting divergence is linearly linked to the SST Laplacian and to the downwind SST gradient. This shows that the response of the MABL wind divergence to a SST front is highly dependent on its spatial scale. The coupling coefficients vary with the ratio of MABL height to turbulence strength, i.e. the inverse Ekman number. We further show different regimes in the rate of variation of the coupling coefficients, depending on the Ekman number value. This can result in qualitatively different vertical winds, having potential implications for the coupling of the MABL with the free troposphere.