A comparative study of linear and step forcing temperature profiles in horizontal convection

We present a numerical study of horizontal convection using both step and linear forcing profiles as the imposed surface thermal driving, the former of which is commonly used in many studies but the latter is closer to the real ocean. Our investigation focuses on how forcing profiles can affect the theoretical and experimental results previously obtained based on the step forcing profile. We show that both types of forcing profiles exhibit similar behaviors in terms of global heat transfer (Nu ), mean flow strength (Re ), and boundary layer thickness. However, these two forcing profiles show significant differences for some other physical quantities, such as the generation rate of available potential energy G APE and global thermal dissipation rate, for which exact relationships with Nu can be derived for the step forcing profile but not for linear forcing. We show that a relationship between Nu and G APE can be developed for linear forcing by defining an effective forcing area, and such area is found to be approximately constant. We further derive an approximate relationship between Nu and the global thermal dissipation rate for linear forcing by assuming that the boundary heat flux is proportional to the local temperature deviation from the mean, which works best under the condition of moderate plume emissions. Differences are also found for the viscous dissipation rate; however, these differences diminish for sufficiently high Ra . Though step forcing is obviously less appropriate for modeling ocean circulations, we show that most differences arising from the forcing profile difference can be reconciled. Considering its convenience in experiment control and theoretical analysis, step forcing is a good configuration for most problems, but one should also be aware of the differences between step and other forcing types in quantities like the global thermal dissipation rate when extrapolating the results to more general situations.