We have investigated properties of the tidally induced bottom boundary layer (TBBL) in a shallow sea under a surface heating, by scale argument and direct numerical simulation (DNS) experiment. Applying the existing scalings of the boundary layer, it is found that the height of TBBL Htbbl\documentclass[12pt]{minimal}
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\begin{document}$$H_{\rm tbbl}$$\end{document} and the efficiency of tidal mixing ϵ\documentclass[12pt]{minimal}
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\begin{document}$$\epsilon$$\end{document} are scaled to (u∗4H/|σ+f|Bs)1/3\documentclass[12pt]{minimal}
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\begin{document}$$(u_{*}^{4} H /|\sigma +f| B_{\rm s})^{1/3}$$\end{document} and Hhom/Htbbl\documentclass[12pt]{minimal}
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\begin{document}$$H_{\rm hom}/H_{\rm tbbl}$$\end{document}, respectively, where u∗\documentclass[12pt]{minimal}
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\begin{document}$$u_{*}$$\end{document} is the friction velocity, σ\documentclass[12pt]{minimal}
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\begin{document}$$\sigma$$\end{document} the tidal frequency, f the inertial frequency (the Coriolis parameter), Bs\documentclass[12pt]{minimal}
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\begin{document}$$B_{\rm s}$$\end{document} the surface buoyancy flux, H the water depth, and Hhom=u∗/|σ+f|\documentclass[12pt]{minimal}
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\begin{document}$$H_{\rm hom}=u_*/|\sigma +f|$$\end{document} the height of TBBL in a homogeneous ocean. Results of DNS experiment agree with these scalings for fairly wide ranges of u∗\documentclass[12pt]{minimal}
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\begin{document}$$u_*$$\end{document} (or tidal amplitude Utide\documentclass[12pt]{minimal}
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\begin{document}$$U_{\rm tide}$$\end{document}), H, Bs\documentclass[12pt]{minimal}
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\begin{document}$$B_{s}$$\end{document}, and |σ/f|\documentclass[12pt]{minimal}
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\begin{document}$$|\sigma /f|$$\end{document}. In exceptional cases with slower Earth’s rotations, weaker tidal flows, and shallower water depths, turbulence occurs intermittently and the scaling underestimates Htbbl\documentclass[12pt]{minimal}
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\begin{document}$$H_{\rm tbbl}$$\end{document} and ϵ\documentclass[12pt]{minimal}
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\begin{document}$$\epsilon$$\end{document}. The efficiency of tidal mixing ϵ\documentclass[12pt]{minimal}
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\begin{document}$$\epsilon$$\end{document} varies from less than 1 to 7 % for the experimental range. This variation can partly explain the reason why the critical value of Simpson–Hunter parameter, which is an index of the position of tidal mixing front is different from place to place around the world.
