| Summary: | Flux–profile relationships are crucial for parametrizing surface fluxes of momentum and heat, that are of central relevance for applications such as climate modelling and weather forecast. Nevertheless, their functional forms are still under discussion, and a generally accepted formulation does not exist yet. We reviewed the four main formulations proposed in the literature so far and assessed how they affect the theoretical behaviour of the kinematic heat flux (<inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mi>H</mi><mn>0</mn></msub></mrow></semantics></math></inline-formula>) and the temperature scale (<inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mi>T</mi><mo>*</mo></msub></mrow></semantics></math></inline-formula>) in the stable boundary layer, as well as their consequences on the existence of critical values for both the gradient and the flux Richardson numbers. None of them turned out to be fully consistent with the literature published so far, with two of them leading to very unreliable expressions for both <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mi>H</mi><mn>0</mn></msub></mrow></semantics></math></inline-formula> and <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mi>T</mi><mo>*</mo></msub></mrow></semantics></math></inline-formula>. All considered, a convincing description of flux–profile relationships still needs to be found and seems to represents a considerable challenge.
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