## 1. Introduction

[2] The Bowen ratio,

partitions the total turbulent surface heat flux into contributions from sensible heat (*H _{s}*) and latent heat (

*H*). As a result, the Bowen ratio occurs repeatedly throughout micrometeorology [e.g.,

_{L}*Panofsky and Dutton*, 1984, pp. 92ff., 132, 186;

*Garratt*, 1992, pp. 36, 130ff.;

*Lewis*, 1995].

[3] A common use for the Bowen ratio is in the *Bowen ratio and energy budget method* [e.g., *Fleagle and Businger*, 1980, p. 290ff.; *Brutsaert*, 1982, p. 210; *Arya*, 1988, p. 191; *Stull*, 1988, p. 274; *Drexler et al*., 2004; *Guo et al*., 2007]. We represent the surface energy budget as

Here *Q _{S}* and

*Q*are the shortwave and longwave radiative fluxes at the surface, respectively. A down-arrow represents incoming radiation; an up-arrow, outgoing radiation.

_{L}*G*is the conductive flux and is positive downward from the surface. In (2), the radiative terms are all taken as positive;

*H*and

_{s}*H*are positive when the flux is from surface to air. Positive terms in (2) thus warm the surface; negative terms cool it.

_{L}[4] If we represent the sum of the radiative terms as the net radiation, *R _{net}* ( ), (1) and (2) let us partition the available energy at the surface, , into sensible and latent heat fluxes:

That is, this method provides the turbulent fluxes without turbulence measurements.

[5] *Oncley et al*. [2007] and *Foken* [2008], among others, have reported that the energy budget, (2), is often not balanced over land. If this were the general case, (3) would not be useful. In contrast, for our domain, which is sea ice and the open ocean, *Persson et al*. [2002], *Cronin et al*. [2006], and *Persson* [2012], for example, found that the energy budget is balanced within the limits of the experimental uncertainty. Hence, (3) should apply.

[6] Besides its relevance to the energy budget, other uses of the Bowen ratio are in interpreting sonic anemometer data [*Schotanus et al*., 1983; *Andreas et al*., 1998] and in specifying the Obukhov length, the stratification parameter in the atmospheric surface layer, when the latent heat flux is unknown [e.g., *Busch*, 1973; *Andreas*, 1992]. *Wesley* [1976], *Kunkel and Walters* [1983], *Andreas* [1988], and *Green et al*. [2001] showed how electromagnetic propagation in the surface layer is sensitive to the Bowen ratio.

[7] In bulk flux algorithms (also known as the bulk aerodynamic method), the turbulent surface heat fluxes are usually parameterized as [e.g., *Fairall et al*., 1996, 2003; *Andreas et al*., 2008, 2010a, 2010b]

Here *ρ* is the air density; *c _{p}*, the specific heat of air at constant pressure;

*L*, the latent heat of vaporization or sublimation;

_{v}*S*, an effective wind speed at reference height

_{r}*r*; Θ

*and*

_{r}*Q*, the potential temperature and specific humidity at

_{r}*r*, respectively; and

*C*and

_{H}_{r}*C*, the transfer coefficients for sensible heat and latent heat, respectively, appropriate for height

_{E}_{r}*r*.

[8] Finally in (4), Θ* _{s}* and

*Q*are the potential temperature and specific humidity at the surface. In this work, our data come from open and ice-covered oceans. These are

_{s}*saturated surfaces*such that

*Q*is computed as the saturation specific humidity at temperature Θ

_{s}*. Other saturated surfaces include large lakes and reservoirs, extensive snow fields, and large glaciers. Our results apply to all such saturated surfaces.*

_{s}[9] From (4) and (1), we can also represent the Bowen ratio as

Thus, if we know the differences Θ* _{s}* − Θ

*and*

_{r}*Q*−

_{s}*Q*and have measured either

_{r}*H*or

_{s}*H*, we can calculate the other flux by knowing the Bowen ratio (if we also know

_{L}*C*and

_{H}_{r}*C*or assume they are equal). Notice also that the signs of Θ

_{E}_{r}*− Θ*

_{s}*and*

_{r}*Q*−

_{s}*Q*dictate the signs of

_{r}*H*,

_{s}*H*, and

_{L}*Bo*.

[10] Over saturated surfaces, the Bowen ratio is constrained. *Philip* [1987] established the theoretical constraint on the Bowen ratio for the case *H _{s}* > 0 and

*H*> 0 under the assumption that the near-surface humidity is not above its saturation value, i.e., no fog.

_{L}*Andreas*[1989; see also

*Philip*, 1989] extended Philip's ideas to also formulate constraints for the cases

*H*< 0,

_{s}*H*< 0 and

_{L}*H*< 0,

_{s}*H*> 0.

_{L}*Andreas and Cash*[1996] subsequently tested all three of these constraints using data collected over surfaces such as the open ocean, Arctic and Antarctic sea ice, Lake Ontario, marginal seas, and snow-covered ground.

*Andreas and Jordan*[2011] continued this type of analysis but used two large data sets collected over sea ice.

[11] Here we add to the *Andreas and Jordan* [2011] data sets a comparably sized data set comprising 13 distinct sets collected over various open ocean regions. In our combined data set, surface temperatures range from −44°C to 32°C and govern the value of the Bowen ratio. For well over 90% of the time in all the individual data sets, the measured sensible and latent heat fluxes collect into one of the three regimes: *H _{s}* > 0 and

*H*> 0,

_{L}*H*< 0 and

_{s}*H*< 0, and

_{L}*H*< 0 and

_{s}*H*> 0.

_{L}[12] As in *Andreas and Cash* [1996], we define a Bowen ratio indicator function *Bo*_{*} that depends approximately exponentially on surface temperature. In each of the three heat flux regimes, is similar in magnitude to *Bo*_{*} and has the same dependence on surface temperature. Moreover, the data, on average, support the three constraints formulated by *Philip* [1987], *Andreas* [1989], and *Andreas and Cash* [1996]. That is, when *H _{s}* > 0 and

*H*> 0, 0 <

_{L}*Bo*≤

*Bo*

_{*}; when

*H*< 0 and

_{s}*H*< 0, ∞ >

_{L}*Bo*≥

*Bo*

_{*}; and when

*H*< 0 and

_{s}*H*> 0,

_{L}*Bo*≈ −

*Bo*

_{*}. Finally, we make these constraints formal by developing simple, empirical relations that predict

*Bo*from

*Bo*

_{*}in each of the three regimes.