The Relationship between the Surfa Res

The surface stress is computed by the method of geostrophic departures using 23 days of double theodolite wind observations at Shilo, Manitoba. The main results are as follows. Relations between the stress and the geostrophic wind fit these data better than do similar expressions using the surface wind. The stress varies linearly, rather than quadratically, with wind speed. For a given surface or geostrophic wind speed, the stress increases with increasing Richardson number and warm air advection. This paper is concerned with computing the stress ‘ P ~ (force per unit area) exerted by the atmosphere on the earth. Some simple relationships that could readily be incorporated into a numerical weather prediction model are examined. This study was to some extent motivated by a recent study of one of the authors (Danard 1969). In the latter papor, it was demonstrated that a cyclone would experience s i m c a n t changes in its velocity fields and rate of frictional filling as it moved over a surface of varying roughness (for example, land to sea). Consequently, if ti numerical model is to simulate the behavior of such cyclones, the surface stress must be carefully calculated. In section 2, an independent method of computing 1p0 is discussed (the method of geostrophic departures). The data used and sources of error are discussed in sections 3 and 4, respectively. Section 5 is concerned with relationships between ‘ P ~ and Vo (surface wind) and V, (geostrophic wind). For the data sample of this study, 8, is a better predictor than Vo for ‘P~ , and a linear dependence of ‘ P ~ on wind speed is preferable to a quadratic one. Dependence on Richardson number and temperature advection is examined in section 6. For a given surface or geostrophic wind speed, the stress is larger for warm air advection or high Richardson number than for cold air advection or low Richardson number. The basic data used in this study are presented in the appendix. 9. METHOD OF COMPUTING 7 0 The method is that of “geostrophic departures” used by Sutcliffe (1936), Sheppard and Qmar (1952), and others. A short description is relevant here. Ignoring lateral diffusion, one may write the equation of motion as where V is the horizontal wind and T is the horizontal stress vector. Integrate (1) from the earth’s surface (2=0) to some height z=h: T O = l , Blfkx(v-vg)+z d7 dz+Th* (2) In (2), ‘Ph is the stress at z=h. The height k will now be identiiied with the top of the ]Ekman Payer, and ‘Bh d be assumed zero. Furthermore, the acceleration term will be assumed negligible compared to the others. Then equation (2) .reduces to