The Quasigeostrophic Assumption and the Inertial-Aective Wind The full equation of horizontal motion relates the changes in the horizontal wind vector d V h to pressure gradient and Coriolis accelerations (discounting friction). The two component equations of horizontal motion in Cartesian coordinates that result are du = g z x + fv (1) x component = g z y fu (2) y component The u-component of the geostrophic wind is obtained from (2) u g = g f z y (3)
divide both sides of equation (2) by f and substitute (3) into the result and solve for the acceleration. = f (u g u) (4) This equation states that there will be northward or southward acceleration of the air parcel if the real wind differs from the geostrophic wind. Let s assume that we have initially only zonal flow (a jet stream in the upper troposphere that lies along a line of latitude) that is unaccelerated (i.e., geostrophic flow). Then equation (1) is zero. The issue now is whether we can anticipate the development of any north-south motions given the fact that we have balanced (geostrophic flow). Equations (2) and (4) say that we can. The real wind can always be broken into a geostrophic and an ageostrophic component u = u g + u a v = v g + v a (5a) (5b) Substitute (5a) into (4) = fu a = f (u g u) (6) 2
Equation (6) gives the ageostrophic wind that occurs if the wind is not in geostrophic balance. Note that the far right hand side of (6) mathematically says that subgeostrophic flow will be associated with northward accelerations and vice versa. This ageostrophic wind is called the inertial aective wind (you ll see why below). The quasigeostrophic assumption is that we try to add back some of the acceleration to the left hand side of the full equation of horizontal motion in which the acceleration is assumed to be zero for geostrophic flow. To do that, we replace V in the full equation of horizontal motion on the left hand side with the geostrophic wind. This gives, when one expands out the Lagrangian derivative V g dv d = V g t + V g (7) in which the subscript h (denoting horizontal) is dropped. Note that we replaced the real wind with the geostrophic wind and did not include the ageostrophic portion of the wind. By replacing the real wind with the geostrophic wind on the right hand side, we re saying that the geostrophic wind almost is the same as the real wind. This adds back a bit of the acceleration (which we have found out is related to divergence) that the geostrophic assumption strictly does not allow (since, except for the effect of the northward variation of the Coriolis parameter, the geostrophic wind is non-divergent). Note that even if we assume that the geostrophic wind is unaccelerated, there can be local changes in the geostrophic wind. 3
V g t = V g The term to the right of the equals sign is essentially the aection of the geostrophic wind by itself. In other words, as an air parcel in geostrophic balance moves into a region with a different pressure gradient, it will initially have its initial speed that will be out of balance with the pressure gradient. (This is sort of like a ball rolling down an inclined plane to a flat surface, proceeding along the flat surface, where there is no gradient, by its own momentum). The first term to the right of the equals sign in (7) is the local change of the geostrophic wind. This is due to changes in pressure gradients (in turn due to isallobaric effects), and the motion of troughs and ridges, for example. If we start with the assumptions that the wind flow is zonal, this term is small or zero. With these assumptions, equation (7) becomes d V V g (8) Putting (8) back into (6) we get 4
! V g! V g = fu a = f (u g u)! V g! V g = fu a = f (u g u) (9a,b) which states that the magnitude of the aection of the horizontal geostrophic wind by itself is associated with an acceleration (even though the wind is geostrophic) that can be estimated by a term that is proportional to the difference of the real wind from the geostrophic wind. The conceptual interpretation of (9b) is that positive aection of inertia (when winds are supergeostrophic) is associated with southward accelerations. This occurs when faster winds move into regions of lower pressure gradients (in exit regions of jet streaks). Negative aection of inertia (when winds are subgeostrophic) is associated with northward accelerations. This occurs when slower winds move into regions of higher pressure gradients (in entrance regions of jet streaks). This ageostrophic wind in equation (9a,b), u a, is called the inertialaective wind because it is proportional to the magnitude of the aection of the geostrophic wind by itself, essentially the aection of the wind s intertia. It can be viewed as what the air parcel has to do to get into geostrophic balance as it moves into a region of pressure gradient different than that which it was in initial balance. The left hand side of (9a,b) is quasigeostrophic because we see that there are some accelerations that develop even though we make the assumption that the wind is geostrophic. We ll now apply (9a,b)) this to the real atmosphere by taking your first look at jet streak dynamics. 5