dx; dxi where P is the thermodynamic pressure, n and n' are the kinematic and volume viscosities* and S; is the Kronecker delta (Hinze, 1975). In the last term on the left-hand side of (2.28), the divergence operator div = Y^jd /dxj is supposed to act on all three factors v'jp v'i following in (2.46).
It is clear from the equation of motion (2.46) that the large-scale motion of the atmosphere, v*, cannot generally be described by an equation that only depends on averaged quantities (for a discussion of early attempts to achieve such a description, see Lorenz, 1967). The divergence term on the left-hand side describes the influence of small-scale turbulent motion on the large-scale motion. It is usually referred to as the eddy transport term.
Apart from the well-known pressure gradient term, the contributions from the stress tensor describe the molecular friction forces. These may be important for calculations of energy transformations, but are often left out in calculations of general circulation (Wilson and Matthews, 1971). According to Boussinesq (1877), it may be possible to approximate the eddy transport term by an expression of the same form as the dynamic viscosity term [see (2.59)], introducing an effective "eddy viscosity" parameter similar to the parameter k considered above in connection with the turbulent transport of heat from the Earth's surface to the boundary layer [i.e. the so-called Prandtl layer, extending about 50 m above the laminar boundary layer and defined by an approximately constant shear stress T in expressions such as (2.33)].
The difference between the scales of vertical and horizontal motion makes it convenient to define separate velocities w and V, v = (w$ez)ez + (v - (vez)ez) = wez + V, and split the equation of motion (2.46) into a horizontal part, d V "
and a vertical part, which as mentioned can be approximated by the hydro-
The viscosities are sometimes taken to include the factor p.
static equation, dp dz
In (2.47), the molecular friction forces have been left out, and only the vertical derivative is kept in the eddy term. This term thus describes a turbulent friction, by which eddy motion reduces the horizontal average wind velocity V*. The last term in (2.47) is due to the Coriolis force, and f = 2 Q sin where Q is the angular velocity of the Earth's rotation and Q is the latitude.
As a lowest-order approximation to the solution of (2.47), all the terms on the left-hand side may be neglected. The resulting horizontal average wind is called the geostrophic wind, and it only depends on the pressure gradient in a given height and a given geographical position. The smallness of the left-hand terms in (2.47), relative to the terms on the right-hand side, is, of course, an empirical relationship found specifically in the Earth's atmosphere. It may not be valid for other flow systems.
The large-scale horizontal wind approaches the geostrophic approximation when neither height nor latitude is too small. At lower heights the turbulent friction term becomes more important (up to about 1000 m), and at low latitudes the Coriolis term becomes smaller, so that it no longer dominates over the terms involving local time and space derivatives.
In analogy to (2.46), the averaging procedure may now be applied to the general transport equation for some scalar quantity, such as the mixing ratio for some minor constituent of the atmosphere. Equation (2.46) then becomes d A * d A * -+(V * -grad )A *+w *-
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