Vulnerability to Suicide

Consistent approximate models of the global atmosphere: shallow, deep, hydrostatic, quasi‐hydrostatic and non‐hydrostatic

“…One such formulation, which we call the r = a model, retains the cos φ Coriolis terms and the spherical metric terms but sets r = a wherever it occurs in undifferentiated form in the NHD equations; it is briefly and transitorily discussed in the textbooks of Gill (1982) (see p. 93), Holton (1992) (p. 37) and Dutton (1995) (p. 233). The PV conservation principle of the r = a model, noted here in appendix C (see (C.13)), contains terms that have no analogue in the unapproximated case.…”

“…Equation (2.9) is Ertel's theorem; see Gill (1982), Pedlosky (1987) and Dutton (1995). The governing equations and the relations that follow from them are vector invariant forms that hold irrespective of coordinate system.…”

“…Rather than via the Lagrange route adopted here (in appendix A), (2.17)-(2.19) are more usually derived by separating the zonal, meridional and vertical components of the momentum equation (2.2) after appeal to the spherical geopotential approximation; see Gill (1982), Holton (1992) and White (2003). With the understanding that they incorporate the spherical geopotential approximation, we call (2.6), (2.7) and (2.16)-(2.20) the non-hydrostatic, deep (NHD) equations.…”

“…The terms on the right-hand side represent the forces acting (per unit mass): Coriolis (−2 × u), centrifugal (− × ( × r)), Newtonian gravity (−∇ N ), pressure gradient (−α∇p), and all other forces (G). For discussion of (2.1) as an expression of Newton's second law of motion see, for example, Lorenz (1967), Phillips (1973, Gill (1982), Pedlosky (1987), Holton (1992 and Dutton (1995). As described in these references, it is usual to combine the centrifugal force (− × ( × r)) with Newtonian gravity (−∇ N ) to give apparent gravity, −∇ A , where A = N − 2 s 2 /2 is a new scalar potential, s being perpendicular distance from the axis of rotation of J.…”

“…In the deep oceans the buoyancy frequency is typically an order of magnitude less than in the atmosphere (see Gill (1982), p. 52), so the cos φ Coriolis terms are correspondingly more important because the ratio R is larger. The ocean model of Marshall et al (1997) can include in a consistent fashion all of the Coriolis terms as well as non-hydrostatic effects.…”