4 Primitive Equations

Size: px

Start display at page:

Download "4 Primitive Equations"

Owen Quinn
5 years ago
Views:

1 4 Primitive Eqations 4.1 Spherical coordinates Usefl identities We now introdce the special case of spherical coordinates: (,, r) (longitde, latitde, radial distance from Earth s center), with , - /2 6 6 /2, 0 6 r 6 1. Figre: Taken from Figre 2.3 of Vallis. The nit vectors of local Cartesian coordinates on the srface of the sphere, x (x, y, z) are defined in the direction of increasing (,, r) respectively: x, x, r r x (4.1) where r a + z (sm of the radis of the earth and altitde). The three-dimensional velocity is defined as v î + vĵ + wˆk x. We have in differential notation: dx r cos d, dy rd, dz dr (4.2) and therefore r cos, v r, w r z (4.3) E. A. Barnes 39 pdated 15:17 on Tesday 22 nd September, 2015

2 The divergence of a vector v in spherical coordinates is: î r v r + ĵ + ˆk@ r (î + ĵv + r cos (v cos ) r(r 2 w) r cos r 2 (4.4) The gradient of a scalar is: r r î + ˆk (4.5) Finally, recall that: î î ĵ ĵ ˆk ˆk 1 and î ĵ î ˆk ĵ ˆk 0 (4.6) Other identities can be fond on pages of Vallis Material derivatives In spherical coordinates the material derivative of a scalar qantity t + r (4.7) Inserting the expressions for, v, w in place of the material derivatives of,, r leads to the material derivative of a scalar t + r + v + w@ r (4.8) For a vector qantity (sch as velocity), the material derivative becomes more complicated since it now involves the material rate of change of the nit vectors themselves: î + ĵ + w ˆk + î + v ĵ + wˆk (4.9) For the material rate of change of, say, î, we first realize that the local rate of change is t î 0 (and of corse the same is tre for ĵ and ˆk). Frther, î does not change when moving p or down (this also holds for ĵ and ˆk), and neither does it change when changing latitde (this is not tre for ĵ or ˆk). Therefore, î r î r cos (ĵ sin - ˆk cos ) (4.10) The first term in the expression î reslts from the fact that latitde circles are crved poleward from the perspective of or Cartesian coordinate system attached locally to the srface of the sphere. This effect becomes stronger nearer to the poles bt vanishes at the eqator hence the sin factor. E. A. Barnes 40 pdated 15:17 on Tesday 22 nd September, 2015

3 The second term in the expression î reslts from the fact that the local vertical, except at the poles, has a component pointing radially away from the axis of rotation (perpendiclar to it) and hence rotates arond with a change in longitde. At the eqator this radial component eqals the local vertical the effect is maximized, whereas at the poles this radial component away from the axis of rotation is perpendiclar to the local vertical the effect vanishes; hence the cos factor. Now, since the nit vectors î, ĵ and ˆk are perpendiclar to each other at all times, the same terms that appear in the expression î mst also appear in the opposite sense in the corresponding expressions ĵ ˆk, ĵ ˆk î cos (4.11) There are no extra terms de to changes in ˆk in the expression ĵ or changes in ĵ in the expression ˆk. For the material rate of change of ĵ and ˆk we still need expression ĵ ˆk. These are given ĵ ˆk ĵ (4.12) Note that these terms do not involve latitde or longitde since meridians are always great circles and are circlarly symmetric. We therefore have: î ĵ ˆk r î r cos (ĵ sin - ˆk cos ) (4.13) r ĵ + v ĵ - r cos î sin - v r ˆk (4.14) r ˆk + v ˆk r cos î cos + v r ĵ (4.15) (4.16) and, hence, for the material derivative of the velocity vector in spherical coordinates: - v r tan + w r î r tan + vw r w ĵ v 2 ˆk (4.17) r where the î component is associated with w, the ĵ component with and the ˆk component with. The terms involving 1/r are called metric or crvatre terms. E. A. Barnes 41 pdated 15:17 on Tesday 22 nd September, 2015

4 4.2 Exact Primitive Eqations We now, finally, write-down the complete set of eqations of motion in spherical coordinates inclding rotational effects: - v r tan + w r + 2 (w cos - v sin ) - r (4.18) + 2 r tan + vw r + 2 sin (4.19) w v 2-2 cos - - g (4.20) t + r + v + w@ r + r v + r cos (v cos ) r(r 2 w) r cos r 2 0 (4.22) Q, T c p T R/cp p0 (4.23) p p p(, T) or p p(, T, S) e.g. p RT (4.24) This set of eqation is known as the exact primitive eqations. 4.3 Approximations to the Primitive Eqations Typically, when yo will hear the phrase primitive eqations, it will not be in the context of the exact primitive eqations discssed above. Instead, this phrase is sed to represent the set of eqations of motion with three related approximations: (1) the hydrostatic approximation, (2) the shallow-flid approximation, and (3) the traditional approximation. When applying approximations to the eqations of motion, it is important to ensre that the fndamental conservation laws (e.g. energy, anglar momentm conservation) are not violated by the approximated versions of the eqations. We will not go throgh this rigorosly here, however, come talk with me if yo are interested in learning how. 1. The hydrostatic approximation: In the vertical momentm eqation, the gravitational term is balanced by the pressre gradient term - g (4.25) Ths, the advection of vertical velocity, the vertical Coriolis term, and the metric term ( 2 + v 2 )/r are all neglected. E. A. Barnes 42 pdated 15:17 on Tesday 22 nd September, 2015

5 2. The shallow-flid approximation: We will re-write r a+z and assme that a >> z. The coordinate r is then replaced by a except where it is sed as the differentiating argment. Ths, for example, 2 w) @z (4.26) 3. The traditional approximation: Coriolis terms and metric terms in the horizontal momentm eqations involving the vertical velocity are neglected. The second and third of these approximations mst be taken (or not) together - as they both presme a small aspect ratio of the motion (vertical scales are mch smaller than horizontal scales). If we make one approximation and not the other, momentm and energy will not be conserved. Applying all three of these approximations to the exact primitive eqations leads to the qasi-static primitive eqations or often simply the primitive eqations. - v a tan - fv - a where + 2 tan + f - r (4.27) - g t + a + v + w@ r + r v + a cos (v cos ) r w 0 (4.31) a cos Q, T c p T R/cp p0 (4.32) p p p(, T) or p p(, T, S) e.g. p RT (4.33) E. A. Barnes 43 pdated 15:17 on Tesday 22 nd September, 2015

Momentum Equation. Necessary because body is not made up of a fixed assembly of particles Its volume is the same however Imaginary

Momentum Equation. Necessary because body is not made up of a fixed assembly of particles Its volume is the same however Imaginary Momentm Eqation Interest in the momentm eqation: Qantification of proplsion rates esign strctres for power generation esign of pipeline systems to withstand forces at bends and other places where the flow