2. Temperature, Pressure, Wind, and Minor Constituents.

Transcription

1 2. Temperature, Pressure, Wind, and Minor Constituents. 2. Distributions of temperature, pressure and wind. Close examination of Figs of MS reveals te following features tat cry out for explanation: In te zonal mean cross sections (Figs..7 and.8), zonal mean wind (positive from te west, or westerly) increases wit eigt were temperature decreases toward te pole. Seasonal mean and instantaneous wind direction at mid-troposperic and stratosperic eigts follows contours of constant eigt on isobaric suaces (Figs..9-.2). Seasonal mean and instantaneous wind speed at mid-troposperic and stratosperic eigts tends to be inversely proportional to spacing of contours (Figs..9-.2). As a consequence of te preceding two bullets, te relationsip between te wind and eigt contours on an isobaric suace tends to resemble flow between walls of a cannel; owever, since te contours evolve over time, tis flow is more aptly described as flow canneled by "flexible membranes". Several oter features are noteworty in Fig..7, e.g., decrease of tropopause eigt toward te poles, increase of lower stratosperic temperature toward te poles, decrease of mid-stratosperic temperature from summer to winter pole at solstice seasons, and increase of mesopause level temperature from summer to winter pole at solstice seasons. Because of te first bullet above, tese termal features are associated wit te following noteworty features in Fig..9 and.0: jets (closed contour wind speed maxima in te latitude eigt plane) at te mid-latitude tropopause, and stratospere-mesospere summer easterlies and winter westerlies wit jets in te upper mesospere. Figs. 8 and 9 reveal equatorward and poleward meanders of te prevailing westerly winds wit sarper features of smaller scale on te daily map. On te montly mean map, ridges (poleward meanders) tend to occur near nort-sout mountain ranges or continental west coasts, and trougs (equatorward meanders) tending to occur over continental east coasts and western oceans. Te patterns depicted in Fig. 8 are typical of autumn, winter, and spring. Weaker patterns, displaced toward iger latitude, occur in summer. Figs..-.2 sow similar features, but te orizontal scale of te meanders tends to be larger in te stratosperic maps. In te next two sections we will investigate te mecanism responsible for te features noted in te four bullets above. 2.2 Geostropic balance. If te planetary rotation rate is fast enoug, pressure and orizontal wind fields are in approximate geostropic balance as a result of te coriolis force. To analyze tis balance,

2 2 we will use a cartesian coordinate system tat is locally tangent to te planet's suace wit coordinate directions, unit vectors, and wind velocity components as follows: Coordinates Unit vectors Velocity components x à i u eastward y à j v nortward z à k w upward Te orizontal wind is v = Ãiu+ Ãjv. (2.) On te large scales of te troug ridge systems in MS Figs.9-.2, te vertical wind component is at least two orders of magnitude smaller tan te orizontal wind components, so we consider ere only te orizontal components. Tese are driven by te orizontal pressure gradient, wic is a force per unit volume Ñ p = à p p i + à j x y. (2.2) 2p - - If W= = x0 4 s for Eart and j= latitude, geostropic rotation - period balance is given in terms of te coriolis parameter, f º 2Wsinj: 2Wsinj kxv à = - Ñ p (2.3) r geostropic balance (Note tat rotation period is te period of te sidereal day, not te solar day). Rearranging (2.3), v = kx à Ñp p u =- y p v =. x (2.4)

3 3 Tis geostropic wind as te properties tat: (a) orizontal wind is parallel to pressure contours on a geopotential suace, (b) wind speed is approximately inversely proportional to contour spacing (proportional to pressure gradient), and as a consequence of (a) and (b), (c) as pressure contours evolve over time, te flow evolves wit it (approximately) as if te pressure contours act like "flexible membranes", canneling te flow. Wy is geostropic balance a good approximation? To answer tis, we proceed in two steps, first deriving te coriolis effect in te equation of motion using a euristic approac (see MS Cap. 0 for a more formal derivation), and second peorming a simple scaling analysis to identify te dominant terms in te equation of motion. For our euristic derivation, consider separately te coriolis force acting on east-west and on nort-sout flow relative to a coordinate frame rotating wit te planet. Te effects on an arbitrary orizontal flow can be obtained by superposing tese two effects. (a) On eastward (westward) flow, te coriolis force appears as excess (deficit) centrifugal force relative to te centrifugal force acting on a resting object due to planetary rotation. Te magnitude of tis force is obtained by expanding te total centrifugal acceleration, recognizing tat atmosperic wind speeds are small compared wit te rotational speed of te planet. Tus total centrifugal acceleration is 2 2 u WtotR W W W R R 2 = æ + ö = R + 2 è ø u + HOT 2 were W tot is total rotation rate, due to bot planet and zonal wind, R = acosj is radial distance outward from te planetary rotation axis, u is relative eastward velocity, and HOT represents a iger order term in te expansion tat is muc smaller tan te explicitly expressed terms. Term W 2 R is te planetary centrifugal acceleration, acting radially outward from te rotation axis. Tis term as already been incorporated in apparent gravity and in our use of geopotential suaces as vertical coordinate suaces. Term 2Wu is te coriolis acceleration acting on te eastward relative motion speed u. Since it acts radially outward, it as two components in te tangent plane (x,y,z) coordinate system: 2Wucosjupward and - 2Wusinj poleward (note tat since sin j canges sign between emisperes, tis acceleration is in te -y direction in te nortern emispere, and in te +y direction in te soutern emispere). Te HOT are real but small effects due to curvature of orizontal motion on te (nearly) sperical planet. Tey need not concern us ere. Since we are only concerned wit orizontal motions and accelerations in te orizontal, te vertical component of te coriolis force (wic is typically 4 orders of magnitude smaller tan gravitational acceleration) also need not concern us ere. We can write te partial y-component equation of orizontal motion including inertial and coriolis accelerations as dv =- 2Wusin j º-fu. (2.5)

4 4 (b) On equatorward (poleward) motion, coriolis force appears as a westward (eastward) acceleration arising as an effect of angular momentum conservation. Imagine looking down on Eart's Nort pole from space. Te projection on te equatorial plane of te displacement vector for a soutward moving object would be seen to be radially outward from te rotation axis. Te tendency for suc an object to conserve angular momentum as it increases its radial distance from te axis would produce a westward velocity increase (decrease in angular velocity) in proportion to te outward displacement of te object dr. Te corresponding cange in velocity over a small time interval would appear as westward acceleration in proportion to te rate of radial displacement, in oter words, in proportion to radial velocity. Tis can be analyzed beginning wit te angular momentum conservation principle tat implies tat W tot R 2 is conserved following te motion of te object. Expanding tis expression for a small radial displacement dr from initial values (, R ), W init init u WinitR 2 æ d ö init = Winit + ( init Winit init init Winit init R R + d R ) 2 = R 2 + R d u + 2 è ø R d R + HOT. Dividing by and taking te limit 0, since du dr =- 2W = 2Wvsinj- 2Wwcosj» fv (2.6) dr dy dz =- sinj + cosj =- vsinj+ wcosj and te contribution from te very small vertical motion can be neglected. [Note tat te time derivatives in tese equations are derivatives following te fluid flow or "substantial derivatives"]. Adding te pressure gradient contributions to (2.5) and (2.6) and dividing by density to convert from force per unit volume to force per unit mass (acceleration), we obtain te inviscid orizontal equations of motion (except for small terms due to Eart's curvature as mentioned above; inviscid implies neglect of any terms due to eiter molecular viscosity or te effects of small scale turbulence). du p - fv = - r x dv p + fu = - r y dv + fkxv à = - Ñp r (2.7) I II III

5 5 We can now do a simple scale analysis on te vector form of tese equations (last equation in te box of eq. 2.7). If any of te tree terms in te equation (inertial acceleration, coriolis acceleration, pressure gradient acceleration) can be sown to be muc smaller tan eiter of te oter two terms, ten te two larger terms must be in approximate balance. To see wic is te smaller term under te conditions sown in te large scale maps in MS Figs. 9-2, we need only compare terms I and II above by assuming velocity, lengt, and time scales V, L, and T. For large-scale atmosperic motions, te time-scale caracterizing canges in te system is T = L/V. Tis is te time required to cange te flow by advection (transport of fluid properties by te flow). Ten te scale of term I in (2.7) is V 2 /L, and te scale of term II is fv. Te ratio of tese terms is Ro V =. fl Tis is an important parameter caracterizing te flow called te Rossby Number. For - large-scale motions in Eart's atmospere (except at very low latitudes), V» 0ms, L» m, f» 0 4 s so tat Ro V = <<. (2.8) fl Tat is, coriolis acceleration is muc larger tan inertial acceleration and tere must be an approximate balance between terms II and III in eq In oter words, eq. 2.4 olds wit an error of order Ro, and te coriolis acceleration on te rapidly rotating Eart is strong enoug to ensure tat Ro<< for large-scale motions except in very low latitudes. Te resulting approximate balance between pressure and coriolis accelerations defines te geostropic wind of eqs. 2.3 and 2.4. We will use v g to denote te geostropic wind. Note tat, to te extent tat spatial variations in r and f can be neglected, te geostropic wind is approximately non-divergent, Ñ vg» Ñ ( kx à Ñp)= 0. Tis as two implications. First, te geostropic wind tends to follow contours on orizontal suaces of a scalar field (in tis case pressure), wic acts as a streamfunction for te flow, i.e., te contours of stream-function act like "flexible membranes". Second, since vertical velocity is connected wit divergence of te orizontal wind, large-scale vertical velocity is constrained to be small (even smaller tan te geometric constraint of a large ratio of orizontal to vertical scale would imply).