Contents7.1 Kinematics of the large-scale horizontal flow............ 17.1.1 Elemen tary kinematic properties of the flow........ 37.1.2 Vorticityanddivergence .................. 37.1.3 Deformation ......................... 67.1.4 Streamlinesversustrajectories ............... 87.2 Dynamics of horizontal flow..................... 97.2.1 Apparentforces........................ 107.2.2 Realforces .......................... 157.2.3 Thehorizontalequationofmotion ............. 177.2.4 Thegeostrophicwind .................... 177.2.5 The effectoffriction..................... 197.2.6 Thegradientwind ...................... 217.2.7 Thethermalwind ...................... 227.2.8 Suppression of vertical motions by planetary rotation . . 277.2.9 Aconservationlawforvorticity............... 287.2.10 Potentialvorticity ...................... 327.3 Theprimitiveequations ....................... 357.3.1 Pressureasaverticalcoordinate .............. 357.3.2 Hydrostaticbalance ..................... 367.3.3 Thethermodynamicenergyequation............ 367.3.4 Inference of the vertical motion field ............ 397.3.5 Solutionoftheprimitiveequations............. 427.3.6 Anapplicationoftheprimitiveequations ......... 447.4 Theatmosphericgeneralcirculation ................ 467.4.1 Thekineticenergycycle................... 487.4.2 Theatmosphereasaheatengine.............. 517.5 Numericalweatherprediction .................... 517.6 Exercises ............................... 56iii CONTENTSAtm ospheric Dynam icsThis chapter introduces a framework for describing and interpreting the struc-ture and evolution of large-scale atmospheric motions, with emphasis on theextratropics. We will be considering motions with horizontal scales1of hun-dreds of kilometers or longer, v ertical scales on the order of the depth of thetroposphere, and time scales on the order of a day or longer. Motions on thesescales are directly and strongly influenced by the earth’s rotation, they are inh ydrostatic balance, and vertical component of the three-dimensional velocityvector is nearly four orders of magnitude smaller than the horizontal compo-nen t but is nonetheless of critical importance. The first section introduces someconcepts and definitions that will be useful in characterizing horizontal flow pat-terns. It is followed by a more extensive section on the dynamics of large scalehorizontal motions on a rotating planet. The third section introduces the sys-tem of primitive equations, which is widely used for predicting ho w large scalemotions evolve with time. The chapter concludes with brief discussions of theatmospheric general circulation and numerical weather prediction.7.1 K inematics of the large-sc a le horiz ontal flowKinematics deals with properties of flows that can be diagnosed (but not nec-essarily predicted) without recourse to the equations of motion. It providesa descriptive framework that will prove useful in interpreting the horizontalequation of motion introduced in the next section.On any “horizontal” surface (that is, a surface of constant geopotential Φ,pressure p, or potential temperature θ), it is possible to define a set of stream-lines (arbitrarily spaced lines whose orientation is such that they are everywhereparallel to the horizontal velocity vector V at a particular level and at a par-ticular instant in time) and isotachs (contours of constant scalar wind speedV )thatdefine the position and orientation of features such as jet streams. Atany point on the surface one can defineapairofaxesofasystemofnatural1The scale of the motions denotes the distance over w hich scalar fie lds, su ch as pressureor the meridional w ind com p onent, vary over a range comparable to the am plitude of thefluctuations. For exam ple, the h orizontal scale of a one-dimensional sinusoidal wave along alatit u d e circle is on the order of 1/ 2 π wavelength and the scale of a circular vortex is on theorder of its radius.12 CONTENTScoordinates (s, n), where s is arc length directed downstream along the localstreamline, and n is distance directed normal to the streamline and toward theleft, as shown in Fig. 7.1. It follows that at any point in the flow, the scalarwind speedV =dsdtanddndt=0The direction of the flow at any point is denoted by the angle ψ, whic h is definedrelative to an arbitrary reference angle. In this section we will refer to the unitvector s as the local downstream direction and n as the transverse direction.Fig. 7.1 Natural coordinates (s, n)defined at point P in a horizontal wind field.The curved arrows represent streamlines.—Table 7.1 Definitions of properties of the horizontal flow. The signs of prop-erties relating to vorticity are defined as positive for the northern hemisphere.vectorial natural coords. cartesian coords.Shear∂V∂nCurvature V∂ψ∂sDiffluence V∂ψ∂nStretching∂V∂s––––Vorticity ζ k · ∇ × V V∂ψ∂s−∂V∂n∂v∂x−∂u∂yDivergence DivHV ∇ · V V∂ψ∂n+∂V∂s∂u∂x+∂v∂yDeformation −V∂ψ∂n+∂V∂s∂u∂x−∂v∂y;∂v∂x+∂u∂y7.1. KINEMATICS OF THE LARGE-SCALE HORIZONTAL FLOW 37.1.1 Elem e ntary kinema tic properties of the flowAt any point in the flow, it is possible to define the kinematic properties listed inTable 7.1, all of which have units of inverse time (s−1). The topmost four rowsare elementary properties or building blocks, that have precise mathematicaldefinitions in natural coordinates. Shear is defined as the rate of ch ange of thevelocity in the direction transverse to the flow. Curvature istherateofchangeof the direction of the flow in the downstream direction. Shear and curvatureare labeled as cyclonic (anticyclonic) and are assigned a positive (negative)algebraic sign if they are in the sense as to cause an object in the flo w to rotatein the same (opposite) sense as the earth’s rotation Ω, as view ed looking downon the pole. In other words, cyclonic means counterclockwise in the northernhemisphere and cloc kwise in the southern hemisphere. Diffluence / confluenceis the rate of change of the direction of the flow in the direction transverse tothe motion, defined as positiv e if the streamlines are spreading apart in thedownstream direction. Dilation / contraction relatetotherateofchangeofthe speed of the flo w in the downstream direction, with stretc hing defined aspositive.7.1.2 Vorticit y and dive rge nceVorticity and div ergence are scalar quantities that can be defined, not only innatural