Simple Mathematical Concepts for Meteorology
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Appendix A Simple Mathematical Concepts for Meteorology A.1 Kinematics of a Fluid Let r.t/ D .x.t/; y.t/; z.t// be the trajectory of an air parcel. It is determined by the coordinates of the air parcel and the motion field v.r;t/. For a fixed initial location the vector dr D .dx; dy; dz/ describes the incremental displacement of the air parcel during the time interval dt. In general, the velocity is defined D dr D as change of position per time given by v dt with the components v D dx dy dz .u; v; w/ . dt ; dt ; dt /. At any time the velocity field defines the streamlines that are everywhere tangential to v.r;t/. A.2 The Material Derivative Acceleration is defined as the change in velocity per time. The acceleration in x-direction, for instance, is given by d 2x @u dx @u dy @u dz @u D C C C (A.1) dt2 @x dt @y dt @z dt @t D @x By applying the definition of velocity at a fixed location (e.g. u @t ), the above equation can be written as d 2x @u @u @u @u D C u C v C w : (A.2) dt2 @t @x @y @z Using the definitions of the nabla-operator,  à @ @ @ rD ; ; (A.3) @x @y @z N. Mölders and G. Kramm, Lectures in Meteorology, Springer Atmospheric Sciences, 537 DOI 10.1007/978-3-319-02144-7, © Springer International Publishing Switzerland 2014 538 A Simple Mathematical Concepts for Meteorology and a vector, v D .u; v; w/, yields d 2x @u du D C v ru ” : (A.4) dt2 @t dt The term rv is a scalar product. Analogously, the y- and z-components of acceleration can be derived. In vector form, acceleration reads d 2r @v dv a D D C .v r/v D : (A.5) dt2 @t dt The identity (Euler operator) d @ D C v r (A.6) dt @t is the material derivative or substantial derivative. @ In contrast to the rate of change at a fixed location, @t , (local derivation), the material derivative represents the rate of change with respect to time following the motion. The term .v r/v contributes to the material derivative nonlinearly. Therefore, the behavior of the atmosphere is difficult to forecast as nonlinear systems behave chaotically. Example. Assume a cold air mass flows down from an ice cap.1 Determine the wind speed the air mass will reach at the bottom a minute later if its acceleration is 0:5 ms2. D dv Solution. Acceleration is given by a dt where t is time and v is velocity. D 0:5 ms 2 D 1 Rearranging and inserting the values yields v 60 s 30 ms . A.2.1 Divergence and Convergence Sometimes only the horizontal components of the wind field are of interest and the following identity can describe the horizontal divergence2 @u @v rv D C : (A.7) H @x @y 2 In many books divvH is used to indicate the divergence. A.3 Rotation 539 A divergence of the horizontal wind field, for example, will exist if u increases with x and v grows with y,orifu increases with x and v decreases with y,butthe @u C @v net-effect @x @y being positive. Imagine that the divergence of v is accompanied by a change in mass. In our example, this would mean a loss of mass. A convergence would mean the opposite. Example. Assume winds, v D .5;3;0/, v D .5; 3; 0/, v D .3; 5; 0/, and all with a specific mixing ratio of 10 gkg1 moving over a distance of 2 km. Is there moisture convergence or divergence? Solution. As the vertical wind component is zero and wind speed remains @u C @v D 5 ms1 C constant with time, the moisture flow is given by . @x @y /q . 2;000 m 3 ms 1 1 1 1 1 1 1 1 2;000 m /10 gkg 0:04 gkg s , 0:01 gkg s , and 0:01 gkg s for v D .5;3;0/, v D .5; 3; 0/ and v D .3; 5; 0/, respectively. The positive values mean moisture divergence, the negative value moisture convergence. A.3 Rotation Another frequent form of motion is rotation,3 which can be mathematically expressed by using the definition of the nabla-operator and the unit vectors rv D i C j C k (A.8) with @w @v D (A.9) @y @z @u @w D (A.10) @z @x @v @u D : (A.11) @x @y Tornados and tropical storms are examples where rotation becomes important. 3Many books use rotv or curlv instead of rv. 540 A Simple Mathematical Concepts for Meteorology N 0Њ v 90Њ NW NE u Њ W E 180 0Њ 270Њ 90Њ SW SE Њ S 270 180Њ Fig. A.1 Schematic view of wind directions in degrees (left) and counting of degrees in geometry (right) for comparison Example. Calculate the rotation of a dust devil with 0:05, 0:08, and 2 min extension in x-, y- and z-direction and a of a wind vector, v D .5;6;2/ms1 in x, y and z direction. r D 2 6 5 2 6 5 Solution. v . 0:8 2 ; 2 0:05 ; 0:05 0:08 /: A.4 Scalar Quantities Scalar quantities (e.g. density, pressure, temperature, etc.) are only defined by an amount, while vectors are quantities that are defined by an amount and a direction. In this sense, we distinguish between wind speed (the amount) and wind direction when taking about wind velocity (the vector). To indicate a vector bold letters, underlining, or an arrow above are used. In Cartesian coordinates, the x-axis points to the east, the y-axis points to the north (Fig. A.1), and the z-axis points upwards (against gravity) according to the traditional definition in meteorology (Chap. 6). The components of the wind vector maybeexpressedbyu, v, and w. By use of the azimuth angle and the zenith angle ' we obtain u Djvj sin 'cos v Djvj sin ' sin w Djvj cos ' (A.12) By using the components the wind vector can be expressed by v D ui C vj C wk (A.13) A.5 Changes of Field Quantities with the Field Coordinates 541 where i, j, and k are the unit vectors given by i D .1;0;0/; (A.14) j D .0;1;0/; (A.15) k D .0;0;1/: (A.16) In this notation, wind speed reads p jvjD u2 C v2 C w2 (A.17) A.5 Changes of Field Quantities with the Field Coordinates Partial differentials describe changes in direction of one field coordinate only. Let us assume an arbitrary property ˛ of an air parcel. We can write for its individual displacement in one direction at constant other directions @˛ j ; (A.18) @x y;z;t @˛ j ; (A.19) @y x;z;t @˛ j ; (A.20) @z x;y;t @˛ j : (A.21) @t x;y;z These derivations are called local derivations or local changes because only one coordinate varies while the other keep being constant. Continuous observation of the concentration of an inert trace gas at a station provides a function C.x;y;z;t/ where x;y;z are constant and t changes. The @C associated registration provides @t . An example of a change in space coordinate is the temperature profile provided by a radiosonde on a calm day, and without temporal change in temperature.4 These @T measurements deliver T.z/ and @z . D dr D C C A velocity v dt with r xi yj zk describes the motion of a particle dr dx dy dz v D D i C j C k (A.22) dt dt dt dt dx D dx D dz D where dt u, dt v, and dt w. 4In nature, radiosondes usually drift. 542 A Simple Mathematical Concepts for Meteorology Another question is how the characteristics ˛ of a parcel will change with time without holding the coordinates constant if the parcel moves. It is the change of ˛ that a moving particles experiences or that we would find if one traveled on the particle. This change is called individual change or derivation and the differential is denoted to as individual or total derivation (or differential). Regarding the change of the characteristic ˛ of a particle on its way from a point P1.x; y; z;t/to a point P2.x C dx;y C dy; z C dz;t C dt/ shows the relationship between the partial and the total differential @˛ @˛ @˛ @˛ d˛ D j dt C j dx C j dy C j dz (A.23) @t x;y;z @x y;z;t @y x;z;t @z x;y;t Using the components of the wind vector and rearranging results in d˛ @˛ @˛ @˛ @˛ @˛ D C u C v C w D C v r˛ (A.24) dt @t @x @y @z @t „ƒ‚… convective changes rD @ C @ C @ d˛ where @x @y @z . The above equation means that the total differential dt @˛ consist of the local differential @t and the velocity components related (convective) changes of ˛. @˛ D When at a place the field quantity does not change with time @t 0, the field quantity behaves stationary.5 The change on a particle path is called Lagrangian d˛ and looks at dt , while the change at a fixed place is named Eulerian and looks @˛ at @t .