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A Two-Dimensional Omega Equation for the 1000-700 Mb Layer with Diabatic Heating

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A Two-Dimensional Omega Equation for the 1000-700 Mb Layer with Diabatic Heating i N PS ARCHIVE 1966 FERRENTINO, P. ;;dffl BnnOOORi""MM WW, A TWO-DIMENSIONAL OMEGA EQUATION FOR THE 1000-700 MB LAYER WITH DIABATIC HEATING PETER S, FERRENTINO $3i ibSBffl i Bff» ismmm mi BOOK m &$Be tfffifi HI : : ; ''!.'' ' - ' ' ' matin WfflH |jj» 1 : : ''''' m GEMS :;'.?;''v7i ' i Wlttl tnUifi msammsM rWP '''''; ',* ''," JK]B$|| . \ : .' BuWMsHHbw ";'','' V 'Nf'^U.: : .. fi mJtHVB iJwfn "', '' itnwi 1 1 i'v - I V ' TJffiW L '' ' H 11 'ii'i : !.''''... m mw ''''• ';-::' ;';',/ NHEK B5K ffissv SdCoct LIBRARY NAVAL POSTGRADUATE SCHOOL MONTEREY, CALIF. 93940 DUDLEY KNOX LIBRARY NAVAL POSTGRADUATE SCHOOL MONTEREY CA 93943-5101 This document has been approved for public release and rale; its distribution is unlimited. 6 &3 NAVAL POSTGRADUATE SCHOOt A TWO-DIMENSIONAL OMEGA EQUATION FOR THE 1000-700 MB LAYER WITH DIABATIC HEATING by Peter S. Ferrentino Lieutenant, United States Navy Submitted in partial fulfillment for the degree of MASTER OP SCIENCE IN METEOROLOGY from the UNITED STATES NAVAL POSTGRADUATE SCHOOL May 1966 ABSTRACT A two-dimensional omega equation is derived by combination of the vorticity and thermodynamic equations. The desired omega is then taken to be the logarithmic average in the 1000-700 nib layer. A diabatic term, after Laevastu, for oceanic areas only is included to deduce the empirical temperature and vapor-pressure changes associated with sensible and latent heating in the maritime layers. Over both continental and oceanic areas a frictional vorticity sink is included in order that excessive energy cannot be generated over the ocean. Among other novel features is the use of the Holl static-stability parameter which affords vertical consistency with analyses prepared by Fleet Numerical Weather Facility. TABLE OP CONTENTS Section Pag© 1. Introduction 11 2. Th© Basic Omega Equation 11 3. Vortical Distribution of Pressure and 13 Omega k. Vertical Integration of the Omega 15 Equation 5. The Lower Boundary Condition 19 6. The Diabatic Heating Term 21 7. State-Change Parameters for the 25 Convective Case 8. Numerical Procedures 30 9. The Computer Program 36 10, Results and Conclusions 38 11, A^mowledgei-nents ';.l 12, Bibliography 57 LIST OP ILLUSTRATIONS Figure Page 1. Pressure Distribution in the Vertical 13 2. The Omega Profile 15 3. Three Cases of the Lower Boundary 20 1|. Model of the Convective-atmosphere Modification 22 5. AT^p as a Function of ATD 26 6. 00Z 28 April 1966 8£0 rab D Field i+2 7. 00Z 28 April 1966 Diabatic Heating [[3 8. 00Z 28 April 1966 Stability kk 9. 00Z 28 April 1966 8£0 mb Adiabatic 1^5 Omega 10. 00Z 28 April 1966 FNWF 8£0 mb Omega I|6 11. 00Z 28 April 1966 8£0 mb Omega l±7 12. 12Z 28 April 1966 8^0 mb D Field 1+8 13. 12Z 28 April 1966 8£0 mb Omega 1+9 ll|. 00Z 29 April 1966 8£0 mb D Field £0 15. 00Z 29 April 1966 8£0 mb Omega 5l 16. 12Z 29 April 1966 8£0 mb D Field $2 17. 12Z 29 April 1966 8£0 mb Omega 53 18. 00Z 28 April 1966 Omega Lower Boundary Sh 19. 00Z 28 April 1966 FNWF Omega Lower ^ Boundary J LIST OP SYMBOLS Scalar Quantities temperature Ta surface air temperature Id surface dew-point temperature hv sea water temperature (surface) |ivb wet -bulb temperature VC. condensation level temperature Q potential temperature Sa surface vapor pressure Qs surface saturation vapor pressure ^v vapor pressure over water Oc condensation level vapor pressure P pressure IA surface pressure re condensation level pressure H height above mean sea level H.T terrain height D Z (actual)- Z (standard atmosphere) I absolute vorticity relative vorticity coriolis force C<J vertical motion of a pressure surface UJlq vertical motion at the lower boundary LJp vertical motion due to frictional effects LIST OP SYMBOLS (continued) LOr vertioal motion due to terrain effects (J* stability parameter G"Ji Holl stability parameter V wind speed VI© 10-meter wind speed YY\ mass P density t time Cd coefficient of drag • Q heating rate per unit mass Qc heat flux for the inversion case O^ cross isobar angle Oft moist adiabatic lapse rate Constants 04* o/Cp dry adiabatic lapse rate QD dew-point lapse rate R gas constant for dry air Cp specific heat at constant pressure latent heat of vaporization I ^ acceleration of gravity 8 LIST OP SYMBOLS (continued) Vector-Dealer, operators \y velocity \^o geostrophic wind velocity WA gradient of A \7 A Laplacian of A vJ(A>d) Jacobian of A and B 1. Introduction Little practical us© has been mad© of vertical motions, CJ« dp/</t, in short-range forecasting due to the complicated and lengthy computations required. Vertical motion computa- tions are usually the by-product of multilevel baroclinic models and single values must be extracted only after going through the entire three-dimensional procedure. In this paper a two-dimensional vertical motion equation is developed using vertically integrated parameters with an assumed vertical motion profile. The layer-mean omega is especially useful for short-range thickness forecasts and the 1000-700 mb layer has been selected for study. Derivation of the CJ -equation is similar to except a that by Thompson |_18J diabatic heating term providing a mechanism for development over oceanic areas has been included. Due to the empiricisms employed the non-elliptic conditions mentioned by Pedersen [l6J do not arise. 2. The Basic Omega Equation The diagnostic omega equation is derived by standard means from a vorticity equation and a thermodynamic equation which retains the diabatic term. Equation (1) is the time derivative of the first law of thermodynamics in (x, y, p, t) coordinates. ii . =- the hydrostatic Substituting, T | $? , which comes from assumption and the ideal gas law; then dividing by, -g/R, yields equation (2). (2) Operating•pfitlno- on equationenuation (2)( ?) with the LaolacianLaplacian gives the final form of the thermodynamic equation as shown by equation (3). Here, O^s— Rp3*^, is the Holl stability parameter to be further discussed in section (i|). The vorticity equation in pressure coordinates is shown by equation (1|) As presented by Thompson llSjand numerous other writers the last term of the left side of (I4) is approximately equal to that on the right side, and the two terms are henceforth deleted. Then, a useful form of the vorticity equation is arrived at (equation 5) by making the geostrophic assumption for vorticity and for velocity, C - and by JLV* > W,*$.WZ f - f * f taking the logarithmic pressure derivative. 12 vorticity. Next subtract Here, 7*5^1 > is the absolute (5) from (3), and the result is the omega equation: -T8^'^)) (6) 3. Vertical Distribution of Pressure and Omega Instead of the distribution of pressure indexes normally considered, that employed here is given by, where V) is an arbitrary pressure level. The 1000-500 mb layer is divided into sub-layers as shown by Figure (1). The resultant pressures, while essentially logarithmic, bear values nearly identical to those of the mandatory levels. -l k ——— - P4 po 2 500.0 rab _2 = 3 ___ ?2> - Pd 2 H 59l[.6 mb x 2 ——— Pz - P^2 707.1 mb 1 __-_- P| - P*2"$ = 81;0.9 mb Pa Po2 1000.0 mb Figure 1. Pressure Distribution in the Vertical 13 // The values of OO- dp/dt, the "vertical velocity in pres- sure coordinates , La assumed to vary parabolically in the vertical with a profile defined by equation (7) which extends nearly to the 5>00 mb level. u>=A(j<ALt+bfA2.\ (7) For boundary conditions it is assumed that^OOat p=p (the kinematic boundary condition at the assumed surface of the earth, p ). Also for convenience it is assumed that at p C a /C^ , i.e. that the 3>00-mb divergence is zero. Then, upon differentiation of equation (7) with the 500-mb diver- gence taken to be zero, equation (8) results. It should be noted that if the ( is value ^%yp)«0 assumed zero at p=p 3 the value of B is altered by only 5$. or. Therefore, B=1.3863A, and the omega profile in terms of A is given by equation (9). cb-mA LA + i.m3^ (9) Since, CO =o, the ratio between 60,, and ^)^, defines the omega profile (equation 10) for the 0-2 layer (see Figure 2) Ik . (10) CO, ^AAz(- £/**+/. 3863) , ^ P 1* p v'&J | pn ^W6 pf6 <UJ-* co^= 1.9/44 <*>i Figure 2. The Omega Profile For the pressure-range considered (see Figure 2), the layer- logarithmic-mean, UJ =0.950dO|,so that U>, , will be used f or U) . Note finally, the profile assumed here applies only to the large-scale adiabatic, frictionlcss component of the vertical velocity (that is terrain irregularities are considered absent ) l\. Vertical Integration of the Omega Equation The Holl stability parameter Lllj is used since it is one of the vertical—consistency requirements used by FNWF (Fleet Numerical Weather Facility) and this study uses FNWF processed data. The Holl parameter is a modification of the standard stability parameter used here, as shown by equations (11), (12), and (13). 15 (id r &dx> \Qp?c?f dpap J gives, Letting, RT= - <3j ^|L a S£Tp (12) Equation (12) may now be finite differenced over the 0-2 layer. <r« §-(**-*•) -(To-Tl) (13) Equation (13) represents the Holl stability parameter in finite-difference form. In effect, Oh , by (13) gives the i,logarithmic-pressure 'average' over the 0-2 layer. This \ - use of O^ is compatible with the assumed logarithmic pressure distribution and will be used throughout the analysis which follows.
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