2.11 please help me solve problem 2.11 ty! 2.4. Derive the expression given in Eq. (2.13) for the rate of change of k2.5. Suppose a 1-kg parcel of dry air is rising at a constant vertical velocity. Ifwhich was used in Eq. (2.7).following the motion.the parcel is being heated by radiation at the rate of 10-'w kg-1, whatmust the speed of rise be to maintain the parcel at a constant temperature?Derive an expression for the density p that results when an air parcelinitially at pressure ps and density ps expands adiabatically to pressure p.- An air parcel that has a temperature of 20°C at the 1000-hPa level is lifteddry adiabatically. What is its density when it reaches the 500-hPa level?28. Suppose an air parcel starts from rest at the 800-hPa level and rises verti-cally to 500 hPa while maintaining a constant 1°C temperature excess overthe environment. Assuming that the mean temperature of the 800- to 500-hPa layer is 260 K, compute the energy released due to the work of thebuoyancy force. Assuming that all the released energy is realized as kineticenergy of the parcel, what will the vertical velocity of the parcel be at 500hPa?2.9. Show that for an atmosphere with an adiabatic lapse rate (i.e., constantpotential temperature) the geopotential height is given byZ = H, [1 – (p/po)**]where po is the pressure at Z = 0 and He = C, 0/go is the total geopotentialheight of the atmosphere.2.10. The azimuthal velocity component in some hurricanes is observed to havea radial dependence given by vx =given by r 2 ro. Letting Vogeopotential difference between the far field (r → 0) and r = r0, assuminggradient wind balance and fo = 5 x 10-5 s1. At what distance from thecenter does the Coriolis force equal the centrifugal force?2.11. In the isentropic coordinate system (see Section 4.6), potential temperatureis useabatic flow is conserved following the motion, isentropic coordinates areuseful for tracing the actual paths of travel of individual air parcels. ShowVo (ro/r)2 for distances from the center= 50 ms-1 and ro = 50 km, find the totalas the vertical coordinate. Because potential temperature in adi- dy-state Temperatarevertical Monentum EqenmentimCHAPTER 2 Basic Conservation Lawsthat the transformation of the horizontal pressure gradient force from z to ecoordinates is given by-vp = VeM%3Dwhere M = c, T+ ® is the Montgomery streamfunction.2. French scientists have developed a high-altitude balloon that remains atconstant potential temperature as it circles Earth. Suppose such a balloon isin the lower equatorial stratosphere where the temperature is isothermal at200 K. If the balloon were displaced vertically from its equilibrium level bya small distance 8z, it would tend to oscillate about the equilibrium level.What would be the period of this oscillation?3. Derive the approximate thermodynamic energy equation (2.55) using thescaling arguments of Sections 2.4 and 2.7.LAB Exercises1. The MATLAB script standard.T-p.m defines and plots the temperature amthe lapse rate associated with the U.S. Standard Atmosphere as functionsheight. Modify this script to compute the pressure and potential temperure and plot these in the same format used for temperature and lapse rat[Hint: To compute pressure, integrate the hydrostatic equation fromface upward in increments of 8z.] Show that if

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Recall that the potential temperature in adi abatic flow is conserved following the motion,…