Slide 1Review: Kinematic of the horizontal flowNatural Coordinates:DefinitionsSlide 5What is going on here?Forces in the AtmosphereApparent forces:Coriolis force:Coriolis force:Slide 11Slide 12Slide 13Slide 14Slide 15Slide 16Winds and geopotential height: example: sea breezeFriction or Viscosity ForceFriction or Viscosity ForceIn summarySlide 21In a tangent plan we have (this is important to remember):Test your understanding:Geostrophic windSlide 25Friction effect:Geostrophic windsTridimensional viewGradient WindSlide 30Slide 31Thermal WindSlide 33Slide 34InterpretationSlide 36Relationships with horizontal temperature gradientBaroclinic atmosphere:Equivalente Barotropic:Exercise 7.3Cold and warm temperature advectionSlide 42Slide 43Slide 44Slide 45Slide 46Slide 47Slide 48Atmospheric DynamicsLeila M. V. CarvalhoDept. Geography, UCSBReview: Kinematic of the horizontal flowStreamlines: lines parallel to the horizontal velocity V at a particular level and at a particular instant in timehttp://weather.unisys.com/surface/sfc_con_stream.htmlNatural Coordinates:YXnnssn and s are natural coordinates (perpendicular and parallel to the flowDefinitionsSheared with no curvature, no diffluence, stretching or divergenceRotation with cyclonic curvature (NH) and cyclonic shear, no diffluence or stretching (and divergenceRadial flow with velocity directly proportional to radius. Diffluence, stretching, divergence and NO CURVATURE (or vorticity)Hyperbolic flow: difluence and straching, no divergence (terms cancel). Shear and curvature cancel (vorticity free)What is going on here?YXnnssForces in the Atmosphere•Equation of motion: (First and second Laws of Newton)•Real forces (independent on the rotating system): gravity, pressure gradient force and frictional force•Apparent forces due to rotation: apparent centrifugal force (affects gravity) and Coriolis (correction for horizontal movements).Apparent forces:•Centrifugal force: Where RA is vector perpendicular to axis of rotation and is angular velocity of earthCombine with gravity to define "effective" gravityCoriolis force:•Coriolis force takes care of rotational effects caused by motion relative to surfaceΩAt rest over the Earth surface will have cetrifugal acceleration= Ω2R. Suppose it moves eastward with speed u: the centrifugal force would increase to:Centrifugal force=RRu2RCoriolis force:22222RRuRuRRRRuExpanding the equation we have now:Centrifugal force due to rotation of the Earth (independent of the relative velocity Deflecting forces that act outward along the vector RSynoptic scale motions u<< ΩR:Last term can be neglected in a first approximationCoriolis ForceCoriolis force can be divided into vertical and meridional components :RφRuR2φcos2 usin2 uA relative motion along the east-west coordinate will produce an acceleration in the north-south direction given by:sin2 udtdvCoAnd vertical acceleration given by:cos2 udtdwCoTo the right of the movement in the NHSuppose now that a particle initially at rest on the Earth is set in motion equatoward by impulsive forcesΩRAs it moves equatorward it will conserve its angular momentum in the absence of torques: a relative westward velocity must developR + δRIf we expand the right hand side and neglect second order differentials (and assume that δR<<R and solve for δu, we get:oaRusin22 aa= Earth’s Radiusoovdtdadtdusin2sin2 dtdavNorthward velocity componentReal forces in the AtmospherePressure gradient ForceLow Pressurep2High Pressurep1wind directionPressure GradientREMEMBER THAT A “GRADIENT” ALWAYS POINT TOWARD THE HIGHEST MAGNITUDES OF THE SCALAR.Pressure Gradient Force xpP 1)8.7(1;1ypPxpPyxHydrostatic Equation:gzpgdzd Definition of Geopotential Geopotential HeightzoogdzggzZ01)( ZgzgpPo1zy>0 for sureSee Holton, 1979, second Ed. Chap1, pg. 21 ZgzgpPo1Surface of constant PressureChanges in geopotential height1212ppvodpdpTgRZZChanges in geopotential height imply in the existence of pressure gradient forcesZgoppzppzyyzgypxxzgxp1;1Winds and geopotential height: example: sea breezeZgPForceGradiento:ZLAND OCEANHigh PressureW Eppzxxzgxp1Friction or Viscosity ForcezF1τ is the shear stress and is the rate of vertical exchange of horizontal momentum N/m2τs at the surfaceFriction or Viscosity ForcezF1τzx is the shear stress in the horizontal direction x due to the stress acting verticallyzuzxsubscripts indicate that τzx is the shear stress in x direction due to vertical shear and μ is the dynamic viscosity coefficientIn summaryFrictionVery small outside boundary layerDepends on vertical gradient in vertical component of shear stress•Actual processes very complex, with turbulence playing key role•Approximate shear stress in surface boundary layer:•Shear stress depends on strength of vertical shear in horizontal wind. Empirically:ν = viscosity coefficient = μ/ρ ~10-5m2s-1Drag coefficient, CD, depends on surface roughness and static stabilityHorizontal Equation of MotionNewton’s Law in vectorial form per unity of mass:FVkV fpFCPdtd1In a tangent plan we have (this is important to remember):xFfvxpdtdu1yFfuypdtdv1Remember that Friction is defined as a negative component that is supposed to decrease (decelerate) the speed We can eliminate density by using the relationship between pressure gradient and geopotentialFVkV fdtdTest your understanding:Estimate pressure gradient and, coriolis parameter in Kansas ~38oNRepresent winds around the Low and High pressure systemsGeostrophic windBy using scale analysis of horizontal equations it can be shown that:Horizontal velocity scale:Length scale:Depth scale:Horizontal pressure fluctuation scale:Time scale (advective):