Part 4. Atmospheric Dynamics

Part 4. Atmospheric Dynamics We apply Newton s Second Law: ma =Σ F i to the atmosphere. In Cartesian coordinates dx u = dt dy v = dt dz w = dt 1 ai = F m i i du dv dw a = ; ay = ; az = x dt dt dt 78

Coordinate Systems x = distance east of Greenwich meridian y = distance north of equator r = distance from center of earth Then, dx = rdλcos φ; dy = rdφ R E = mean radius of earth = 6 6.37 10 km z φ φ0 = g 0 φ φ 0 = mean sea level geopotential g 0 = sea level gravity 79

Velocity Components u dx dλ = = RE cosφ (zonal velocity component) dt dt dy dφ v= = RE (meridional component) dt dt dz dr w =! dt dt Forces 1 p 1 1 x ; p p = pz = ; p p y = ρ x ρ z ρ y 80

Friction Friction represents the collective effects of all scales of motion smaller than scales under consideration. Friction has its greatest impact near the earth s surface. A simple model: F x 2 = CDu slows F 2 y = CDv wind 81

Pressure as a Vertical Coordinate Why use p? Large scale motions are hydrostatic so p monotonically decreases with height. P-surfaces are nearly horizontal and are useful in analysis. Vertical velocity: ω = dp dt ω > 0 ; ω < 0 dp ω =! ρgw o dt 82

Other Vertical Coordinates σ = p/ p s ; Advantage is lower boundary σ = 1 is tropographic surface. θ : (isentropic coordinate) horizontal motions tend to follow isentropic surfaces. Also terrain-following: z$ z z s = H zs H Where z s is height of topography, and H is the depth of model atmosphere. 83

Natural Coordinates Defined with respect to stream lines. ds dn v = and 0 dt dt = 84

Apparent Forces The coordinate system we are used to rotates with an angular velocity Ω= 2π rad day = 7.292 10 s 1 5 1 The effects of the rotation create apparent forces that are due solely to the fact that the coordinate system rotates. 85

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Earth s rotation vector ( Ω ur ) can be resolved into two comp s, a radial component and a tangential component. The linear velocity of a point fixed on the earth is u =Ω R cosφ. E 87

Effective Gravity The force per unit mass called gravity or effective gravity g is the vector sum of the true gravitational attraction g* that draws objects to the center of mass of earth and apparent centrifugal force that pulls objects outward from the axes of rotation with a force Ω R =Ω R cos φ. 2 2 A E Ω R =Ω R cosφ 2 2 A E 88

Coriolis Force (Farce?) As a parcel moves toward or away from the axis of rotation its angular momentum is conserved: d dt 2 2 ( RA ωra) Ω + = 0 Where ω is the relative angular velocity due to an air parcel moving at the surface of the earth, hence Thus, d dt ω = u/ R A. 2 ( RA ura) Ω + = 0. Differentiating we find: 2Ω RR + ur& + Ru & & = 0 or du dt A A A A u = 2 Ω+ R& A. RA 89

But R& R& A A d = ( REcos φ) dt dφ = REsinφ dt R E dφ is the linear velocity on a meridian circle or dt R E dφ dy = = v. dt dt thus, & and R = vsinφ A du dt or du dt u =+ 2Ω+ vsinφ RA uv = 2Ω vsinφ + sinφ R A Except near poles the first term dominates, or du! 2Ωvsin φ. dt 90

A northward moving parcel will be turned to the east and a southward to the west. For a parcel moving along a latitude circle, the parcel experiences a relative acceleration a 2 y = u / RA and in an absolute reference frame it experiences an acceleration a = ( U + u) 2 / R A. 91

But, or relative( a ) = absolute( a ) + apparent ( a ) 2 u U u R A y y y 2 + = + apparent ( ay ) RA 2 2 2 u U 2Uu u = + + + apparent ( ay ) R R R R A A A A Let U =Ω R A. 2 Apparent ( ay) = 2 Ωu Ω RA. Coriolis force due to u-motion What s added to form effective gravity g. Since second term is incorporated in g, ay = 2 Ω u. 92

R A 2Ω u cosφ 2Ω u 2Ω usinφ 2Ω u Thus dv dt = 2Ω usinφ Horizontal Equations of Motion du dt dv dt p = α + 2Ω vsinφ + F x p = α 2Ω usinφ + F x x y 93

Above the atmospheric boundary layer (ABL), friction is unimportant, the air flow approaches equilibrium, such that or du dt = 0 = dv dt p α = fv x p α = fu y Called geostrophic equilibrium. 94

p At a given latitude, for large, n Vg is large. At low latitudes, f 0, Vg must be larger for a given pressure gradient in order to maintain geostrophic flow. Geostrophic balance is rarely achieved at low latitudes. It can be readily shown: p p dp = dn + dz. n z On a constant pressure surface dp = 0. 95

Thus, or } g ρ p p dz = n t dn 1 p dz = g ρ n dn p p or in terms of geopotential heights: 1 p Z = g0 ρ n n or V g 1 p g0 Z = = ρ f n f n Tighter height grad- Stronger the winds. 96

Thermal Wind Geostrophic wind equations: 1 p 1 p (1) fvg =, (2) fu = ρ x ρ y Hydrostatic eq. Eq. of state 1 p (3) g =, (4) ρ = ρ z p * R T 97

Substitute (4) into (1), (2), (3) * * (5) R T p g ; (6) R T fv = fu p g = p x p y (7) g R* T = p p z Cross differentiate between (5) and (7): 2 fv g = R* n z T z x 2 g ln p = R * x T z x Adding above eqs. yields ( l p) (8) fvg g = z T x T 98

Cross differentiate between (6) and (7) 2 fug ( lnp) = R* z T z y 2 g ( lnp) = R * y T z y Subtracting 2 nd from 1 st : fu g (9) = ( g/ T) z T y Completing differentiation of (8) and (9): (10) (11) vg g T v T = + z ft x T z ug g T u T = + z ft x T z 99

Terms in are corrections for slope of isobaric surfaces and are small compared with 1 st terms on RHS of (10) and (11). Thus, thermal wind eqs. vg g T ug g T! ;! z ft x z ft y Vertical shear of horizontal wind is large where there are strong horizontal gradients in temperature (i.e., across polar front). 100

In Natural Coordinates V g ) Vg) 2 2! const. T f n where T is the mean temperature of layer 1 2 Also, g0 Vg ) 2 Vg) 2 = ( Z2 Z1) f n 101

Gradient Wind Sharp troughs are often associated with the subgeostrophic flow. The flow can be balanced because of the large centripetal acceleration. Apparent centrifugal force helps balance P-grad. 102

Gradient Balance 103

Consider the equations of motion in cylindrical coordinates: v v r r v v r r θ r ρ r 2 vr θ vr θ 1 p + = fvθ vθ vθ vθ vθvr 1 p + + = fvr r r θ r ρr θ p Consider circular concentric isobars with centers at r = 0. Then = 0, θ vr vθ and for circular symmetry, = = 0, and V r = 0. Then the first θ θ equation can be written 2 c 1 p + fc = 0, R ρ r where c the above are = v θ and R is the radius of the cyclone/anticyclone. Solutions to when p r 2 2 fr f R R p c = ± + 2 4 ρ r is positive (a low) the square root can never become imaginary so that all values of pressure gradient are permitted. There is no theoretical restriction on the magnitude of the pressure gradient for a low. However, p when < 0 r (a high) the square root can become imaginary. For C to be real, 104

p r ρ 2 f R 4 or a high may not exceed a value determined by the latitude and radius of curvature. 105

Ekman Balance The conditions for balance in the Ekman layer are that: 2 c R 1 p 2 + fc + C 0. { Dc = ρ r F We see that friction decelerates the flow and turns the wind towards low pressure. This results in low-level divergence out of anticyclones and lowlevel convergence into cyclones. In summary: Frictional acceleration acts directly opposite to the direction of the wind. Coriolis acceleration is perpendicular to the wind direction. Centripetal acceleration is also perpendicular to the instantaneous wind direction. 106

Continuity Equation The atmosphere behaves as in incompressible fluid 107

Continuity Equation ρ ρu ρv ρw = + + t x y z As an incompressible fluid: u v w + + = 0 x y z or Hor. Div. 64748 u v w + =. x y z 108

Pressure Tendency Equation The pressure at any height (z) is given by the weight of the air column above it: 0 p p 0 z dp = dp = p = g ρdz. The pressure tendency at z is p ρ = g ρdz = g dz t t t z z But, ρ v = DivH V w t z ( ρ H) ( ρ ) where v V = u + v H 2 2. 109

or p v = g divh( ρvh) g ( ρw) dz t z z z z 0 ρ w ) ρw) z At the surface: ( w) 0 0 p v = g div V + g w t z z ρ =. p t z z= 0 z= 0 ( ρ ) ( ρ ) H H z v = g div V H ( ρ H). 110

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Example of Baroclinic Atmosphere K.E. is generated Baroclinic systems: Cold fronts Sea breeze fronts Mtn. slope flows 112

Barotropic Atm. Baroclinic 1) ρ and p surfaces coincide 1) ρ and p surfaces intersect 2) p and T surfaces coincide 2) p and T surfaces intersect 3) p and θ surfaces coincide 3) p and θ surfaces intersect 4) No geostrophic wind shear 4) Geostrophic wind shear 5) No large-scale w 5) Large-scale w 113

Vorticity Analogous to solid body angular momentum. Vertical component: Consider equations of motion v u ζ = x y. u u u u p + u + v + w = α + fv+ Fx t x y t x v v v v p = u + v + w = α fu+ Fy t x y z y Take vector cross product or take partial derivative with respect to x of 2 nd equation, and subtract partial derivative with respect to y of the 1 st equation and rearrange: ζ ζ ζ ζ 2 f + u + v + w + v t x y z 2y ( divergence) ( tilting) u v w v w u = ( f + ζ ) + x y x z y z solenoidal or baroclinic friction α p α p Fy F x + x y y x x y 114

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Examples of rotational and shear cyclonic vorticity illustrated in natural coordinates. 117

Schematic illustration of the inferred change of vorticity and resultant motion (b) as an air parcel in gradient wind balance moves through a constant pressure gradient wind field in the upper troposphere given in (a). 118

The Omega Equation We desire to form a single equation that combines the vorticity equation and the first law of thermodynamics to describe the vertical motion pattern above the surface associated with extratropical cyclones. We follow procedures used to derive the vorticity equation previously except that we do so on a constant pressure surface. For compactness, we write in vector notion: ζ v p ω + V p( ζ p + f), t p where ζ is the relative vorticity on a constant pressure surface. We have p also neglected tilting and fraction terms. The quantity u v + x y on a constant pressure surface is replaced by ω p, where ω = dp / dt = ρgw. 119

Introducing the concept of geostrophic vorticity, ζ g, and giving the vector form of geostrophic velocity as v g v Vg = kx pz, f then v v g 2 ζ g = Vg = pz. f Substituting ζ into the vorticity tendency equation and letting ζ p = ζ g, g g 2 v ω pz + V p( ζ g + f) = ( ζ g + f). t f p Writing the first law of thermodynamics as: dlnθ Cp = Q/ T, dt where Q represents sensible heating. 120

C p [ ln θ t + u ln θ x + v ln θ y + ω ln θ ] = Q/T p Since using the gas law: θ = T [1000/p] R d/c p = pα R d ( 1000 p ) Rd /C p then ln θ x = ln α x ; ln θ y = ln α y ; ln θ t = ln α t on a constant pressure surface and [ α C p t + u α x + v α ] ln θ + ωα = α y p T Q. From the hydrostatic relation in a pressure coordinate framework (i.e., z p = α/g): α = g z p so that the above can also be written as: [ C p ( g z ) u ( g z ) v ( g z ) + ωα ln θ ] = α t p x p y p p T Q) By convention: σ = α ln θ p = g z ln θ p p is defined so that the above becomes, after rearranging: ( g z ) ( V t p p g z ) ωσ = p α C p T Q = R d pc p Q. 1 121

Performing the operation / p: g 2 p z t p + f p [ V p (ξ g + f) ] = f (f + ξ g ) 2 ω p 2 ; performing the operation 2 p and assuming that σ is a function of pressure only yields: ( ) ( g 2 z p [ 2 p V p g z )] σ 2 p t p p ω = R d 2 p pc Q. p Adding the last two equations produces: f p [ V p (ξ + f) ] 2 p ( [ V p g z )] σ 2 p p ω = R d C p p 2 p Q+f (f + ξ g) 2 ω p. 2 Since z/ p = α/g = RT/gp, this relation can also be written as: σ 2 pω+f (f + ξ g ) 2 ω p 2 = f p [ V p (ξ g + f) ] + R [ d V p 2 p p T ] R d C p p 2 pq. This equation is called the Omega equation and represents a diagnostic second order differential equation for dp. dt The three terms on the right side represent the following: [ V p p (ξ g + f) ] vertical variation of the advection of absolute vorticity on a constant pressure surface. 2 p [ V p T ] the curvature of the advection of temperature on a constant pressure surface. 2 pq the curvature of diabatic heating on a constant pressure surface. 2 122

These three terms can be interpreted more easily. Using the relation between / p and / z, and our observation that 2 ω w, w [ V p (ξ g + f) ] [ V p (ξ g + f) ] p z In most situations in the atmosphere, the vorticity advection is much smaller in the lower troposphere than in the middle and upper troposphere since V and ξ g are usually smaller near the surface.we have shown that on the synoptic scale, cold air towards the poles requires that V becomes more positive with height. Using this observation of the behavior of V and ξ g with height: w V p (ξ g + f) In other words, vertical velocity is proportional to vorticity advection.since upper-level vorticity patterns are usually geographically the same as at midtropospheric levels (since troughs and ridges are nearly vertical in the upper troposphere, the 500 mb level is generally chosen to estimate vorticity advection.this level is also close to the level of nondivergence in which creation or dissipation of relative vorticity is small, so that the conservation of absolute vorticity is a good approximation. Thus for the Northern Hemisphere where ξ g > 0 for cyclonic vorticity, w>0if V p (ξ g + f) > 0 positive vorticity advection (PVA) w<0if V p (ξ g + f) < 0 negative vorticity advection (NVA) To generalize this concept to the southern hemisphere, PVA should be called cyclonic vorticity advection; NVA should be referred to as anticyclonic vorticity.the curvature of the advection of temperature on a constant pressure term can be represented as: 2 p [ V p T ] k 2 B sin kx 3 123

where B is a constant.therefore, V p T B sin kx Since: [ w 2 V p p T ] then w V p T. Thus, w>0if V p T>0 warm advection w<0if V p T<0 cold advection The 700 mb surface is often used to evaluate the temperature advection patterns since the gradients of temperature are often larger at this height than higher up and the winds are significant in speed.the 850 mb height can be used (when the terrain is low enough) although the values of V are often substantially smaller. Finally, since 2 pq k 2 C sin kx can be assumed in this form, w 2 pq,and Q w results. Therefore, w > 0 diabatic heating w < 0 diabatic cooling 4 124

An example of diabatic heating on the synoptic scale is deep cumulonimbus activity.an example of diabatic cooling is longwave radiative flux divergence. In summary, the preceding analysis suggests the following relation between vertical motion, vorticity and temperature advection, and diabatic heating. positive vorticity advection w>0 warm advection diabatic heating. negative vorticity advection w<0 cold advection diabatic cooling. When combinations of terms exist which would separately result in different signs of the vertical motion (e.g., positive vorticity advection with cold advection), the resultant vertical motion will depend on the relative magnitudes of the individual contributions.also, remember that this relation for vertical motion is only accurate as long as the assumptions used to derive the Omega equation are valid. Using synoptic analyses the following rules of thumb usually apply: i) vorticity advection: evaluate at 500 mb. ii) temperature advection: evaluate at 700 mb; at leevations near sea level, also evaluate at 850 mb. iii) diabatic heating: contribution of major importance in symoptic weather patterns (Particularly cyclogenesis) are areas of deep cumulonimbus.refer to geostationary satellite imagery and radar for determination of locations of deep convection. Petterssen s development equation The vorticity equation can be written as: (ξ z + f + V t H p(ξ z + f) =0 if vertical advection of absolute vorticity, the titlting term and the solenoidal term are ignored.we assumed that the above equation is valid at the level of nondivergence ( 500 mb). V H is the wind on the pressure surface.since, if the wind is in geostrophic balance: 125

V H500 = V HSFC + V g where V g is the geostrophic wind shear.thus, (ξ z + f) 500 =(ξ z + f) SFC +(ξ z + f) T since V H500 equation as: = ( V HSFC ) + ( V ).We can write the vorticity (ξ z + f) SFC t From the thermal wind equation, = V H500 p (ξ z + f) 500 (ξ z + f) T t V g = g f 2 p ( z) where z = z 500 z G with z 500 the 500 mb height and z G the surface elevation so that, (ξ z + f) T t = g ( z) f 2 p t Integrating between the surface pressure, p SFC, and 500 mb yields, after rearranging: g 500 p SFC t ( ) z dp = g p t z 500 z G dz = g ( z) t = 500mb p SFC ( V p ( g z p ) + ωσ + R ) Q dp pc p Performing 2 p on the above equation, substituting into the vorticity equation yields: (ξ z + f) SFC t = V H500 p (ξ z + f) 500 + g f 2 p 500 p SFC VH p ( ) z dp p + 2 p f 500 p SFC ωσ dp + R 2 p fc p 500 p SFC Q p dp 126

This is the Petterssen development equation for the change of surface absolute vorticity due to: V H500 p (ξ z + f) 500 : horizontal vorticity advection at 500 mb. g f 2 p 500 p SFC VH p ( ) z p dp = R 500 f 2 p p SFC V H p p (T ) dp : proportional to a pressure-weighted horizontal temperature advection between the surface and 500 mb. 2 p f 500mb p SFC R 2 p Q fc p p dp : proportional to a pressure-weighted diabatic heating pattern. σω dp : proportional to vertical motion through the layer. 127

The Q Vector In order to keep the mathematical development as simple as possible we will consider the Q-Vector formulation of the omega equation only for the case in which β in neglected. This is usually referred to as an f plane because it is equivalent to approximating the geometry by a Cartesian planar geometry with constant rotation. On the f plane the quasi-geostrophic prediction equations may be expressed simply as follow: Du g Dt Dv g Dt g g DT g Dt fv = (Q1) 0 a 0 + fu = (Q2) 0 a 0 Sρω = 0 (Q3) These are coupled by the thermal wind relationship ug R T vg R T p =, p = p f y p f x 0 0 (Q4) We now eliminate the time derivatives by first taking R p ( Q1) ( Q3) p f y to obtain 0 128

ug ug ug R T T T p + ug + vg f0va + ug + vg Spω = 0 p t x y f0 y t x y Using the chain rule of differential equations, this may be rewritten as RS p ω va ug R T f0 p = + ug + vg p f0 y p t x y p f0 y ug ug vg ug R ug T vg T p + + + p x p y f y x x y 0 But, by the thermal wind relation (Q4) the term in parenthesis on the righthand side vanishes and ug ug vg ug R T ug T ug p + = p x p y f y x x y 0 Using these facts, plus the fact that u / x+ v / y = 0 g g we finally obtain the simplified form ω 2 v σ f a 0 = 2Q2 y p where R ug T vg T R Vg Q2 T p + y y y y = p y Similarly, if we take R p ( Q2) + ( Q3) p f x 0 (Q5) 129

followed by application of (Q4) we obtain where Q 1 2 u a 0 2Q1 ω σ f = x p R u T v T R V p x x x y p x g g g + = T (Q6) If we now take ( Q6) / x+ ( Q5)/ y and use the continuity equation to eliminate the ageostrophic wind, we obtain the Q-vector form of the omega equation: 2 2 2 ω σ ω + f0 = 2 Q 2 p where R Vg R Vg Q ( Q1, Q2) = T, T p x p y This shows that on the f plane vertical motion is forced only by the divergence of Q. Unlike the traditional form of the omega equation, the Q- vector form does not have forcing terms that partly cancel. The forcing of ω can be represented simply by the pattern of the Q-vector. Hence, regions where Q is convergent (divergent) correspond to ascent (descent). 130

Q vectors (bold arrow) for idealized pattern of isobars (solid) and isotherms (dashed) for a family of cyclones and anticyclones. (After Sanders and Hoskins, 1990). Orientation of Q vectors (bold arrows) for confluent (jet entrance flow. Dashed lines are isotherms. (After Sanders and Hoskins, 1990). 131

Potential Vorticity 132

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