Fronts And Frontogenesis

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Fronts and Frontogenesis

Definition The definition of a front varies:* from classical polar-front theory, and inpopular usage, it is the boundary betweentwo air masses. Media people often talkabout “clash” between air masses; indeed,usage of term “front” was likely influenced byWWI being contemporary with itsdevelopment (Bjerknes 1919). This ideasuggests the front approaches a“discontinuity” in some atmospheric property

* Other treatments have tended to view thefront as a broader zone of transition, or asa finite region of strong gradients. Theimplication here is that the front does NOTapproach a discontinuity. In fact, it is observed that fronts may fitinto either of these models. Some frontshave been observed as neardiscontinuities while others have not. Afront may evolve through a life cycle froma broad baroclinic zone to a neardiscontinuity and then decay

Frontogenesis Terminology: frontogenesis – creation orintensification of a front (front genesis,birth, creation, formation, Genesis, gene,generate .)frontolysis- destruction or weakening of afront (front lysis, dissolution, destruction,paralysis, analysis .)

Structure of fronts We observe that fronts slope with height,and that they almost always slope towardthe cold air. We can derive a simpleformula that does a reasonable job ofrepresenting this slope warm cold First, we note that pressure must becontinuous across the front. For thetypical case of cold air to the north, andwarm air to the south, the gradients are inthe y-direction.

Structure (cont)The reason the pressure must becontinuous is that for a discontinuity, thepressure gradient would be infinite, thatis dp/dy Δp/ Δy; for Δy 0 is Δp If the pressure gradient were infinite, thecorresponding accelerations in the eqns ofmotion would be infinite, leading to aninfinitely strong wind, which of course isnot observed.

Since pressure is continuous, then both temperature anddensity must be discontinuous, or neither temperatureand density are discontinuous .if the ideal gas lawapplies p ρRTTwTcfrontρwρcρwρcColder air has lowertemperature andthus higher density For the continuous pressure field, we can express thepressure differential asdp p/ y dy p/ z dz(This is just the equation of a line). We recognize p/ z asphysically meaningful and can apply the hydrostaticapproximation

p/ z -ρg to getdp p/ y dy – ρgdzThis equation must apply on both sides ofthe front. Then on the cold side we havedp ( p/ y)cold dy – ρcold g dzand on the warm side,dp ( p/ y)warm dy – ρwarm g dzEquating the rhs of both eqns:( p/ y)c dy – ρc g dz ( p/ y)w dy – ρw g dzCollecting terms in dy and dz gives

[( p/ y)c – ( p/ρy)w]dy (ρc-ρw)gdzOr dz/dy [( p/ y)c – ( p/ρy)w]/g (ρc-ρw)We thus see that non-zero dz/dy requiresthat [( p/ y)c – ( p/ρy)w] also be non-zero.That is, if the front slopes, then there mustbe a discontinuity of the pressure gradient.This is the reason why fronts should beanalyzed with a kink in the isobars!

Let us now assume that the component of thewind parallel to the front is in geostrophicbalance: u ug -1/ρf p/ y Solving for the pressure gradient, p/ y -ρfug Substitute into our eqn for frontal slope: dz/dy [(ρfug)w – (ρfug)c]/g (ρc-ρw) For a narrow frontal zone, we observe that theproportional difference in the Coriolis parameteris very small; e.g., for a frontal zone 10 km wideat 40 N, we have (fw-fc)/f 0.002 (0.2%) Similarly, density differences are small ( 1%)

But the differences in the geostrophic windcan be of similar magnitude to the winditself, i.e.,(ugc – ugw )/ 0.5*(ugc ugw) 0.1 – 1Then the number in the numerator in thefrontal slope eqn is dominated by thechange in geostrophic wind, so we canrewrite the eqn as:dz/dy ρf(ugw – ugc)/g(ρc-ρw)

By the ideal gas law, p ρRT. Then forp constant (continuous across front), thediscontinuous step increase (or decrease)of ρ must be balanced by a correspondingstep decrease (or increase) of T; that is,Δρ/ρ ΔT/T or in terms of our problem,(ρc-ρw)/ρ (Tw – Tc)/T Then we can rewriteour frontal slope eqn as:dz/dy fT/g (ugw – ugc)/(Tw-Tc)

Insights from the equation Velocity difference (ugw – ugc) across a frontalzone of width Δy can be expressed as (ugw –ugc)/Δy Recall the vertical component of vorticity isζ v/ x - u/ yThen, if u decreases northward (i.e., u decreasesas y increases), the front is a zone of positivegeostrophic vorticity, ugw – ugc 0 soU 0U 0This is consistent with the observed kink in theisobars.

Strong fronts (large temperature contrast)do not necessarily slope more than weakones, since (ugw – ugc) also is likely toincrease for a strong front If the shear is cyclonic (ugw – ugc) 0, thendz/dy 0. So for cold air to the north, thefront slopes toward the cold air.

Typical evolution of sea breeze Assume atmosphere at rest, early in themorningJust after sunrise, land heats up. Initially theperturbations are small so the response is linear.Contour ofwind speedOnce perturbations become large, thenonlinear effects cause a front to form (we willstudy this in detail) on the inland side.

Strong narrowupdraftExtensive region of weaksubsidenceAntitriptic flow (Jeffreys,1922 QJRMS) Sea breeze is deeper on the inland sidebecause stable stratification over the watersuppresses the vertical extent.

Frontogenesis in the sea breeze We will begin with a 2D framework, and tryto create an equation for θ/ t *If we define the front as θ/ x, we can get / t(- θ/ x) - / x( θ/ t) -( u/ x)( θ/ x) –( ω/ x) ( θ/ p) -1/cp(p0/p)k / x(dQ/dt) We could also define the front in otherways such as with the convergence ofwind. In that case, we start with umomentum equation

u/ t -1/ρ p/ x -u u/ x -w u/ z fv - / z(u’w’)If we put a minus sign in so that positive values give us astronger front, then. / t(- u/ x) 1/ρ 2p/ x2 u/ x u/ x u 2u/ x2 w/ x u/ z w 2u/ z x -f v/ x / x( / zu’w’) [-u / x (- u/ x) - w / z(- u/ x)] adv. of frontal character 1/ρ 2p/ x2 requires non-constant PGF (2nd derivative curvature) u/ x u/ x convergence (nonlinear) w/ x u/ z tilting of vertical shear into horizontal-f v/ x differential coriolis force – this becomes frontolyticlater in day / x ( / z u’w’) differential friction (usually frontolytic)

Frontogenesis The classical definition of thefrontogenetical function isF D/Dt θ This is just a generalization of our earlierexpression used in discussion of seabreeze frontogenesis. Here we considergradients in any direction (i.e., θ θ/ x θ/ y) and of any sign (as per theabsolute value).

Isentropes along front Consider a case with frontal zone along xaxis and isentropes parallel to front, withno wind variations along front. Also,temperature decreases toward north(increasing y). Then F D/Dt (- θ/ y) ( v/ y)( θ/ y) ( ω/ y)( θ/ p) – 1/cp(p0/p)k / y(dQ/dt) Here – the gradient is in y-direction (not x)and vertical coordinate is pressure

Role of deformation Pure deformation flow is defined as: u/ x v/ y 0 (non-divergent)Or equivalently, u/ x - v/ y Therefore, if we have divergence in xdirection, it has to be exactly balanced byconvergence in y-direction, and vice-versaAxis of dilatation

In this case, the x-axis would be called the axisof dilatation and the y-axis the axis ofcontraction. Qualitatively, we can diagram the effect ofdeformation on the gradient as follows: Consider a control area defined as a rectangle.If the long edge of the rectangle is aligned withthe axis of dilatation, the rectangle getsstretched out longer : since there is nodivergence, area is unchangedθ1θ1θ2θ2

If we assume the long sides of the rectanglecorrespond to isotherms (or adiabats), thenthe effect of deformation in this case isfrontogenetic. Conversely if the long edge of the rectangleis along the axis of contraction, therectangle becomes more of a square, and ifthe long sides are isotherms, thedeformation is frontolytic. For other orientations, the effect ofdeformation will depend on the relativeangle of the axis of dilatation and theisotherms

The effect of horizontal convergence, aswe have seen, is frontogenetic; converselydivergence is frontolytic. Combining theeffects of deformation and divergence inthe along wind direction, F θ /2 (Dcos2b – δ) where b anglebetween axis of dilatation and isotherms.b 0, cos2b 1 (frontogenetic)b 90, cos 2b -1 (frontolytic)b 45, cost2b 0

tilt

Structure of fronts We observe that fronts slope with height, and that they almost always slope toward the cold air. We can derive a simple formula that does a reasonable job of representing this slope First, we note that pressure must be continuous across the front. For the typical case of cold air to the north, and

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