1 such that v k g. v g. u g

Transcription

1 Mesoscale Meteoroloy: Quasi-Geostrohic Theory 4, 6 February 7 Wait this is a mesoscale class why do we care about a tenet o synotic meteoroloy?? On the synotic-scale, scale analysis o the orcin terms in the equations o motion leads us to tyically nelect horizontal arcel accelerations in curved low (the eostrohic aroximation) and vertical arcel accelerations altoether (hydrostatic balance). Parcel accelerations cannot be nelected on the mesoscale, however: they are key asects o mesoscale dynamics. So, why do we send a week coverin rinciles that are oten violated on the mesoscale? Mesoscale eatures orm and evolve within the backround synotic-scale environment. Smallerscale variations in boundary lacement, toorahy, etc. modulate recisely when and where mesoscale henomena occur, but such variability only matters i the larer scales are suortive o a iven henomenon in the irst lace. O articular interest are the ositions and intensities o trouhs and rides (and thus surace boundaries) and the sin and manitude o vertical motion, and quasi-eostrohic theory oers several tools to hel us with such assessments. Note that these notes are not meant to be comrehensive. They assume backround knowlede o the scalin assumtions that enter into eostrohic balance, or instance. They do not cover all stes within the derivations that result in the equations below. Textbooks in synotic or dynamic meteoroloy cover these in more detail. Alternatively, you may wish to review rimers on these toics available at: Geostrohic balance: htt://derecho.math.uwm.edu/classes/syni/geosarox.d Thermal wind: htt://derecho.math.uwm.edu/classes/syni/thermalwind.d Intro to Q-G theory: htt://derecho.math.uwm.edu/classes/synii/qgvorticity.d Q-G heiht tendency: htt://derecho.math.uwm.edu/classes/synii/qghttend.d Q-G omea equation: htt://derecho.math.uwm.edu/classes/synii/qgomea.d Q-vector ormulation: htt://derecho.math.uwm.edu/classes/synii/qvectors.d Thermal Wind Formal Deinition Geostrohic balance allows us to exress the horizontal equations o motion as: v x u y such that v k where Φ = z is the eootential and z is heiht. I we take the artial derivative o u and v with resect to ressure, substitute with the hydrostatic relationshi and ideal as law, and interate between two isobaric levels and (where >, such that has hiher altitude), we obtain: v R T ln R T ln d v d v T v v u x T u u y

2 Or, in vector orm, v T v Rd v k T ln v Here, vt = (ut, vt) deines the thermal wind, and T v is the virtual temerature averaed over the layer between and. The thermal wind deines the relationshi between how the eostrohic wind chanes with heiht and the horizontal radient o layer-mean virtual temerature (which is directly roortional to the thickness). It imlies a symbiotic relationshi between vertical wind shear and the horizontal layer-mean virtual temerature radient. From insection o the equations above, it can be shown that the thermal wind blows arallel to the isotherms (o layer-mean virtual temerature) with warm air to the riht o the wind. We can rove this with a thouh exeriment. Consider only a north-south layer-mean virtual temerature radient, such that vt =, with warm air to the south. Layer-mean virtual temerature decreases with the north, such that its meridional radient is neative. Since Rd is ositive, is ositive in the Northern Hemishere, and the natural loarithm is always ositive, the leadin neative on ut ensures that ut >, deinin an eastward-directed wind since vt =. Alication to Temerature Advection and Hodorahs I we know the eostrohic wind or, aroximately, the ull wind at two ressure levels, we can deine the thermal wind over that layer usin vector subtraction: vt() - vt(). Grahically, this can be accomlished by lacin the oriin o both vectors at a common location and drawin a vector rom the end o vt() to the end o vt(), as deicted in Fi. below. Note that this is identical to the rocess used to construct a hodorah. Fiure. (let) A thermal wind vt associated with cold air advection. (riht) A thermal wind vt associated with warm air advection. Isotherms o layer-mean Tv (or, nearly equivalently, T) are deicted by the dashed black lines arallel to vt. Desite identical direction and manitude to the thermal wind in each anel, horizontal temerature advection sin diers because o dierences in how the eostrohic wind varies with heiht (and is oriented with resect to the isotherms) in the layer between and.

3 The let anel o Fi. deicts backin winds, winds that turn counterclockwise with heiht. The riht anel o Fi. deicts veerin winds, winds that turn clockwise with heiht. From Fi., backin winds and counterclockwise-turnin hodorahs are associated with layer-mean cold virtual temerature advection. Veerin winds and clockwise-turnin hodorahs are associated with layer-mean warm virtual temerature advection. We aroximately state that veerin winds are associated with warm air advection and backin winds are associated with cold air advection. The imortance o temerature advection to synotic and mesoscale meteoroloy will be made clearer later in this lecture. Geostrohic Vorticity and Diverence The eostrohic vorticity and diverence have the ollowin deinitions: v x u y D u x v y I we substitute in the deinitions or u and v iven earlier and let = (a constant value o the Coriolis arameter), we obtain or ζ: x x y y x y The subscrit o on the Lalacian oerator indicates that it is evaluated on an isobaric (constant ressure) surace. This equation indicates that ζ is related to the Lalacian o the eootential. Where the eootential is a relative minimum (maximum), ζ is a relative maximum (minimum). Thus, cyclonic ζ is maximized in the base o a trouh, while anticyclonic ζ is maximized in the aex o a ride. Likewise, i we substitute in the deinitions or u and v iven earlier in this lecture and let =, we obtain or D: D x y y x In other words, assumin a constant Coriolis arameter, the eostrohic wind is non-diverent! A Note on Advection and Partial Derivatives Throuhout the remainder o this and subsequent lectures, we will encounter terms o the orm v a, where a is some eneric scalar quantity. These reresent advection, or the transort o some quantity by the wind. Consider a two-dimensional advection term in comonent orm: a a v a u v x y We can aroximate the artial derivatives with centered inite dierence aroximations, i.e., 3

4 a x a x a x x and a y a y a y y where (x, y) denotes the location where the inite dierence is evaluated, x+ is a oint alon the ositive x-axis a distance Δx away rom x, x- is a oint alon the neative x-axis a distance Δx away rom x, and y+ and y- are deined similarly excet relative to y. Qualitative interretation does not require us to deine Δx or Δy, just to deine equally-saced oints on each side o (x, y). Let us consider the two examles iven in Fi. below. Fiure. (let) An east-west radient o a, with lower (hiher) values to the west (east). (riht) A local minimum o a. The reen and old vectors denote two wind vectors. Please see the text or urther details. Focus irst on the examle at let. At the location o the reen arrow, v = and u >. Let the x+ oint be located at a+δa and the x- oint be located at a-δa. Thus, a a x x a x x a a a a x a x Since Δa is ositive and Δx is deined as ositive, this term is ositive. For u ositive, the leadin neative indicates that v a, deinin neative advection. For neative advection, the wind transorts or advects lower values o a to a iven location. Conversely, at the old arrow, v = and u <. Settin the x+ and x- oints at the same locations as beore, we ind that a/ x is the same value as in the revious examle. But, u is neative, so the leadin neative indicates that v a, deinin ositive advection. For ositive advection, the wind transorts hiher values o a to a iven location. Let us consider the examle at riht, with a westerly wind blowin throuh a local minimum o a. At the location o the reen arrow, v = and u >. I we let the x+ oint be located at a+δa and the x- oint be located at a, we obtain: 4

5 a a x x a x x a a a x a x This quantity, as in the other examle, is ositive. For u >, the leadin neative aain indicates that v a. At the old arrow, v = and u >. I we let the x+ oint be located at a and the x- oint be located at a+δa, we obtain: a a x x a x x a a a x a x This quantity is neative. For u >, the leadin neative indicates that v a. The indins or this examle can be interreted in liht o how the wind transorts the local minimum. Areas downwind see the local minimum transorted toward them, thus decreasin values by advection, while areas uwind see the local minimum transorted away rom them, thus increasin values by advection. Toether, these examles let us state the ollowin uidelines, indeendent o wind direction: Wind that blows rom lower toward hiher values deines neative advection. Wind that blows rom hiher toward lower values deines ositive advection. Wind blowin throuh a local minimum will have neative advection downwind (where the wind blows toward) and ositive advection uwind (where the wind blows rom). Wind blowin throuh a local maximum will have ositive advection downward and neative advection uwind. As we are oten interested in qualitative rather than quantitative uidance, these uidelines hel us quickly interret the sin o advection. For temerature-related variables, warm advection is analoous to ositive advection. For vorticity-related variables, cyclonic advection is analoous to ositive advection in the Northern Hemishere. In the ollowin, we will also encounter artial derivatives o the sort a/, where ressure is the vertical coordinate. Thus, these terms describe vertical variation in a quantity a. These can be aroximated usin centered inite dierences as beore, i.e., a a a where the + oint is deined above and the - oint is deined below the level at which the inite dierence is evaluated. However, or this deinition, note that Δ is neative because decreases with heiht: at + is smaller than at -. Thus, i a increases with heiht, a/ is neative, while i a decreases with heiht, a/ is ositive. 5

6 Quasi-Geostrohic Heiht Tendency Equation Formal Deinition It can be shown that the quasi-eostrohic orms o the vorticity and thermodynamic equations can be written in terms o two unknowns: vertical motion ω (artial derivative o ressure with time, so ositive values indicate downward motion) and eootential Φ. The quasi-eostrohic heiht tendency equation is obtained by maniulatin these equations to eliminate ω. Doin so, one obtains the ollowin equation: h v v R dq K c dt This is a artial dierential equation describin the local time-rate o chane o the eootential, where χ = Φ/ t, on an isobaric surace. It is written with ressure as the vertical coordinate. In the above, σ deines static stability and is assumed to be constant in the vertical, ζ + deines the eostrohic absolute vorticity, h is a unction o ressure and is ositive-deinite, K is a rictional coeicient and is ositive-deinite, and dq/dt is the diabatic heatin rate and is ositive or diabatic warmin. All other variables have their standard meteoroloical meanin. It contains our orcin terms on the riht-hand side: Geostrohic absolute vorticity advection Dierential (in the vertical) otential temerature advection Friction Dierential (in the vertical) diabatic heatin The let-hand side o the equation exresses χ in terms o its second artial derivative in x, y, and. Where a variable is a local min (max), the second artial derivative is a local max (min). As a result, we can exress the quasi-eostrohic heiht tendency equation as: h R dq v K v c dt A ositive (neative) eootential heiht tendency χ corresonds to risin (allin) heihts. Interretation We can interret the orcin terms to the quasi-eostrohic heiht tendency equation as ollows, ocusin on allin heihts. The oosite holds or risin heihts. Cyclonic eostrohic absolute vorticity advection on an isobaric surace results in allin eootential heiht on that isobaric surace. In eneral, this term only controls the motion o the trouh/ride attern; it enerally does not inluence trouh/ride amlitude. 6

7 Cold advection decreasin with heiht, or warm advection increasin with heiht, over a inite ressure layer centered on an isobaric surace results in allin eootential heiht on that isobaric surace. This term can inluence both trouh/ride motion and amlitude. Friction acts to cause heihts to all near the surace or anticyclonic eostrohic absolute vorticity. This term acts only near the surace and is o comaratively small manitude. Diabatic coolin decreasin with heiht, or diabatic warmin increasin with heiht, over a inite ressure layer centered on an isobaric surace results in allin eootential heiht on that isobaric surace. This term is imortant when diabatic rocesses are imortant but is neliible otherwise. While the quasi-eostrohic heiht tendency equation can be used to redict eootential heiht, it is instead oten used as a dianostic equation. It is tyically alied in the middle trooshere. The dierential otential temerature advection term can be evaluated usin thermal wind or by evaluatin otential temerature advection on isobaric levels above and below the isobaric level on which the quasi-eostrohic heiht tendency equation is evaluated. The ability to use thermal wind or this evaluation means that, i you aroximate the eostrohic wind with the ull wind, the local eootential heiht tendency can be evaluated rom a vertical soundin! Direct measurements o diabatic heatin rate are tyically not available. Instead, we can iner the sin and relative manitude o this term usin other data and knowlede o atmosheric hysics. For instance, the resence o clouds above and a relatively dry layer below a iven isobaric level iners condensation and latent warmin in the cloud and otential evaoration and latent coolin in the layer below. Quasi-Geostrohic Omea Equation Formal Deinition I one maniulates the quasi-eostrohic vorticity and thermodynamic equations to eliminate Φ instead o ω, a dianostic equation or ω the quasi-eostrohic omea equation is obtained: R dq v h v K c dt This is a artial dierential equation or vertical motion ω on an isobaric surace. It contains our orcin terms on the riht-hand side: Dierential (in the vertical) eostrohic absolute vorticity advection Potential temerature advection Dierential (in the vertical) riction Diabatic heatin The let-hand side o the equation exresses ω in terms o its second artial derivative in x, y, and. Thus, as we did with the quasi-eostrohic heiht tendency equation, we can exress the quasi-eostrohic omea equation in terms o a roortionality: 7

8 R dq v h v K c dt Interretation Usin the same deinitions introduced with the quasi-eostrohic heiht tendency equation, we can interret the orcin terms to the quasi-eostrohic omea equation as ollows, ocusin on ascent (ω < ). The oosite holds or descent. Cyclonic eostrohic absolute vorticity advection that increases with heiht over a inite ressure layer centered on an isobaric surace results in ascent across the isobaric surace. Warm advection on an isobaric surace results in ascent across the isobaric surace. Because K ~ above the surace, cyclonic eostrohic absolute vorticity near the surace results in lower troosheric ascent. This is known as Ekman umin. This occurs only in the lanetary boundary layer. Diabatic warmin on an isobaric surace results in ascent across the isobaric surace. Vertical motion is tyically maximized in the middle trooshere, and so the quasi-eostrohic omea equation is tyically alied on middle troosheric isobaric suraces (7-3 hpa). It is only a dianostic equation; there are no time derivatives in its ormulation. Because o the relationshi between eostrohic absolute vorticity and eootential rom earlier, there is a symbiotic link between the quasi-eostrohic heiht tendency and omea equations. I the eootential heiht on an isobaric surace alls, then the eostrohic absolute vorticity on that isobaric surace increases. This imlies reater cyclonic eostrohic absolute vorticity advection on that isobaric surace and, deendin on its vertical structure, a otential or reater dierential cyclonic eostrohic absolute vorticity advection and thus stroner orcin or ascent. As with the quasi-eostrohic heiht tendency equation, otential temerature advection can be evaluated usin thermal wind or rom a satial analysis on the chosen isobaric level. The ability to use thermal wind or this evaluation means that, i you aroximate the eostrohic wind with the ull wind, the sin and relative manitude o vertical motion can be evaluated rom a vertical soundin! The oosite o Ekman umin is known as Ekman suction. Inerences o the sin and relative manitude o diabatic heatin rate ollow rom those or the quasi-eostrohic heiht tendency equation; e.., stron latent warmin in thunderstorms rovides urther orcin or ascent. The Q-Vector Form o the Quasi-Geostrohic Omea Equation Formal Deinition The two rimary orcin terms o the quasi-eostrohic omea equation, dierential eostrohic absolute vorticity advection and otential temerature advection, can and oten do have oosite sin to each other. This is the rimary motivation or develoin an alternative equation, one that does not suer rom this roblem: the Q-vector equation. As we will ind, it also has roerties that make it attractive or assessin rontoenesis, a toic which we will discuss next week. 8

9 To obtain the Q-vector equation, one starts with the quasi-eostrohic horizontal momentum and thermodynamic equations. Substitutions rom thermal wind and eostrohic diverence, couled with considerable maniulation, are required to obtain the Q-vector equation: where Q = (Q, Q) and: Q or, roortionally, Q Q R v x T Q R v y T Thus, Q-vector converence Q is associated with orcin or synotic-scale ascent, while Q-vector diverence Q is associated with orcin or synotic-scale descent. Interretation Most analyses o the Q-vector equation make use o automated routines to comutin Q-vectors rom ridded analyses o atmosheric data. However, one can estimate Q-vectors as ollows. Let us erorm an axis transormation, deinin the x-axis to be arallel to an isotherm (o T, in an aroximate sense) with warm air to the riht. The y-axis is deined erendicular and to the let o the x-axis. An examle is rovided in Fi. 3 below. Fiure 3. Idealized deiction o the axis transormation described above. North (east) is to the to (riht) o the iure; however, the new x-axis is deined arallel to the isotherms with warm air to the riht, with the new y-axis erendicular to the let o the x-axis. In this coordinate system, T/ x is zero. This allows us to simliy Q and Q. I we do so and aly the non-diverence o the eostrohic wind, we obtain: R Q T y v k x 9

10 Rd and are both ositive constants on a iven isobaric surace. These terms, alon with the manitude o the cross-isotherm temerature radient, control the manitude but not direction o the Q vector. I we are rimarily interested in the Q vector s direction, then we can aroximate the above with: v v Q k, where x x v x v x To evaluate Q, we irst ind the vector chane in v alon the isotherm (the rotated x-axis). Next, we aly the -k x oerator, which indicates a 9 clockwise rotation (i.e., to the riht) o this vector. Let us aly this to the idealized trouh-ride attern iven in Fi. 4 below. x Fiure 4. Idealized trouh-ride attern, as deined by the reen streamlines, with a ride (H) to the west and a trouh (L) to the east. Dashed black lines indicate isotherms, with colder air to the north. Solid dark rey arrows indicate Q-vectors at three locations, ride, trouh, ride rom west to east. Consider the H in Fi. 4. Here, the new x-axis is ointed just slihtly south o due east, alon the isotherm. Let the x+ oint lie at the intersection o isotherm T with the reen streamline to the east and the x- oint lie at the intersection o isotherm T with the reen streamline to the west. In this case, the eostrohic wind at x+ is out o the north and the eostrohic wind at x- is out o the south. I we lace them at a common oriin and draw a vector rom the end o that at x- to the end o that at x+, we obtain a vector ointed rom north to south. Alication o the -k x oerator rotates this vector 9 clockwise, such that it oints toward the west. Consider the L in Fi. 4. Here, the new x-axis is ointed just slihtly south o due east, alon the isotherm. Let the x+ oint lie at the intersection o isotherm T with the reen streamline to the east and the x- oint lie at the intersection o isotherm T with the reen streamline to the west. In this case, the eostrohic wind at x+ is out o the south and the eostrohic wind at x- is out o the north. I we lace them at a common oriin and draw a vector rom the end o that at x- to the end o that at x+, we obtain a vector ointed rom south to north. Alication o the -k x oerator rotates this vector 9 clockwise, such that it oints toward the east.

11 I we evaluate Q-vector diverence at a oint between the H and L, we et ositive diverence, such that ω >, indicatin orcin or descent. The oosite is true downstream o the L, ahead o the next H: neative diverence (converence), such that ω <, indicatin orcin or ascent. Connection to Frontoenesis A key advantae o the Q-vector ormulation versus the quasi-eostrohic omea equation is its connection to rontoenesis, indicatin how the manitude o the horizontal temerature radient (a metric o rontal strenth) chanes with time. The quasi-eostrohic thermodynamic equation can be maniulated to show that: d dt R T Q ˆi ˆ Q j Here, the total derivative is written with a subscrit o, indicatin that the wind that enters into its deinition is the eostrohic rather than the ull wind. This equation indicates that the rate o chane o the horizontal temerature radient ollowin the eostrohic wind is a unction o the Q-vector. Perhas a more useul exression, however, would be one or the manitude o the horizontal temerature radient, i.e., d T dt This deinition can be exanded to show that: d dt T R T T Q This indicates that the rate o chane o the manitude o the horizontal temerature radient is related to the orientation o the temerature radient (always ointed rom cold to warm air) and the Q-vector. I these two vectors are erendicular to each other, their dot roduct is zero and the manitude o the horizontal temerature radient does not chane ollowin the low. I these two vectors are arallel to and in the same direction as each other, their dot roduct is ositive and the manitude o the horizontal temerature radient increases ollowin the low. Finally, i these two vectors are arallel to but in oosite directions rom each other, their dot roduct is neative and the manitude o the horizontal temerature radient decreases ollowin the low. Alyin these rinciles to Fi. 4, we ind that the Q-vector is aroximately erendicular to the horizontal temerature radient at each location where the Q-vector was evaluated. Thus, the manitude o the horizontal temerature radient does not chane with time in this examle. Let us consider a dierent examle rom an intense nor easter that develoed in late March 4, as in Fi. 5 below.

12 Fiure 5. 7 hpa eootential heiht (black lines every 3 dam = 3 m), temerature (reen dashed lines every 3 C), Q-vectors (black vectors; reerence vector at bottom let, units: Pa m - s - ), and Q-vector converence (shaded with warm colors denotin orcin or ascent; units: - Pa m - s - ), derived rom GFS model analysis data valid at UTC 6 March 4. Fiure obtained rom htt:// Focus on the orane-circled area just o o the northeast United States coastline. Here, Q-vectors oint rom cold to warm air arallel to and in the same direction as the horizontal temerature radient. This deicts a rontoenetical situation. South o the orane-circled area, the oosite is indicated: Q-vectors that oint rom warm to cold air arallel to and in the oosite direction as the horizontal temerature radient. This deicts rontolysis: weakenin o a ront with time. Fi. 5 illustrates that orcin or both vertical motion and rontal evolution can be identiied with Q-vectors. Areas o rontoenesis oten overla with reions o ascent. While the release o uriht (CAPE) or slantwise instability may be imortant, banded reciitation in the cold sector o mid-latitude cyclones oten occurs where stron rontoenesis and ascent overla, articularly in the 85-5 hpa layer. In next week s lecture, we will more ormally deine rontoenesis and revisit some o these concets.