Über die Autorin bzw. den Autor

Jonathan E. Martin is a Professor in the Department of Atmospheric and Oceanic Sciences at the University of Wisconsin-Madison where he has taught since 1994.  He has received numerous accolades for his teaching including the Underkofler Excellence in Teaching Award and is a Fellow in the Teaching Academy of the University of Wisconsin.  His teaching excellence is allied with  research expertise in the study of mid-latitude weather systems. Professor Martin has published extensively in scholarly journals and was awarded the distinction of being named a Mark H. Ingraham Distinguished Faculty Member by the College of Letters and Science at UW-Madison.

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Mid-Latitude Atmospheric Dynamics: A First Course provides an introduction to the physical and mathematical description of mid-latitude atmospheric dynamics and its application to the diagnosis of extratropical cyclones. Requiring a background in physics and calculus but no prior knowledge of meteorology, this student-friendly text places the emphasis on conceptual understanding.

Written in a conversational tone, this text is an ideal companion for a first course in the subject, delving into greater depth as the book, and the student, progresses. Real weather examples are woven through the more mathematically focused early chapters, while later chapters introduce a range of case-studies from around the globe to illustrate theoretical and phenomenological aspects of the mid-latitude cyclone life cycle.

• features end of chapter bibliography and problems
• takes a conceptual building block approach
• includes numerous real weather examples from around the globe

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Mid-Latitude Atmospheric Dynamics

John Wiley & Sons

Chapter One

Objectives

Regions of upward vertical motion are often associated with clouds and precipitation since rising air cools by expansion. This cooling increases the relative humidity of the air which can eventually lead to condensation and cloud formation. Regions of rising air are also often associated with mass divergence in an atmospheric column and, consequently, surface pressure falls and cyclogenesis. Regions of downward vertical motion are often cloud free as air dries and warms upon being compressed as it sinks to higher pressure. Mass convergence into an atmospheric column, characteristic of regions of downward vertical motion, results in surface pressure rises and surface anti-cyclogenesis. As a result of the fundamental nature of these relationships, it is not an exaggeration to say that determination of where, when, and to what degree the air is rising or sinking is of fundamental importance for accurately diagnosing the current weather or predicting its future state. In this chapter we will investigate a number of different methods for diagnosing synoptic-scale vertical motions in typical mid-latitude weather systems.

Some of these diagnostic methods will derive from careful consideration of the ageostrophic wind vector itself. Several others (the Sutcliffe development theorem as well as the traditional and [??]-vector forms of the quasi-geostrophic omega equation) will arise from simultaneously solving the quasi-geostrophic vorticity and thermodynamic energy equations for the vertical motion, [omega], and will make reference only to the instantaneous mass distribution. Taken together, the collection of diagnostics to be developed in this chapter will provide us with a formidable set of tools for understanding the synoptic-scale behavior of mid-latitude weather systems. We begin our investigation by considering the ageostrophic wind.

6.1 The Nature of the Ageostrophic Wind: Isolating the Acceleration Vector

Recall that the geostrophic wind is non-divergent on an f plane. In fact, under such conditions only departures from geostrophy contribute to horizontal divergence and, through the continuity of mass, to vertical motions as shown in (4.9). For this reason it is extremely important to examine means by which the ageostrophic motions in the mid-latitude atmosphere might be diagnosed. We begin with the frictionless equation of motion

d[??]/dt = -f [??] [??] - [nabla][empty set], (6.1)

and take the vertical cross-product of this expression to obtain

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (6.2a)

The right hand rule dictates that [??] [??] [??] = -[??], and [[??].sub.g] = ([??]/ f) [nabla][empty set, so

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (6.2b)

The famous British meteorologist R. C. Sutcliffe reasoned that surface pressure falls resulted from vertical differences in mass divergence in a column. Larger mass divergence aloft than at the surface resulted in surface pressure falls and vice versa for surface pressure rises. Such differences in divergence could be related to differential accelerations at the surface and aloft through application of (6.2b). Thus, Sutcliffe argued that isolation of the acceleration vector could give insights into the sense of the vertical motion in an atmospheric column. Before presenting the elegant theory of Sutcliffe (1939), let us endeavor to isolate the acceleration vector, and its ageostrophic consequences, in two rather simple cases. These cases correspond to the two broad classes of circumstances in which geostrophic balance is violated: the presence of along-flow speed change and curvature in the flow.

The canonical synoptic example of along-flow speed change is the isolated jet streak. Shown in Figure 6.1 is the isotach distribution associated with an isolated wind speed maximum at 300 hPa in the northern hemisphere. The dashed line drawn perpendicular to the jet axis divides the jet into the so-called entrance region to its left and the exit region to its right. A parcel of air located on the western edge of the entrance region (indicated by the solid circle in Figure 6.1) would quite obviously experience an acceleration in the direction of the flow at that location. Hence, the vector d[??]/dt points eastward toward the center of the jet streak. Consequently, the ageostrophic wind vector, [[??].sub.ag], points northward at the indicated point. The result of this distribution of ageostrophic winds in the entrance region of the jet is that there is convergence of air at 300 hPa to the north of the indicated position and divergence of air at 300 hPa to the south of the indicated position. Given that 300 hPa is nearly at the top of the troposphere, upper-level divergence (convergence) is associated with upward (downward) vertical motion in the intervening column and so a thermally direct vertical circulation generally exists in the entrance region of a straight jet streak.

A parcel of air located on the eastern edge of the exit region (indicated by the open circle in Figure 6.1) would quite obviously experience a deceleration in the direction opposite the flow at that location. Hence, the vector d[??]/dt points westward toward the center of the jet streak. Consequently, the ageostrophic wind vector, [[??].sub.ag], points southward at the indicated point. The result of this distribution of ageostrophic winds in the exit region of the jet is that there is convergence of air at 300 hPa to the south of the indicated position and divergence of air at 300 hPa to the north of the indicated position. Upward vertical motion occurs in the column beneath the upper divergence maxima and, thus, a thermally indirect vertical circulation generally exists in the exit region of a straight jet streak.

Curvature in the flow is also a circumstance that violates the geostrophic assumption. Consider flow through an upper tropospheric trough-ridge couplet where the wind speed is constant and everywhere parallel to the geopotential height lines as shown in Figure 6.2. Under such circumstances, the acceleration of the wind will be entirely a consequence of directional changes. Thus, between points A and B in Figure 6.2, a southwestward-directed acceleration is required to turn the wind from westerly at point A to northwesterly at point B. There is no direction change between points B and C and, thus, no acceleration vector. A northeastward-directed acceleration is required to turn the northwesterly wind at point C to a westerly direction at point D. In order to turn the westerly at point D to a southwesterly at point E, a northwestward-directed acceleration is required. No change in direction exists between points E and F but a change from southwesterly at F to westerly at point G requires a southeastward-directed acceleration as shown. Given the four acceleration vectors drawn in Figure 6.2, it is simple to draw the ageostrophic winds in this trough-ridge couplet. The ageostrophic winds clearly converge on the western side of the upper trough (on its upstream side) leading to downward vertical motion in the column in that location. Meanwhile, the divergence of the ageostrophic winds on the downstream side of the upper trough is associated with upward vertical motions in the column in that location. This result provides a first insight into the physical reason why inclement weather is often found downstream of upper-level trough axes while clear skies are often found downstream of upper-level ridge axes. This basic relationship lies at the heart of understanding the distribution of sensible weather in the middle latitudes.

6.1.1 Sutcliffe's expression for net ageostrophic divergence in a column

Having examined the distribution of the ageostrophic winds in these canonical synoptic examples, let us now turn our attention to the insightful work of Sutcliffe (1939). We begin by considering the surface wind [[??].sub.0], the wind at some upper tropospheric level, [??], and the vertical shear between the two layers, [[??].sub.s]. Based upon these simple definitions, it is clear that [??] = [[??].sub.0] + [[??].sub.s] and therefore

d[??]/dt = d [[??].sub.0]/dt + d [[??].sub.s]/dt ( 6.3)

where

d/dt = [partial derivative]/[partial derivative]t + [??] * [nabla]

is the Lagrangian operator used to describe d[??]/dt. Given these definitions, (6.3) can be expanded into

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (6.4)

Alternatively, (6.4) can be written as

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (6.5)

Recognizing that the first two terms on the RHS of (6.5) describe the acceleration of the wind at the surface, [(d[??]/dt).sub.0], an expression for the differential acceleration in the layer arises:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (6.6)

This expression relates the fact that if there is shearing over the surface wind ([[??].sub.s] * [nabla][[??].sub.0]) or a change in the shear vector (d[[??].sub.s]/dt) then there must be a difference in acceleration between the upper tropospheric wind and the surface wind. Based upon (6.2b), this implies that there must be some net divergence in the column and therefore, by continuity, vertical motions. Let us now examine the physical significance of the two terms on the RHS of (6.6). As we will do with every other diagnostic expression, we will consider the effect of each term in isolation, beginning with the shearing over the surface wind.

(a) Shearing over the surface wind: d[??]/dt - [(d[??]/dt).sub.0] = [[??].sub.s] * [nabla][[??].sub.0]

Our expression begins by first expanding this term into its full component form given by

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (6.7)

Figure 6.3 depicts a schematic sea-level pressure minimum and some 1000-500 hPa thickness contours. Considering the value of this term at the center of the low-pressure center greatly reduces the mathematical complexity of applying (6.7). We will assume that the winds are geostrophic everywhere, which dictates that a thermal wind vector be directed along the positive y-axis in the northern hemisphere. At the center of the low, therefore, there is no x-direction vertical shear so that [u.sub.s] = 0. It is also clear that there is no [partial derivative][v.sub.0]/[partial derivative]y at the center of the low-pressure center. Thus, (6.7) reduces to

[[??].sub.s] * [nabla][[??].sub.0] = [v.sub.s] [partial derivative][u.sub.0]/[partial derivative]y [??]

for the scenario illustrated in Figure 6.3. We have already found that [v.sub.s] is positive in this case. We now discern that [partial derivative][u.sub.0]/[partial derivative]y is negative so that the product [v.sub.s] [partial derivative][u.sub.0]/[partial derivative]y is negative. Consequently, the vector [[??].sub.s] * [nabla] [[??].sub.0] points in the negative x direction as indicated. Since [[??].sub.s] * [nabla][[??].sub.0] represents the acceleration at the top of the column minus the acceleration at the bottom of the column, taking the vertical cross-product of [[??].sub.s] * [nabla][[??].sub.0] indicates the direction of the column-differential ageostrophic wind. Figure 6.3 shows that, in this case, there is greater ageostrophic divergence (convergence) aloft than at the surface north (south) of the surface low implying ascent (descent) in that location. The surface cyclone will propagate toward the net column mass divergence (i.e. in the direction of the ascending air) as only mass divergence and ascending air will be associated with sustained pressure falls at the surface. Application of similar reasoning to the case of a surface anticyclone (a recommended exercise for the reader) leads us to a general statement: the sea-level pressure perturbation will propagate in the direction of the thermal wind.

(b) Rate of change of the shear vector: d[[??].sub.s]/dt

Figure 6.4(a) shows some 1000-500 hPa thickness isopleths along with the thermal wind vector in the layer in the northern hemisphere at some time T = 0. Some time later (T = [T.sub.1]), the horizontal thickness gradient has been increased by some agency in the atmosphere such as confluent horizontal flow. The result of such an increase in the baroclinicity is a larger thermal wind, still directed to the north as shown in Figure 6.4(b). If we assume the winds are everywhere geostrophic, then the difference in the thermal wind vectors in Figures 6.4(a) and 6.4(b) represents a change in the shear vector and can be represented by the expression d[[??].sub.s]/dt so long as the change has been measured following an individual air parcel. Thus, d[[??].sub.s]/dt is directed in the positive y direction. The vertical cross-product of d[[??].sub.s]/dt (which points directly toward the low thicknesses in Figure 6.4b) represents the column-differential ageostrophic wind for this example. The column of air on the warm (cold) side of the thickness gradient experiences greater divergence (convergence) aloft than at the surface and so it rises (sinks). Thus we find that anytime the horizontal flow acts to increase the thickness (or temperature) gradient, the response is the development of a thermally direct vertical circulation in which warm air rises and cold air sinks. Conversely, any systematic relaxation of the horizontal gradient of temperature by the action of the horizontal flow induces a thermally indirect vertical circulation. This physical insight, a direct consequence of the fact that the rate of change of the shear vector produces divergence in the column, is central to the dynamics of frontogenesis, a topic we will explore in great detail in Chapter 7.

6.1.2 Another perspective on the ageostrophic wind

We now turn our attention to a more formal expansion of the ageostrophic wind relationship (6.2b) which, recall, stated that

[[??].sub.ag] = [??]/f d[??]/dt.

The Lagrangian derivative in the preceding expression can be expanded so that

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (6.8)

The three terms on the RHS of (6.8) represent three contributions to the total ageostrophic wind: (1) the local wind tendency component, (2) the inertial advective component, and (3) the convective component. If we substitute [[??].sub.g] for [??] everywhere in (6.8) then

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (6.9)

Our aim in this development is to diagnose the synoptic-scale vertical motion by first isolating the distribution of the ageostrophic wind. As is clear from (6.9), diagnosis of the convective component of the ageostrophic wind requires a priori knowledge of the vertical motion. Thus, it is not feasible to perform the intended diagnosis on the convective component. For this reason we will consider only the first two terms on the RHS of (6.9) in the foregoing analysis, starting with the local wind tendency component.

The local wind tendency component of the ageostrophic wind ([[??].sub.[ag.sub.T]) can be related to geopotential height or pressure changes since

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (6.10a)

on pressure surfaces or

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (6.10b)

on height surfaces. This component of the ageostrophic wind is known as the isallobaric wind as a result of its dependence on the gradient of isallobars (lines of constant pressure tendency, [partial derivative]p/[partial derivative]t). Knowledge of the isallobaric wind, like any component of the ageostrophic wind, only tells us about the distribution of vertical motion when we know its divergence. Thus, we are most interested in the divergence of the isallobaric wind, given by

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (6.11a)

on pressure surfaces or

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (6.11b)

on height surfaces. It is left as an exercise to show that pressure (or geopotential) falls are associated with convergence of the isallobaric wind while pressure (or geopotential) rises are associated with divergence of the isallobaric wind.

The inertial-advective component ([[??].sub.IA]) of the ageostrophic wind is given by

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (6.12)

(Continues...)

Excerpted from Mid-Latitude Atmospheric Dynamicsby Jonathan E. Martin Copyright © 2006 by John Wiley & Sons, Ltd.. Excerpted by permission.
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