The Equations of Motion Synotic Meteorology I: The Geostrohic Aroimation 30 Setember, 7 October 2014 In their most general form, and resented without formal derivation, the equations of motion alicable on constant height surfaces and osed in Cartesian coordinates are given by: Dv 1 = 2 Ω v + g + F r (1) ρ From left to right, the terms of (1) reresent the total derivative of the three-dimensional wind field v, the Coriolis force, the ressure gradient force, effective gravity, and friction. The total derivative in the first term of (1) has the general form: D ˆ i ˆ = + u + v j + w kˆ t y Equation (1) may be eanded into its comonent forms and recast into sherical coordinates. Presented without derivation, the corresonding equations of motion (for u and v only) are: Du uv tan uw φ + fv + 2Ωwcosφ = 1 + a a ρ 2 Dv u vw + tanφ + + fu = 1 + F ry a a ρ y F r (2a) (2b) where ϕ = latitude, Ω = the rotation rate of the Earth (7.292 10-5 s -1 ), u = onal comonent of the wind, v = meridional comonent of the wind, w = vertical comonent of the wind, = ressure, ρ = density, f = 2Ω sin ϕ = Coriolis arameter, and a = Earth s radius (6.378 10 6 m). The second and third terms of (2) reresent terms related to the curvature of the Earth. The fourth term of (2b) and fourth and fifth terms of (2a) are Coriolis terms. The two terms on the righthand side of (2) are ressure gradient and frictional terms, resectively. These equations, alongside the corresonding equation for w, define the equations of motion. We note that these forms of the equations of motion are resented with height, or, as the vertical coordinate. Scale Analysis of the Equations of Motion The equations of motion, as stated in their full form in (2) above, are quite comle. They describe all tyes and scales of atmosheric motions, including some (e.g., acoustic waves) that are either negligible or altogether irrelevant for the study of synotic-scale motions. The Geostrohic Aroimation, Page 1
We desire to simlify (2) into a form that reflects only the most imortant rocesses on the synotic-scale. In other words, we want to kee only those terms that are large for synotic-scale motions. We do so by erforming a scale analysis on the terms in (2). This is done by relacing each variable with an aroriate characteristic value for that variable based uon observed values for mid-latitude synotic-scale motions. The characteristic values aroriate for the scale analysis of (2) are given by Table 1 below. Variable Characteristic Value Descrition u, v U 10 m s -1 Horiontal velocity scale w W 0.01 m s -1 Vertical velocity scale, y L 10 6 m Horiontal length scale H 10 4 m Deth scale δ/ρ δp/ρ 10 3 m 2 s -2 Horiontal ressure fluctuation scale t L/U 10 5 s Time scale f, 2Ω cos ϕ f 0 10-4 s -1 Coriolis scale Table 1: Characteristic values aroriate for synotic-scale motions for the variables found within Equation (2). Several imortant insights into what is meant by synotic-scale motions can be drawn from the values in the table above: The magnitude of the horiontal velocity is much larger several orders of magnitude than the magnitude of the vertical velocity. Synotic-scale features have horiontal etent of hundreds of kilometers or more. These features also etend through a meaningful deth (~10 km) of the trooshere. Synotic-scale motions evolve slowly, on the order of 1 day (8.64 10 4 s) or longer. We are considering only mid-latitude henomena by stating that f 0 10-4 s -1, which is its value at latitude ϕ = 45 N. If we lug in the aroriate characteristic values from the table above into (2), we obtain: u-mom. Du -fv 2Ωw cosϕ uw a uv tanφ a ρ 1 F r v-mom. Dv Fu vw a u 2 tanφ a ρ 1 y F ry Scale U 2 /L f 0 u f 0 w UW/a U 2 /a δp/ρl DU/H 2 Value (m s -2 ) 10-4 10-3 10-6 10-8 10-5 10-3 10-12 Table 2: Scale analysis of the equations of motion. The Geostrohic Aroimation, Page 2
Note that the D in the scaling of the frictional terms is the eddy diffusivity, a term related to turbulent (or frictional) rocesses within the boundary layer. It is of order D 10-5 m 2 s -1. The Geostrohic Aroimation on Constant Height Surfaces From Table 2 above, it is aarent that for synotic-scale motions, two terms are at least one order of magnitude larger than the other terms. These are the Coriolis and ressure gradient terms. If we retain only these two terms in (2), the following eressions result: 1 fv = ρ 1 fu = ρ y (3a) (3b) Equation (3) reresents the geostrohic relationshi. The geostrohic relationshi gives the aroimate relationshi between the horiontal ressure gradient and the horiontal velocity (as scaled by the Coriolis arameter) in large-scale etratroical/mid-latitude weather systems. Geostrohic balance is defined as the balance between the horiontal ressure gradient and Coriolis forces. This means that the horiontal ressure gradient force and Coriolis force are of equal magnitude to but directed in oosite directions from each other. The horiontal ressure gradient force always oints from high toward low ressure, in contrast to the horiontal ressure gradient itself, which always oints from low toward high ressure. Under the constraint of geostrohic balance, the Coriolis force oints in the oosite direction of the horiontal ressure gradient force, or from low toward high ressure. This balance is deicted in Figure 1 below. Figure 1. Grahical deiction of geostrohic balance for an idealied eamle in the Northern Hemishere with low ressure to the north and high ressure to the south. The ressure gradient force (gf) is denoted in red, the Coriolis force is denoted in blue, and the resultant wind is denoted in grey. Black lines denote idealied isobars, here deicted on a constant height surface. The Geostrohic Aroimation, Page 3
The magnitude of the horiontal ressure gradient force is directly roortional to the magnitude of the horiontal ressure gradient, as one might eect from insection of the right-hand side of (3). Similarly, the magnitude of the Coriolis force is directly roortional to the horiontal wind seed (i.e., magnitude of the horiontal velocity vector), as one might eect from insection of the left-hand side of (3). Thus, a larger horiontal ressure gradient requires a larger Coriolis force to balance it, thus necessitating a faster horiontal wind seed. Note that in the Northern Hemishere, the Coriolis force always oints erendicular and to the right of the wind, as deicted in Figure 1 above. We can rove this from (2). If we retain only the total derivative terms and terms that involve the Coriolis arameter f in (2), we obtain: Du = fv (4a) Dv = fu (4b) The total derivative terms on the left-hand side of (4) reresent accelerations; i.e., changes in u and v with time following the motion. Thus, (4) reresents arcel accelerations due eclusively to the Coriolis force. Note that in the Northern Hemishere, f is always ositive. Consider air that is moving from south to north, such that v > 0. In this case, for f > 0, the lefthand side of (4a) must be ositive. This means that, following the motion, the air arcel is accelerating to the east (u > 0). This is 90 to the right of the air s motion from south to north. Similar arguments can be made for any given wind direction. Thus, the Coriolis force is always directed 90 to the right of the wind in the Northern Hemishere. How is Geostrohic Balance Achieved? Consider air that is initially at rest. Because the Coriolis force is directly roortional to the wind seed, the Coriolis force is initially ero. Now consider there to be a horiontal ressure gradient on the synotic-scale, such as is deicted in Figure 2 with lower ressure to the north and higher ressure to the south. In this case, the horiontal ressure gradient force is directed from south to north. This causes the air to accelerate and, thus, begin to move. This brings about a force imbalance. Initially, as the horiontal ressure gradient force is the only force acting uon the air, it moves from high toward low ressure i.e., to the north. However, as the air begins to move, the Coriolis force becomes non-ero. Very quickly, on the time san of erhas minutes, the horiontal ressure gradient force and Coriolis force come into (near-)balance, and the air is deflected to the right. The Geostrohic Aroimation, Page 4
Figure 2. Schematic illustrating how geostrohic balance is initially achieved, what haens when geostrohic balance is disruted, and the restoration of geostrohic balance through geostrohic adjustment. Figure obtained from Figure 6-17 of Meteorology: Understanding the Atmoshere, 4 th Ed., S. Ackerman and J. Kno. Consider air that is moving but that obeys geostrohic balance. It is ossible for geostrohic balance to be disruted, such as by localied heating, deflection of the flow by terrain features, and/or localied instabilities within the flow. The rocess by which geostrohic balance is restored is known as geostrohic adjustment. Simly ut, during geostrohic adjustment, both the wind and ressure (and, by etension, mass) fields adjust to one another such that balance may again be achieved. The wavy ath that the air takes in Figure 2 after it initially achieves geostrohic balance is an eamle of geostrohic adjustment; the horiontal ressure gradient and Coriolis forces adjust to each other, resulting in a temorarily wavy flow until balance is restored. Balance restoration is tyically associated with gravity and/or inertia-gravity waves, the characteristics and dynamics of which are beyond the scoe of this class. The Geostrohic Relationshi on Isobaric Surfaces To obtain (3), we conducted a scale analysis of the equations of motion alicable on constant height surfaces. Consequently, (3) is alicable only on constant height surfaces. We wish to now obtain a form of (3) that is alicable on isobaric surfaces. To do so, a coordinate transformation from the to the vertical coordinate is necessary. First, consider an idealied eamle of two isobaric surfaces that vary in height only in the -direction, as seen in Figure 3. The Geostrohic Aroimation, Page 5
Figure 3. Idealied deiction of two isobaric surfaces, 0 and 0 +, that vary in height only within the -direction. The distances and reresent the distance between these two isobaric surfaces in the - and - directions, resectively. Consider the change in ressure as one moves from oint A, where = 0, to oint B, where = 0 +. This can be reresented in terms of a artial derivative, where: B A = (5a) The subscrit in (5a) denotes that this artial derivative is evaluated on a constant height surface, which is aroriate because B = A here. If one were to aly a centered finite difference aroimation halfway between oint A and oint B to the artial derivative in (5a), the right-hand side of (5a) would simly become, equal to B ( 0 + ) minus A ( 0 ). Likewise, consider the change in ressure as one moves from oint B, where = 0 +, to oint C, where = 0. This can also be reresented in terms of a artial derivative, where: C B = (5b) The subscrit in (5b) denotes that this artial derivative is evaluated at a constant value of, which is aroriate because B = C here. If one were to aly a centered finite difference aroimation halfway between oint B and oint C to the artial derivative in (5b), the righthand side of (5b) would simly become -, equal to C ( 0 ) minus B ( 0 + ). Because A = C, we can substitute A for C in (5b). If we then multily the resulting equation by -1, an eression for B A results. Equating this to (5a), we obtain: The Geostrohic Aroimation, Page 6
The Geostrohic Aroimation, Page 7 = (6) If we divide (6) by, we obtain: = (7) Uon insection of Figure 3, we can see that / is equivalent to the sloe of an isobaric surface. Taking the limit of this term in (7) as aroaches 0 and alying the definition of the artial derivative to the result, we obtain: = (8) In (8), the subscrit of indicates that the artial derivative is evaluated on an isobaric surface. If we substitute into (8) with the hydrostatic equation, we obtain: g = ρ (9a) If we were to reeat the same rocess, ecet for ressure varying only in the y-direction, we would obtain: y g y = ρ (9b) It is the eressions in (9) that enable us to convert the ressure gradient terms evaluated on a constant height surface (from Equation 3) to their equivalent height gradient terms evaluated on an isobaric surface. If we lug (9) into (3), we obtain: g fv = (10a) y g fu = (10b) Or, making use of the definition of the geootential that we introduced in an earlier lecture, (10) can be written as:
Φ fv = (11a) Φ fu = y (11b) Note that there is no reference to time in (3), (10), or (11). Thus, the geostrohic relationshi is a diagnostic relationshi; it can only be used to diagnose the horiontal velocity as a function of the ressure or height field at a given time. This stands in contrast to a rognostic relationshi, or one that involves a reference to time (such as through the resence of a artial derivative with resective to time), which ermits the rediction of the evolution of the velocity field. The Geostrohic Wind To eress geostrohic balance, u in (3), (10), and (11) is relaced by u g while v is relaced by v g. These two variables, u g and v g, define what is known as the geostrohic wind. The geostrohic wind is a stable, slowly-evolving flow. Contrast this with highly curved, raidly accelerating, or near-surface flows, which can evolve more raidly or may be unstable. Given that density is not routinely measured and that most meteorological analysis is conducted on isobaric surfaces, either (10) or (11) are used to evaluate the geostrohic wind from observations. Because the ressure gradient and Coriolis forces are of equal magnitude but in oosite directions under the constraint of geostrohic balance, the geostrohic wind blows arallel to the isobars on a given height surface. This is deicted in Figure 1 above. The geostrohic wind similarly blows arallel to lines of constant geootential height on an isobaric surface. This relationshi between the wind and the isobars or constant height contours is one of the defining characteristics of geostrohic balance. An alication of this relationshi is the Buys-Ballot law, which states that when the (geostrohic) wind is hitting your back, low ressure is to your left. In the mid-latitudes, the geostrohic wind aroimates the true horiontal velocity to within ~10%. How closely the wind is to being arallel to the isobars or constant height contours on a synotic analysis rovides us a measure for how close the atmoshere is to satisfying geostrohic balance at that location. Deartures from geostrohic balance occur most commonly when the centrifugal and/or frictional forces are non-negligible. The former is associated with curved flow, whereas the latter is imortant rimarily near Earth s surface within the lanetary boundary layer. We will consider how and when the wind is influenced by these forces in later lectures. The Ageostrohic Wind In defining the geostrohic relationshi above, we neglected the total derivative terms on the left-hand side of (2). We now want to revisit (2), keeing these terms (as well as the Coriolis and ressure gradient terms) while continuing to neglect the rest. The resultant equations are: The Geostrohic Aroimation, Page 8
Du 1 = fv ρ Dv 1 = fu ρ y (12a) (12b) However, (3) with u g and v g substituted for u and v gives eressions for the ressure gradient terms that aear in (12). If we substitute for those terms into (12), we obtain: Du ( v ) = fv fv = f (13a) g v g ( u ) Dv = fu + fug = f u g (13b) The terms v v g and u u g define the ageostrohic wind, where v v g = v ag and u u g = u ag. What does (5) mean? Most simly, acceleration is tied to the ageostrohic wind, or dearture of the total wind from the geostrohic wind. Acceleration is imortant on the synotic-scale rimarily in the vicinity of jet streams and jet streaks. Because the geostrohic wind aroimates the full wind to within ~10%, the ageostrohic wind is said to be one order of magnitude smaller than the geostrohic wind. Given a tyical scaling of U ~ 10 m s -1 on the synotic scale, the characteristic scale for the ageostrohic wind is ~1 m s -1. The Rossby Number In our earlier scale analysis, we found that the acceleration terms (the total derivatives) are tyically one order of magnitude smaller than the Coriolis and ressure gradient terms. Just how close this is to being true for any given weather situation can be assessed by determining the ratio between the characteristic scales of the acceleration and Coriolis terms. This defines the Rossby number, named after the famous meteorologist Carl-Gustav Rossby, and is given by: Ro = 2 U L f U U = f L o 0 (14) Note that the U, f 0, and L in (14) are not the same as their values in Table 1. Rather, they reresent values secific to the synotic-scale weather conditions being assessed. For Ro 0.1, the magnitude of the acceleration term is one order of magnitude smaller than the magnitude of the Coriolis term. This describes geostrohic balance. For Ro 1 or Ro > 1, the magnitude of the acceleration term is comarable to or eceeds the magnitude of the Coriolis term. In such situations, geostrohic balance does not hold. Instead, we must consider gradient wind balance or cyclostrohic balance for such situations. The Geostrohic Aroimation, Page 9
For Further Reading Section 1.3 of Midlatitude Synotic Meteorology by G. Lackmann introduces geostrohic balance and the Rossby number. Section 3.2 of Mid-Latitude Atmosheric Dynamics by J. Martin rovides a discussion of the geostrohic aroimation and geostrohic balance. Section 4.1 of Mid-Latitude Atmosheric Dynamics rovides a generalied discussion of how equations osed on constant height surfaces may be transformed into equations osed on isobaric surfaces. The Geostrohic Aroimation, Page 10