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AEP notes relevant for test-2 (1) See the photocopy of the chapter that deals with the derivation of fictitious forces in the rotating frame of reference (from “Classical Mechanics” by Rana and Joag) (2) See the photocopy that deals with temperature changes of rising of parcel of air and that of atmospheric stability (Chapter 14: Meteorology of air pollution pages 765 – 774 of “Atmospheric Chemistry and Physics”) In the notes below I have tried to compile and adapt the information available in your reference books (mainly from Holton’s book) and web resources to match best to what was taught in class: (3) Geostrophic Balance : We have the horizontal equations of motion: F rx and F ry are frictional forces at the boundary. Above the boundary layer, the frictional terms are negligible. Also, a scale analysis shows that the acceleration terms (the left hand side of the equations) is 1 order of magnitude less than the pressure gradient force (PGF) and Coriolis force (CF) terms (10 -4 ms -2 to 10 -3 ms -2 ). Eq. (1) and (2) become: This is called the geostrophic approximation. It states that the PGF is balanced exactly by the Coriolis Force. In geostrophic flow, there are no accelerations acting on the air parcel. If a flow is curved then it has acceleration since the parcel is changing direction. Thus, geostrophic flow is straight line flow. Wind will flow parallel to the isobars. Furthermore, the PGF and the CF must act opposite each other. Since the PGF will always point perpendicular to the isobars and towards lower pressure, the CF must act in the opposite direction:

Transcript of notes-t2 (2)

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AEP notes relevant for test-2

(1) See the photocopy of the chapter that deals with the derivation of fictitious forces in the rotating frame of reference (from “Classical Mechanics” by Rana and Joag)

(2) See the photocopy that deals with temperature changes of rising of parcel of air and that of atmospheric stability (Chapter 14: Meteorology of air pollution pages 765 – 774 of “Atmospheric Chemistry and Physics”)

In the notes below I have tried to compile and adapt the information available in your reference books (mainly from Holton’s book) and web resources to match best to what was taught in class: (3) Geostrophic Balance :

We have the horizontal equations of motion:

Frx and Fry are frictional forces at the boundary. Above the boundary layer, the frictional terms are negligible. Also, a scale analysis shows that the acceleration terms (the left hand side of the equations) is 1 order of magnitude less than the pressure gradient force (PGF) and Coriolis force (CF) terms (10

-4 ms

-2 to 10

-3 ms

-2). Eq. (1) and (2) become:

This is called the geostrophic approximation. It states that the PGF is balanced exactly by the Coriolis Force. In geostrophic flow, there are no accelerations acting on the air parcel. If a flow is curved then it has acceleration since the parcel is changing direction. Thus, geostrophic flow is straight line flow. Wind will flow parallel to the isobars. Furthermore, the PGF and the CF must act opposite each other. Since the PGF will always point perpendicular to the isobars and towards lower pressure, the CF must act in the opposite direction:

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Note that at the Equator, geostrophic balance is not applicable since the horizontal CF = 0. The geostrophic approximation is usually accurate within 10-15% of the true horizontal wind in the mid-latitudes. We can develop equations to calculate the geostrophic wind from equations (3) and (4). If vg and ug are the meridional and zonal geostrophic wind components, (3) and (4) become:

How good is the geostrophic approximation? The Rossby Number Earlier it was mentioned that the geostrophic approximation usually gave you winds within 10-15% of the actual wind in the mid-latitudes. We can use a tool to give us an objective view on whether it is valid to make the geostrophic approximation. This tool is called the Rossby Number (R

o):

Here U, fo, and L symbolize the typical values used in a scale analysis. The Rossby number is dimensionless. It is a ratio of the wind acceleration to the Coriolis acceleration (although that isn’t all too obvious from (9)). The magnitude of the Rossby number indicates whether the geostrophic approximation can be made. If Ro

is small (0.1 or less), then it is usually OK to use it. For a typical

mid-latitude synoptic situation, U ~ 10 m/s, fo ~ 10

-4 s

-1, and L ~ 10

6 m. This yields Ro

= 10

-1 or 0.1.

Summary The geostrophic approximation is used quite frequently in atmospheric science. Be aware of its limitations: i) Neglects frictional effects: As you approach the surface, the approximation becomes increasingly invalid. ii) Assumes no acceleration of the flow: If the flow is changing in direction or speed, the approximation is not valid. iii) Not valid in the tropics: As the CF becomes increasingly small, the Ro becomes larger and the approximation does not hold up. iv) Additionally, See the photocopy of that deals with the derivation of the geostrophic wind speed (Chapter 1, Appendix I, pages 43 – 46 of “Atmospheric Chemistry and Physics”). This

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note also talks about “Ekman Spiral”, which is the variation of wind direction with height at the surface boundary layer.

(4) Thermal Wind: (note:-equation numbers re-start from (1) in this sub-section) The thermal wind is defined as the vector difference between the geostrophic winds at two levels. It is not really a wind at all, just a measure of the shear of the geostrophic wind. But there are good reasons for considering the geostrophic wind; mainly, it provides a convenient way of connecting the structure of the temperature field to the wind field. To see the connection, first we write down the hypsometric equation. This says that the thickness between any two pressure levels is proportional to the mean temperature within that layer:

If the geostrophic wind is increasing with height, then the horizontal pressure gradient must also be increasing with height. If the pressure gradient is increasing in the positive x-direction, the temperature gradient must also be increasing in that direction.

In the figure above, the thickness of the atmosphere at x2 is greater than the thickness at x1. We know through the hypsometric equation that the mean temperature between pressure levels po

and po+2dp

must be greater in the x2 column than the x1 column. This causes the horizontal pressure gradient to change more rapidly over x2 as height increases, resulting in a stronger geostrophic wind (dark arrows). The thermal wind equation describes how much the geostrophic wind will change with height:

where (a) is the local rate of change of the geostrophic wind with pressure, which depends on (b), the horizontal temperature gradient. If the temperature is constant across the pressure level, then the geostrophic wind is independent of height and motion in the atmosphere is constrained. The thermal wind vector points parallel to the isotherms with warm air to the right facing downstream:

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Direction changes of the geostrophic wind with height can be used to find if warm air advection (WAA) or cold air advection (CAA) is occurring in the layer. If the wind is veering (turning clockwise) with height, we have a WAA situation. If the wind is backing (turning counter-clockwise) with height, CAA is occurring. Barotropic vs. Baroclinic Atmosphere A barotropic atmosphere is one in which density is only a function of pressure:

A barotropic atmosphere puts a severe constraint on atmospheric motion. Constant pressure and density surfaces are parallel. Also, constant pressure surfaces are isothermal for an Ideal Gas (which the atmosphere is usually considered). This makes the geostrophic wind independent of height, as shown in the thermal wind equation (thus there would not be a thermal wind, since Vg does not change with height). A baroclinic atmosphere is one in which density is a function of pressure and temperature:

Now that the temperature field can be independent of the pressure field, the thermal wind equation applies and the geostrophic wind can change with height.

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(5) Circulation and Vorticity: (note:-equation numbers re-start from (1) in this sub-section) Circulation refers to the tendency for a chain of air parcels in the atmosphere to rotate cyclonically (counterclockwise in the northern hemisphere). If an area of air is of interest, you would use circulation to address the problem. Vorticity referes to the tendency for the wind shear around a point to produce cyclonic rotation. If a point in the atmosphere is of interest, you would use vorticity to address the problem. Circulation Formally defined as the path integral of the tangential velocity around an air parcel (or closed chain of fluid elements):

Circulation Theorem The same forces that cause horizontal and vertical accelerations (PGF, Coriolis force, gravitational accelerations, friction) also cause changes in circulation. They are related by the circulation theorem:

The term on the left hand side is the total change of circulation (C) with time. The term on the right hand side is called the solenoidal term. Essentially, it represents the effect that the pressure gradient force has on the circulation. If the atmosphere is barotropic (density is a function of pressure only), then the solenoidal term = 0. A solenoid can be thought of as a parallelogram formed by intersecting lines of constant density and pressure:

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This situation arises when, for example, a sea breeze forms due to temperature (and thus density) differences between land and water areas. The warm land creates warmer, and less dense air, in the column above. Vorticity Vorticity is more commonly used in meteorology. It is the microscopic measure of rotation in a fluid around a point. Mathematically, vorticity is defined as the curl of the velocity vector:

The ‘a’ subscripts in the equation above (3) refer to the “absolute” reference frame, or absolute vorticity. The absolute vorticity includes the spin due to the Earth’s rotation. Relative vorticity does not include the spin of the Earth, and is calculated as:

Although the horizontal component of vorticity can be important in a lot of meteorological applications (e.g. tornado formation, boundary layer rolls), the vertical component of vorticity is most commonly applied in large-scale dynamics. The vertical component of absolute vorticity (η) and relative vorticity (ς) are thus defined as:

Hereafter, any mention of absolute and relative vorticities is considered to be the vertical component unless otherwise specified. Circulation, which is a scalar integral quantity, is a macroscopic measure of rotation for a finite area of the fluid. Vorticity, however, is a vector field that gives a microscopic measure of the rotation at any point in the fluid. Interpretation of Vorticity Signs Positive relative vorticity is associated with cyclonic (counterclockwise) circulation in the Northern Hemisphere. Negative relative vorticity is associate with anti-cyclonic (clockwise) circulation in the Northern Hemisphere.

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Planetary Vorticity Planetary vorticity is the difference between absolute and relative vorticity. It is the local vertical component of the vorticity of the earth due to its rotation:

Vorticity in Natural Coordinates A physical interpretation of relative vorticity is most easily obtained by looking at vorticity in natural coordinates. Consider the unit vectors of natural coordinates:

t - Oriented parallel to the horizontal velocity at each point n - Oriented perpendicular to the horizontal velocity and pointing positively to the left of the flow direction k - Directed vertically upward

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The terms on the RHS can be interpreted as follows: a) Rate of change of the wind speed normal to the direction of the flow. This is called the shear vorticity. Thus, vorticity can be created even for straight line flow. Consider the case of a jet stream maximum in the upper troposphere:

A cyclonic (positive relative vorticity) circulation develops north of the jet maximum, while and anti-cyclonic (negative relative vorticity) circulation develops south of the jet. b) The turning of the wind along a streamline. This is called the curvature vorticity. Note that curved flow could have a net zero relative vorticity if the shear vorticity and curvature vorticity are equal and opposite of one another. Potential Vorticity (note:-equation numbers re-start from (1) in this sub-section) Potential Vorticity is defined as a variable that combines the absolute vorticity and some measure of the thickness of a column of air. Derivation: Potential temperature is defined as:

This is called Poisson’s equation. The variable ps is the standard pressure at which we measure the air parcel’s temperature if taken there dry adiabatically from pressure level p (usually ps = 1000 hPa). Using the Equation of State, we can solve for the density ρ:

Thus, on an isentropic surface (temperature constant, or constant θ surface), density is only a function of pressure. In this case, the solenoidal term from the circulation theorem is zero:

In this barotropic situation, the absolute circulation is conserved following the motion. This satisfies what is known as the Kelvin circulation theorem, which can be expressed as:

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Assuming an infinitesimal parcel of air and substituting (5) into (4),

where ζθ is the vertical component of relative vorticity on an isentropic surface and f is the Coriolis parameter (f = 2Ω sinφ). Study the figure below (Fig. 4.7 in Holton – reference book) – it represents an air parcel confined between two potential temperature surfaces, separated by a pressure interval δp :

As the parcel moves from left to right, it must conserve its mass. As the distance between potential temperature lines increases, the air parcel expands vertically. The area of the air parcel must decrease as the parcel expands vertically if mass is to be conserved. So the value of δA is a function of how quickly the lines of potential temperature change with pressure:

Equation (8) is the expression for Ertel’s Potential Vorticity in isentropic coordinates. The units of (8) are 10-6 Kkg-1m2s-1, which are called PVU (Potential Vorticity Units). The minus sign is installed so that PV is generally positive in the Northern Hemisphere. A value of 2 PVU typically signifies the tropopause (boundary between the troposphere and the stratosphere). Interpretation of Ertel PV Eq. (8) essentially describes a ratio of the absolute vorticity of the air parcel (ζθ + f) to the depth of the vortex (−g ∂θ/∂p). PV must be conserved following the motion in adiabatic, frictionless flow. So, if the if either the depth or the spin of the air parcel changes, a compensating effect must occur in the other term. We can further simplify PV by assuming that the atmosphere is incompressible (constant density). If this is the case, the horizontal area δA must be proportional to the depth of the column:

Where M is the mass of air parcel and h is the depth of the air parcel. Both m and ρ are constant in this situation. Substituting for δA into eq (6) yields:

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A simple example of flow over a topographic barrier illustrates the impacts that the conservation of PV has on the absolute vorticity of air parcels. Westerly Flow Example Let’s look at an example of this. Suppose we have an air parcel traveling west to east over a mountain, between two potential temperature levels:

Near the surface, potential temperature lines follow the surface very closely (see θo in fig. above). However at upper levels, potential temperature surfaces change altitude farther upstream and downstream of the mountain. So as an air parcel approaches from the left, it initially is stretched vertically, thus h increases and by Eq. (9) ζ+f must increase to conserve PV. This will cause the air parcel to turn cyclonically as it approaches the mountain, as shown in Fig. 4.9b above (note that as latitude increases, f will increase, thus decreasing the magnitude of ζ necessary to conserve PV). As the air parcel crosses the mountain, the depth (h) decreases and the vorticity ζ+f becomes negative. This causes anti-cyclonic turning and southward displacement. When the air parcel returns to its original depth, it will be south of its original latitude. Thus, f will be less and ζ must be positive to conserve PV. Notes on mountain flows • In the real atmosphere, vertical motions are generally suppressed and most flows do not go over the top of mountain barriers but around them. • These examples illustrate the Rossby PV Conservation Law, which states that a change in depth is dynamically analogous to a change in the Coriolis parameter for a barotropic fluid. Barotropic (Rossby) Potential Vorticity (PV) Equation For an incompressible, barotropic fluid, the absolute vorticity is conserved:

The quantity conserved is called the Rossby Potential Vorticity.

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(6) Some Kinematics The atmosphere is characterized by horizontal (and vertical) variations in the wind field. These horizontal variations can be expressed as partial derivatives of u and v with respect to x and y. Magnitude of the horizontal winds (zonal u, and meridional v, winds) are much higher than vertical wind. Consider the possible sums of derivatives that can be constructed including a derivative of u with respect to one direction and a derivative of v with respect to the other direction:

What do these quantities tell us about the characteristics of the fluid flow? Let’s use a Taylor series expansion for find out.

Neglecting higher order terms:

The combinations of the partial derivatives are named as:

So, the Taylor series above can be re-written as:

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We can use these two expressions to explore what each of these four types of pure motion looks like physically. Although we will do this by examining each of the four types of pure motion by itself, in the atmosphere they can all occur simultaneously. (a) Pure vorticity:

(b) Pure divergence:

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(c) Pure stretching deformation:

(d) Pure shearing deformation:

(7) Equations of motion (momentum, mass, energy conservations) – geophysical fluid

dynamics primer: The fundamental aim is to understand the circulations of the atmosphere and ocean and the observed distributions of physical quantities such as temperature. The temperature distribution can be viewed as the result of a "competition" between the sun, which tries to warm the tropics more than the poles (and so create horizontal contrasts), and gravity , which tries to remove horizontal contrasts and arrange for warmer fluid to overlie colder fluid. This “competition” is complicated by such effects as the rotation of the earth, the variation of the angle between gravity and the rotation axis (the beta effect), and contrasts between the properties of air and water. The atmosphere is thin film of fluid, on a spherical earth, under the influence of gravity, rotation, and differential heating by the sun. The atmosphere is compressible, and thus stratified, and almost inviscid. The oceans are similar, but essentially incompressible and saline.In principle, to understand the atmosphere-climate system we need to understand all of its main components and how they interact. Dynamics depends on the

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distribution of heating (the main energy source and driver of motions). The heating depends on the Radiative transfer properties of the atmosphere. Those depend on water vapor. Water vapor depends on the motion of air- the dynamics. A closed cycle – everything is dependent. In this section, we will touch upon the essential ingredients of atmospheric dynamics.

Geopotential and geopotential height: The gravitational force per unit mass acting on an air parcel is g which his a vector acting vertically- towards Earth's center. We can write the gravitational force as a potential: ∇φ=-g. Since g = – gk, a downward pointing vector, this gives: dφ/dz=g. The geopotential φ(z) is the work required to raise a unit mass to height z from the surface:

Note that dφ = gdz = – dp/ρ = – RTdln(p) where we used the ideal gas law and the hydrostatic balance relation. This is similar to the expression for the height of a pressure surface, only divided by g. We define the geopotential height to be the height corresponding to dz=dφ/g

0

where g0 is the gravitational acceleration at the surface.

In the case of an isothermal atmosphere, this is just the log-pressure height (with the gravitational acceleration held fixed approximation).

What do we want to know about the atmosphere? Density, ρ (mass, Speed,v, Pressure, p, and Temperature, T. The equations of motion describe the forces in the atmosphere that act on a parcel of air. From geophysical fluid dynamics, we have the equations expressing conservation of momentum, mass and energy, and also the equation of state. In geophysical fluid dynamics we generally take what is called the Eulerian point of view, in

which we consider our dependent variables such as fluid velocity and density to be a function of position rather than being identified with the individual parcels making up the fluid. Thus, the velocity at some point refers at different times to different parcels of fluids. The alternative point of view, in which variables are referenced to fluid parcels, is called the Lagrangian point of view. It has occasional uses in fluid dynamics, but is much less commonly seen than the Eulerian representation. Momentum equations are given by the Navier-Stoke’s equations (in general form in fluid dynamics):

In the atmospheric dynamics we consider, Coriolis force, gravitational force and frictional force at the surface. Work can be done on the fluid parcel by both surface forces and body forces. Surface force: pressure gradient. Body forces: gravity and Coriolis. The Coriolis force does no work because it is always perpendicular to motion of the fluid parcel. Force must be along the direction of motion for work to be done.

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For 3-dimensional flow,

(2) is the equation of motion for all 3 directions. Normally we decompose (2) into the equation of motion for each u,v, and w direction. For example, to get the equation of motion in the x-direction, dot multiply (2) by i :

This brings us to a useful technique for making our lives simpler. Frequently, equations like (3-5) contain terms that really aren’t too significant, at least compared to the other terms. We employ a technique called scale analysis to determine which, if any, terms we can neglect. To perform a scale analysis, we need to calculate the average magnitude that all of the terms will have. Some scale magnitudes for mid-latitude synoptic scale systems are: Horizontal velocity (U) ≈ 10 m/s (u,v) Vertical velocity (W) ≈ 10

-2 m/s (w)

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Using scale analysis, it can be shown that the vertical Coriolis Effect in (3) is three orders of magnitude smaller than the other terms. So we can throw it out. Similarly, the frictional terms and the Coriolis term in (5) are 4 orders of magnitude smaller than the other terms, and dw/dt is 8 orders of magnitude less. Simplifying, equations 3-5 become:

Note that (8) simplifies to the hydrostatic approximation. This is justification for assuming that large-scale flow is in hydrostatic balance for mid-latitude, synoptic scale features. Note also that we can’t use (8) to predict vertical velocity in numerical models – other approaches must be used. In summary, momentum equations can be represented as:

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Mass conservation is done by continuity equation. Recall the derivation of continuity equation done in class, essence of which is given below:

Energy conservation: For a system in thermodynamic equilibrium, the first law of thermodynamics states that the change in internal energy of the system is equal to the difference between the heat added to the system and the work done by the system. The first law of thermodynamics applies to a

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moving fluid parcel. The rate of change of total thermodynamic energy is equal to the rate of diabatic heating plus the rate at which work is done on the parcel by external forces. The thermodynamic energy equation takes a basic form:

Temperature tendency equation is:

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In summary, primitive equations for atmospheric dynamics can be written as (simplified for horizontal motion):

(8) Rossby waves and gravity waves

(Please refer to the pdf version of power point presentation supplied)