Chapter 9 Vertical Motion. (1) Divergence in two and three dimensions. The del “or gradient” operator is a mathematical operation performed on something.
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(1) Divergence in two and three dimensions. The del “or gradient” operator is a mathematical operation performed on something which yields the gradient vector of that something in the cartesian dimensions. The del operator in three dimensions is: The del “or gradient” operation on the scalar temperature yields the vector of the “change of temperature” with respect to the x-, y-, and z- directions.
Divergence The del operator can be used to determine divergence of a vector field, such as wind. The dot product of the vector field and del yields divergence. Remember, the dot product is simply the sum of the products of each of the components of two vectors. It yields a scalar quantity. Thus, the divergence of a wind field is: Units will be seconds -1 or s -1 To the whiteboard (Ch. 9 pg 3 fig.)
(2) Divergence theorem Graphically, divergence (and convergence) can be determined by drawing a box (or other shape) in the wind field and determining whether there is more air moving into the box or more moving out.
If we use something like a rectangle, the length of the sides becomes important because the sides are not all the same length. Then divergence (or convergence) must be determined by how much air flow occurs across each side of the rectangle.
Divergence Theorem Equation is: Note: Error, pg. 5, dxdy. This states that the divergence integrated over an area is equal to the line integral of the outward component of wind along the perimeter of the area. Where n is a unit vector point outward from the area (box) and dl represents a one-dimensional integral oriented along the edge of the box. We are integrating all the way around the area with dl (or with dxdy - first with dx, then with dy) and getting a measure of the flow of air outward from the area with The dot product of the wind vector, v, and a unit vector, n, pointing out of the box, gives us the component of the wind directed outward from the box (or divergence from the box).
The left side of the equation can be replaced with average value of divergence from the box (or region). To get the divergence from the box (or region), we need to divide by the area, A. Pg 6: Divergence theorem in words.
(3) Estimating Divergence on a map. To make the determination easier, we can draw the shape such that two sides are parallel to the wind flow (so no air is flowing into or out of the area on these sides) and the other two sides are perpendicular to the air flow. The magnitude of the divergence (or convergence) across the sides perpendicular to the wind flow will be the mean wind across the side times the length of the side.
However, we have another error. We need to divide by the area of the region, (in order to get the magnitude of the divergence or convergence) which in this example would be hard to calculate. We can get a determination of whether divergence or convergence is occurring, without dividing by area as follows:
Lower-right side (out): Speed: 25 knots = 12.86 m/s Length: 1 cm = 0.01m Left side (in): Speed ~12.5 knots = 6.43m/s Length: 3.5 cm = 0.035m
In three dimensions, you would work with the surface of a volume, rather than an area. If net flow in: you have convergence. If net flow out: you have divergence.
Near the center of a low-pressure or high- pressure center, draw a spiral and then a straight line perpendicular to the ends of the spiral. –At ground level, it should show convergence into the low-pressure center and divergence out of a high-pressure center.
(4) Three-dimensional divergence The Law of Mass Conservation states that if you have net divergence of mass from a volume, the density must decrease. Continuity eqn: However, if density decreases, the Ideal Gas Law says that pressure in the volume must decrease (if temp. is constant). If pressure decreases, the pressure gradient force will be directed inward, toward low pressure, and air will start moving back into the volume.
Unless the parcel is ascending or descending, the three-dimensional divergence tends to be small because changes in density are small. For most purposes, the atmosphere can be regarded as essentially incompressible (in the horizontal) because the density doesn’t change (much) in response to horizontal motions. Whatever air comes in one part of a volume must go out another part.
So, for a volume in the atmosphere, if horizontal divergence is occurring, the volume must shrink in the vertical.
The Law of Mass Conservation, (Continuity Equation) in pressure coordinates becomes: Or, Where,, is the vertical motion in pressure coordinates: positive for downward motion. The vertical motion in cartesian coordinates is:
If the vertical derivative of vertical motion is positive, the column is stretching. If is positive, it means that the upward vertical velocity at the top of the column is greater than that at the bottom, so the column is stretching in the vertical. If is positive, it means that the downward vertical velocity at the bottom of the column is greater than at the top, so the column is stretching in the vertical. For z, it is w at top - w at bottom (w at greater height - w at lower height) For p, it is ω at bottom - ω at top. (ω at greater pressure - ω at lower pressure)
(5) Vertical Motion and Convergence The Continuity Equation: Horizontal Divergence: Then, And, stretching of the air column ( is positive) occurs where there is horizontal convergence (negative divergence).
Vertical shrinking of an air column occurs when there is horizontal divergence.
The top part ➵ is moving upward slower than the middle part. ➵ The middle part ➵ is moving up faster than the bottom part. ➵ Upward motion in the mid-troposphere implies vertical stretching below the strongest vertical motion and vertical squishing above the level of strongest vertical motion.
Since the tropopause effectively acts as a cap. Upward motion in middle troposphere and effectively zero at tropopause implies shrinking and divergence. Upward motion over a flat surface implies stretching below (and convergence). The continuity equation shows that:
The anvil spreading from a cumulonimbus cloud shows divergence near the tropopause caused by strong updrafts within the cloud, implying convergence at low levels. Link: Satellite loop of tornadic supercells
(6) Computing Vertical Motion from Divergence If we integrate this equation from low-level (high pressure), P 0 (where ω=ω 0 ), to P 1 (where ω=ω 1 ), then: The difference in vertical velocity (in pressure coordinates) between two pressure levels is equal to the average divergence between the pressure levels multiplied by the pressure interval itself. Integrating from P 0 to P 1 means: P 1 -P 0 will be a negative term.
Example: Let average divergence in the layer be -5x10 -5 s -1. Let pressure interval be 200mb above ground level; then: P 1 -P 0 = -200mb. If the lower level is ground, then the vertical motion “at the ground,” (ω o ) is effectively zero, so:
(7) Vertical Motion at the Ground A slope means that even horizontal wind can cause an upward motion. The magnitude of the vertical wind is due to the magnitude of the component of the wind up or down the slope and the magnitude of the slope itself. This vertical wind due to the sloping terrain can be written as: If only the x-direction is considered, then we could write this as:
(8) Advection and the Total Derivative Consider a volume of air and “A” represents some meteorological variable about that volume; such as, water vapor. The total derivative with respect to time represents all changes with time to the variable as it moves with the air flow. The coordinate system is moving with the air. This total derivative to A can be written as: However, if we want to consider a volume of air “at a particular location,” we have to consider the flow of air into and out of the volume. Then, But, the last term on the right is just the advection of “A” into or out of the volume at a particular location, so the equation can be written as:
Then, Considering water vapor (specific humidity, q), it will change “at a particular location” because of evaporation “E” or condensation “C” - “including deposition and sublimation” - and due to advection of water vapor into or out of the volume “at a particular location.” We have added the advection term.
(9) Thermodynamic Equation, Revisited Changes in temperature to a volume of air moving vertically with the flow of air occurs due to (1) adiabatic expansion or contraction and (2) heat being added or removed from the volume. This can be written as: Where: = dry adiabatic lapse rate. Q = heat added or removed. c p = specific heat of air.
To get this “at a particular location,” add the advection term and write as a partial differential equation: Expanding the advection term into horizontal and vertical components gives: Then, combining the terms involving vertical motion gives:
The change in temperature at a particular location due to vertical motion is proportional to the difference between the actual lapse rate,, and the dry adiabatic lapse rate. When considering the change in temperature at a particular level, what matters is not just the change in temperature of the lifted parcel but the difference between that lifted parcel’s temperature and the environmental temperature at that level.
Since a rising parcel cools at the dry adiabatic lapse rate (if it is not saturated), and dry adiabats are lines of constant potential temperature - then the potential temperature of a rising, not saturated parcel of air does not change. Therefore, the thermodynamic equation, written in terms of potential temperature does not have the dry adiabat term in it.