Homework #4 Key: Physical explanations 1.The way water drains down a sink, counterclockwise or clockwise, is independent of which hemisphere you are in. A draining sink is an example of vortex in cyclostrophic balance, so that the pressure gradient force, directed inward toward the drain, is equally and oppositely balanced by the centrifugal force, directed away from the drain. The magnitude of the Coriolis force at this small spatial scale is very small compared to those two forces, so it plays little if any role how the drain rotates. The direction of the rotation in the swirling sink is more influenced by any initial motion in the water and the shape of the drain. The former is probably the parlor trick used at the Mitad del Mundo museum near Quito, Ecuador, to make the water swirl counterclockwise or clockwise down the drain in their exhibit, as I discussed in class. Examples of atmospheric phenomena that are in cyclostrophic balance include dust devils and tornadoes, very tight vortices with a spatial scale of about a kilometer or less—and these can rotate either way in the northern and southern hemisphere. A mature hurricane is also an example of a vortex in cyclostrophic balance, since the size of the eye is on the order of kilometers. However, as the Coriolis force is important to initiate the rotation of thunderstorms around an area of surface low pressure in the tropics at synoptic scales to create a tropical cyclone, the hurricane retains the direction of this rotation in the mature stage. 2. Mountain-valley winds are meteorological phenomena on the mesoscale, with a spatial scale of approximately 1-50 km. The mountains heat and cool faster relative to the surrounding air. This will cause an area of surface low pressure to form on the mountain slopes during the day due to the land surface heating. In contrast to the synoptic scale (spatial scale of 100s to 1000s of km), the relative magnitude of the Coriolis force at this spatial scale is very small compared to the pressure gradient force, so the resultant wind will not be in geostrophic balance. In the absence of any rotation in the flow, air will simply flow down the pressure gradient from high pressure in the valleys to low pressure on the slopes, and rise over the mountains and possibly form clouds if there is sufficient moisture. 3. Wind traveling around cyclones and anticyclones is in gradient balance, such that the forces governing the wind speed and direction are: Pressure gradient, Coriolis, and Centrifugal. This is described by the gradient wind (Vgr) equation as presented in the text and class notes (see those sources for definitions of all terms):

⎟⎟⎠

⎞⎜⎜⎝

⎛+

∂∂

=T

2gr

gr RV

nf1V Φ

This equation is solved by the quadratic formula, the solution, as shown in the Holton text notes given as part of the class lecture notes:

nR

4Rf

2fRV T

2T

2T

gr ∂∂

−±−=Φ

For normal cyclones and anticyclones, real and positive solutions to the gradient wind are possible ONLY if the square root term is positive—no imaginary solutions are allowed (see lecture notes and Holton text). Therefore, there are two possibilities, corresponding to normal cyclones and anticyclones in the atmosphere: Cyclones: RT = positive and dΦ/dn term = negative For any value of curvature and gradient in geopotential the square root term is positive, so a there is no limit as to the strength of the pressure (geopotential) gradient in an area of low pressure with cyclonic curvature. Anticyclones: RT = negative and dΦ/dn term = negative Since the second term underneath the square root in this case can be negative if the pressure (geopotential) gradient is large enough, there is a limit to the strength of the pressure gradient allowed for solutions of the gradient wind equation in the case of anticyclonic curvature. Physically, this means there will be asymmetry in the pressure gradients associated with areas of high and low pressure on a mid-latitude weather map. Areas of low pressure can have very large and tight pressure gradients, allowing for very intense cyclones to develop. Anticyclones will tend to have pressure gradients that are much weaker, since strong pressure gradients around areas of high pressure are not possible, as they would violate the gradient wind solution. For example, on the weather map for this homework assignment the upper-level low over California is a much tighter circulation than the upper-level high in the Gulf of Mexico. 4. Stars and planets everywhere generally take the shape of spheres because gravitational force pulls the mass of the star or planet directly towards its center equally in all directions. However, strictly speaking, the Earth is not perfectly spherical, but an oblate spheroid that bulges out at the equator. This is explained by the fact that Earth is rotating about an axis extends from the North Pole to the South Pole. Therefore, at any geographic point on Earth off the axis of rotation, there will be a centrifugal force directed away from the axis of rotation. This force is equal to V2/R where V is the speed of rotation at the given point and R is the distance from the axis of rotation. The centrifugal force will be a maximum at the equator, since at this point the speed of rotation about the Earth’s axis is greatest, and it is directed perfectly upward away from the Earth’s surface. The centrifugal force will effectively decrease the amount of gravitational acceleration at the equator as compared to the poles, causing the Earth to bulge out in its middle. As an aside, for this very reason, it is more desirable to launch spaceships into orbit from a low-latitude location, because there is less gravitational force to overcome. In the derivation of the Coriolis force, the centripetal acceleration appears in the derivation of total acceleration in a rotating reference frame, but is so small it is neglected. 5. There are two types of force balances that apply to synoptic-scale motions in the mid-latitudes. In geostrophic balance, the pressure gradient force balances the Coriolis force and the wind motion is perfectly parallel to lines of constant pressure or geopotential

height. Assuming the Coriolis parameter is constant (i.e. variation in latitude is not important) it can be mathematically proven that the geostrophic wind is non-divergent. This was mentioned in the lecture notes and you should prove it to yourself. Since the geostrophic wind is pretty much non-divergent, it will not induce any vertical motion in the atmosphere. In gradient balance, the flow is curved, introducing the effect of centrifugal force on the horizontal wind, directed away from the axis of rotation. In cyclonically rotating flows, the centrifugal force effectively reduces the pressure gradient force, decreasing the wind speed from what it would otherwise be in geostrophic balance (subgeostrophic wind). In anticyclonically rotating flows, the centrifugal force effectively increases the pressure gradient force, increasing the wind speed from what it would otherwise be in geostrophic balance (supergeostrophic wind). Thus, as wind travels eastward from the base of a trough of low pressure to the top of a ridge of high pressure, it must increase in speed—and this increase necessarily implies that air is diverging aloft and there is rising motion from below. Reverse the argument going from the top of a ridge to the base of a low. Please see the figures in the lecture notes on this if you’re having trouble conceptualizing it! Curvature in upper-level flow is a necessary prerequisite for large-scale vertical motion to generate clouds and precipitation. The same arguments would not be true in the tropics, because the magnitude of the Coriolis force is so weak there that tropical winds are typically not in geostrophic balance.

PROBLEM 6PROBLEM 6

PROBLEM 8PROBLEM 8

PROBLEM 9PROBLEM 9

PROBLEM 10PROBLEM 10

5460

5460

38 74

5460

443230

59

Problem Setup for Tucson calculations

ddΦΦRRTTTUCSON

Radius of cyclonic curvature = 1000 km

ddΦΦ

RRTT LA

Radius of cyclonic curvature = 400 km

Problem Setup for LA calculations

PROBLEM 11PROBLEM 11

Points for computing divergence and vorticity

at Tucson

AABB

CC

DD

Points for computing divergence and vorticity

at LA

AABB

CC

DD