Chapter 10

Geostrophic Balance

(1) Wind and Height GradientsOn a contour analysis of a pressure surface:Strong winds

associated with contours close together; weak winds with contours farther apart.

Winds tend to blow (nearly) parallel to height contours.

Winds tend to be oriented such that higher heights are 90o to the right (in the northern Hemisphere) of the direction toward which the winds are blowing. (left in S. H.)

How does this relate to what you learned in ATMO 201 as “Buys Ballot’s Law”?

Winds that follow this are called Geostrophic Winds.

(2) The Geostrophic Equation

Expresses the magnitude of the wind speed as a function of the geopotential height gradient on a constant pressure surface.

Where, go = gravitational acceleration, f = Coriolis parameter = 2Ωsin(lat).

Ω = 7.292 x 10-5 s-1

f ranges from -1.4 x 10-4 s-1 at South Pole to +1.4 x 10-4 s-1 at the North Pole.f ≈ about 1 x 10-4 s-1 for mid-latitudes.

Zf

gv h

og

Consider: If the height gradient is 3 decameters (30 meters) per degree of latitude, how strong would the geostrophic wind be?

Note: 1o latitude ≈ 1.11 x 105 m1o = 60 n.mi., as long as you measure from the latitude lines (parallels, up and down)

v g

go

fh Z

9.8ms 2

110 4s 1 30m

1olat

1olat

1.11105m26.4m

s 1.943kt

1ms

51.29kt

We could orient the x axis parallel to the wind, and take the height gradient orthogonal to wind.The magnitude of the height gradient is the magnitude of:

Z

y

ug go

f

Z

y

vg go

f

Z

x

We can also treat each component separately, and calculate the components.

Magnitude ug2 vg

2 oooDirection 3.3191809031

36arctan

ug go

f

Z

y

vg go

f

Z

x3.85 x 105m)

3.3 x 105m)

Note error in book fig., page7, Ch. 10,backwards

The u-component of the geostrophic wind is dependent on the height gradient with respect to y.The v-component is dependent on the height gradient with respect to x.

ug go

f

Z

y

vg go

f

Z

x

The geostrophic wind is parallel to the height gradient; i.e., 90o to the right of the height gradient (in the northern hemisphere).

Remember cross products.

But, a x b is a vector, and has a direction.

If a horizontal vector is crossed with a vertical vector, the result will be a horizontal vector at right angles with the original horizontal vector.If the vertical vector is a unit vector, the magnitude of the resulting cross product vector will equal the magnitude of the original horizontal vector.

between anglesin|| baba

Vector Form of Geostrophic Wind Equation:

The palm and three fingers are pointing “curled” toward increasing height gradient (the vector pointing toward increasing height gradient).The thumb is pointing out from the page parallel to the geostrophic wind vector.

v g

go

fk h Z

Right hand Rule via Gig ‘Em: All fingers point in direction of k. Curl fingers in direction of increasing height gradient vector. Thumb points in direction of resultant of cross product.

An equivalent geostrophic equation is:

Meaning, crossing the vertical unit vector with the geostrophic winds results in a vector oriented toward lower heights.

f k v g goh Z

(3) Geostrophic DivergenceRemember, divergence is defined as:

And, the two components of the Geostrophic wind are:

Then if we take the derivative of ug with respect to x and the derivative of vg with respect to y, we can get the divergence by adding them together.

h v

u

x

v

y

ug go

f

Z

y

vg go

f

Z

x

ug

x

x

go

f

Z

y

vg

y

go

f

Z

x

Adding the two together gives:

Since go is a constant, it can be brought out and since f is a function only of y (latitude), then using the chain rule gives:

Since x and y are independent variables, the order of differentiation can be reversed and the first two terms cancel each other.

ug

x

vg

y

x

go

f

Z

y

y

go

f

Z

x

ug

x

vg

y

go

f

x

Z

y

go

f

y

Z

x

go

Z

x

y

1

f

Thus, the divergence of the geostrophic wind equation reduces to:

Since f changes so slowly with latitude, the right-hand side is typically much smaller in magnitude than any typical values of observed divergence of the actual wind. Thus, divergence of the geostrophic wind is essentially zero. It is non-divergent.

01

fyx

Zg

y

v

x

uo

gg

Therefore:(1) the primary wind patterns in the atmosphere are not, by themselves, associated with upward or downward motion. (2) departures from geostrophic balance are required for strong upward and downward motion. (3) it is very difficult to see convergence and divergence on constant pressure maps. (4) Because geostrophic winds are not divergent or convergent, constrictions on contours means the air must flow faster since the air, in geostrophic balance, is essentially confined between the contours.

If the wind obeys geostrophic balance, air between two height contours at a particular level must stay between those two height contours and stay on that level as well.

(4) Geostrophic AdvectionRemember, advection can be expressed as a dot product of two vectors with one of the vectors being wind (v)and the other the “del” of a scalar quantity (A) such as temperature (the gradient of A).The result is a scalar quantity.

v h A v h A cos

This can be written as:

Meaning, advection is proportional to the projection of the wind (⎮v⎮) onto the gradient (⎮∇hA⎮) multiplied by the magnitude of that gradient(⎮∇hA⎮).

v h A

v cos h A

For geostrophic wind, it is written:

Meaning, the projection of the gradient (⎮∇hA⎮)onto the direction of the geostrophic wind (⎮vg⎮) multiplied by the magnitude of that wind.

cosAvAv hghg

So, the advection of “A” by the geostrophic wind, vg, is directly proportional to:

(1) the magnitude of the geostrophic wind - which is inversely proportional to the spacing between height contours.(2) the magnitude of the gradient of “A” - which is inversely proportional to the spacing between successive contours (isopleths) of “A.”

cosAvAv hghg

Then, the smaller the areas bounded by the height contours and isopleths of “A”, the greater the advection.

Thus, we could say that the advection of “A” by the geostrophic wind is inversely proportional to the area of the parallelogram formed by the height contours and the isopleths of “A.”

We could also say that the advection is greatest when the “spatial density of intersections” of contours and isopleths of “A” is greatest.

(5) StreamfunctionsAny vector that is non-divergent can be expressed in terms of a scalar quantity known as a stream function; Ψ.

v k

v g

go

fk h Z

g go

fZ

The geostrophic wind vector is written as:

Therefore, the geostrophic stream function must be:

If “f” is considered constant.

Then, if the wind is non-divergent as it is for geostrophic wind:

It is not necessary to plot the wind vectors to see the wind pattern. Contours of the stream function will represent the wind.

Where the stream function contours are close together, the wind speed is strong; and farther apart where the wind is weak.All stream function contours (streamlines) should parallel the height contours.Adding arrows to the height contours produces streamlines (stream function contours).

(6) Nature of Geostrophic BalanceConsider the horizontal forces acting on a volume of air above the near surface region (where friction becomes significant).

The forces are the Pressure Gradient Force,

The Coriolis Force.

Fx m

dP

dx

Fy m

dP

dy

The Pressure Gradient Force is directed away from high pressure and toward low pressure.

In other words, it is opposite to the pressure gradient (which is directed from low pressure toward high pressure).

A gradient vector always points in the direction of thegreatest rate of increase.

The Coriolis Force - keeps the air from moving directly from high pressure to low pressure.

Its magnitude is proportional to the Coriolis parameter:Its magnitude is proportional to the horizontal wind speed.Its direction in the northern/southern hemisphere is 90o to the right/left of whatever direction the horizontal wind is blowing. Its magnitude is so (relatively) weak it would take several hours for the Coriolis force to cause a substantial change in the wind direction if it were the only force present.

2 sin

v

Consider a parcel of air moving parallel to the contours on an upper-air chart.

The pressure gradient force is oriented roughly 90o to the direction the parcel is moving and toward lower heights.The Coriolis force is oriented 90o to the direction the parcel is moving and toward higher heights.If the speed of the wind is just right, the pressure gradient force and the Coriolis force are of the same magnitude and pointed opposite to each other.

(7) Attaining Coriolis Nirvana

If the forces are not in balance, if something changes; e.g., the pressure gradient force, the parcel will accelerate in the direction of the stronger force. How does the parcel stay in geostrophic balance?

Consider the ways the wind might be out of geostrophic balance.

(1) Component of wind across contours is zero but the component along the contours is too weak.(2) Component of wind across contours is zero but the component along the contours is too strong.

(3) Component of wind along the contours is just right but there is a component of wind across contours from high to low pressure (heights).Component of wind along contours is just right, but (4) there is a component of wind across contours from low to high pressure (heights).

Consider case (1), the wind speed is too weak.

Departures from geostrophic balance cause a reorientation of the forces which causes the air motion to speed up or slow down and move back toward geostrophic balance. The time spent going to fast is the same as going to slow, so in the average, the air is in geostrophic balance. There are forces which will dampen this oscillation, and eventually lead to balance if p is not changing rapidly.

(8) Nature of the Coriolis Force

What we call the Coriolis force, the apparent force resulting from the earth’s rotation, is really a collection of mechanisms all resulting in a force towards the right in the N.H.

Rotating coordinate systemEasiest to visualize with the artillery exampleGives a good explanation for (part of) Coriolis when moving N or South

Variations in Centrifugal ForceExplains movement toward right with pure E/W motion

Conservation of Angular MomentumAlso a centrifugal force variation (R is changing)

To the whiteboard…

(9) AccelerationThe Forces acting on an air parcel are, then:

Pressure Gradient ForceCoriolis ForceFriction (turbulent mixing)

These can be expressed as accelerations (Force/unit mass).The net effect (net force/accelerations) of these expressed as components in the x- and y-direction are:

Du

Dt,

Dv

Dt

Since these are the result of the various forces/accelerations on the parcel, (Newton’s 2nd Law) they can be written as:

If acceleration and friction are zero, these then show geostrophic balance - Balance between the Pressure Gradient acceleration and the Coriolis acceleration.

Du

Dt go

Z

x 2v sin Fx

Dv

Dt go

Z

y 2u sin Fy

Acceleration = Pressure Gradient acceleration + Coriolis acceleration + Friction

Du

Dt fv go

Z

x Fx

Dv

Dt fu go

Z

y Fy

Since wind is a vector, the equations (Newton’s 2nd law) can be written in vector notation as one.

Where, “h” means the horizontal components of the vectors only.

Dv h

Dt f k

v h goh Z

F h

Consider the change in the wind in time.The acceleration of the wind (change in speed or direction, or both) can be represented as the difference between two wind vectors, at different times.The change in the wind vector is equal to a vector drawn from the end of the first vector to the end of the second vector (second vector minus the first vector).

The acceleration is obtained by dividing the difference vector (blue) by the time interval.

If only the direction is changing, not the speed, the direction of the acceleration is essentially at right angles to the original direction. If only the speed is changing, the direction of the acceleration is parallel to the original direction.

(10) Gradient and Cyclostrophic Winds

If the acceleration (net force/unit mass) is always at right angles to the wind direction, (e.g., to the left), the air will constantly curve to the left.Then the Pressure Gradient force is greater than the Coriolis force (since the Coriolis force - in the northern hemisphere- tries to make the air move to the right. This is how air flows about a low pressure (low height) region in the absence of friction.

This is Gradient Wind - the air flowing parallel to curved isobars (contours).

It is called “balanced” because it flows parallel to the contours, even though the forces are not in balance.

The winds about the low are weaker than in straight isobar / contour flow.In straight flow, the wind speed must be strong enough for the Coriolis force to balance the Pressure Gradient force.

In curved flow about a low, the Coriolis Force is weaker than the Pressure Gradient force, so the winds are weaker.The winds are Subgeostrophic.The contours / isobars about a low will be close together, signifying a strong Pressure Gradient force.

About a high pressure / height region, the Coriolis force must be greater than the Pressure Gradient force.The wind speed is greater than it would be in a straight isobar / contour flow. The winds are Supergeostrophic.The center of a high will be broad and flat, signifying a weak PGF.

From Wallace and Hobbs, 2nd, Fig 7.12© 2006 Elsevier, Inc.

An alternative view of the “balance”: Read 7.2.6 of W&H

From Wallace and Hobbs, 2nd, Fig 8.51© 2006 Elsevier, Inc.

Cyclostrophic Balance