Conversion to polar coordinates

1
Conversion to polar coordinates

• Standard low pressure systems and tropical
  cyclones are two examples of in meteorology that
  clearly have a physical structure compatible with
  polar coordinate system

2
Polar coordinates - defined

• Polar coordinates are defined in a 2-D
  coordinates system which measures positions in
  space based on
• 1 - radial distance
• 2 An azimuthal angle from a reference axis
  (usually the axis)
• Relationship between polar and rectangular
  coordinates

3
Polar coordinates

• From the figure we can see that rectangular
  coordinates and polar coordinates are related as
  follows
• And additionally from these relationships we find

4
Polar coordinate basis vectors

• Using the definitions of a vector from chapter 2
  we observe that the depicted radial vector has a
  magnitude and direction.
• Magnitude
  as already observed
• Direction

5
Polar coordinate basis vectors

• Notice that depends on q, this is a major
  difference from rectangular coordinates!
• To find the other basis vector, we wish to
  maintain an orthogonal coordinate system so
  simply find a vector perpendicular to .
  Assume a basis vector,
• And recall that perpendicularity requires the dot
  product of the two basis vector are zero so
• By inspection, we can see that
  is a sufficient choice for a
  perpendicular unit basis vector

6
Polar coordinate basis vectors

• Another way to derive is to utilize the
  q-dependence of by adding 90 degrees to angle
  of interest
• This yields the same result for as before.
  Notice that depends on q as well.
• -- The orientation between the two polar basis
  vectors is shown below.

7
Polar coordinate basis vectors

• We know have two polar basis vectors expressed in
  terms of the rectangular basis vectors.
• We can solve for and in the above two
  equations (2 equations 2 unknowns)

8
The gradient operator defined in polar coordinates

• Now that we have related the components and basis
  vectors of the polar and rectangular coordinates,
  we can express various operators in polar
  coordinates.
• By use of the chain rule, the horizontal gradient
  is expressed in terms of partial or r and q as

9
The gradient operator defined in polar coordinates

• From our previous relationships we observe that
• To find and , utilize the
  fact that

10
The gradient operator defined in polar coordinates

• Finally, recall that
• Substitution of all of these terms in the above
  expression, we obtain
• Which simplifies to

11
Velocity defined in polar coordinates

• Let us now express, the radial vector
  specifically in terms of our new basis vectors,
• Grouping basis terms together,
• We obtain the radial vector again

12
Velocity defined in polar coordinates

• To express velocity in polar coordinates,
  Consider a system where both r and q depend on
  time and take the time derivative of the radial
  vector (recall that the basis vector has q
  dependence)
• Or
• Where and

13
Advection on a scalar

• Now that we have the velocity field and gradient
  representations in polar coordinates we can
  derive other quantities such as the advection or
  divergence.
• First let us examine the advection on a scalar
  quantity, f(r,q).
• Using the representation of the velocity field
  and gradient in polar coordinates and the
  orthogonal nature of the basis vectors, the
  advection of f in polar coordinates is simply

14
Advection on a vector

• Now let us examine the advection on a vector
  which is of critical use since we can then
  express the equation of motion in polar
  coordinates
• Substitution of the gradient and velocity vector
  in polar coordinates yields
• The key difference in expanding the above
  expression is that we have to consider
  derivatives of the basis vectors now!
• Let us examine each term individually.

15
Advection on a vector

• The first term The first term is
  straightforward since we are looking at
  derivatives with respect to r and the basis
  vectors depend on q.

16
Advection on a vector

• The second term Now we have to consider the
• q-dependence of the basis vectors.
• Grouping terms of the same basis,

17
Advection on a vector

• Combining both the first and second terms and
  grouping basis terms, we see that

18
Equation of motion

• The equation of motion is
• Use of polar coordinates in the above yields the
  following two
• Equations
• Radial component ( )
• Azimuthal component ( )
• Notice the addition of the centripetal terms,
• that account for effects of curvature in
  circular flows.

19
Gradient Wind

• The 2-D equations of motion and 2-D
  incompressible conservation of mass in polar
  coordinates are

20
Gradient Wind- Assumptions

• 1) The flow is steady
• 2) The flow is axi-symmetric
• 3) Flow is strictly tangential

21
Gradient Wind

• The above assumption simplify the equations of
  motion to one radial equation
• If we know the pressure field distribution then
  we can solve for the azimuthal flow in the
  gradient wind equation by the use of the
  quadratic equation

22
Gradient Wind-terminology

• If the observed curved flow is greater than the
  theoretically predicted gradient wind is consider
  to be a super-gradient flow.

23
Gradient Wind-exercise

• The gradient wind is expressed in rectangular
  coordinates as
• What type of flow do we obtain if we consider the
  limit
• ?

24
Gradient Wind

• The gradient wind can represent various physical
  circumstances depending on direction of the
  azimuthally flow and the sign of the pressure
  gradient. Let us examine a few of these
  possibilities
• 1) Normal cyclonic flow
• This is a standard ideal low pressure system with
  cyclonic flow.
• From a force relationship perspective, consider
  first a geostrophic balance. The straight flow
  is a balance between an inward directed pressure
  gradient and an outward directed Coriolis force.
  In the case of the gradient wind, there is a
  force imbalance where the pressure gradient force
  is larger than Coriolis force. This leads to net
  force directed inward and a resultant centripetal
  acceleration causing a curved cyclonic flow.

25
Gradient Wind

• 2) Normal anti-cyclonic flow
• The decreasing pressure from the origin indicates
  a high pressure system. Since the aziumthal flow
  is less than zero, the flow is antic-cyclonic and
  thus the flow represents a standard high pressure
  system.
• There is a force imbalance where the outward
  directed pressure gradient force is less than
  inward directed Coriolis force. This force
  imbalance leads to net force directed inward and
  a resultant centripetal acceleration causing a
  curved anti-cyclonic flow.

26
Exercise

• For a normal anti-cyclonic flow,
• Determine if there is a limit on the magnitude of
  the pressure gradient.

27
Gradient Wind

• 3) Anomalous anti-cyclonic flow about a high
  pressure system
• This system has the same force balance as the
  previous example but considers the negative root
  of the quadratic solution.
• Calculations show that the pressure gradient for
  the above solution to exist would be extremely
  small.
• Although this flow is theoretically possible, it
  is not really empirically seen.