Skewness of the Prior Through Position Errors and Its Impact on Data Assimilation - Data Assimilation for Atmospheric, Oceanic and Hydrologic Applications - page 168

where p D p.x;t/ is the wavefield being advected at the speed c.x/ . We attach to

( 7.57 ) an unbounded domain in as well as a localized initial condition such as the

functions ( 7.42 ), which was our prototype for the surface pressure field of members
of a prior distribution of tropical cyclones. By the method of characteristics we know
that the solutions to ( 7.57 ) move away from their initial location according to,

x

dX
dt D c.X/;

(7.58)

where is the location of, say, the minimum central pressure of the

function ( 7.42 ). We may immediately note that even though equation ( 7.57 )is
linear in the amplitude of the disturbance, equation ( 7.58 ) may be non-linear if
the function

X D X.t/

is non-linear. Hence, the motion of the location of the minimum
central pressure may evolve non-linearly and therefore even if the initial distribution
of phase uncertainty is Gaussian it still may evolve into a non-Gaussian phase
distribution owing to the non-linearity in ( 7.58 ). Two examples follow:

c.x/

(1) Linear Shear

In the case where the shear is a linear increasing function of distance from the origin,
i.e.

c.x/ D c 0 x

L ;

(7.59)

where c 0 is a characteristic phase speed and L is characteristic length scale. By

inserting ( 7.59 )into( 7.58 ) and solving finds
X.t/ D X 0 exp c 0 t
L

; (7.60)

where X 0 is the initial location of the minimum central pressure. In Sect. 7.3 the

parameter and was normally distributed. Notice that if the

location of the minimum central pressure is normally distributed with mean

X 0 was denoted as '

x and

2 , then at a later time, variance t , the phase distribution will be normal with

exp c 0 t
L

; 2 exp 2c 0 t
L

X N x : (7.61)

Hence, linear shear produces disturbances that move away from their initial location
exponentially with time. Nevertheless, disturbances in linear shear preserve the
Gaussian character of their initial phase uncertainty.

(2) Quadratic Shear

In the case where the shear is a quadratic function of distance from the origin, i.e.

x
L

2

c.x/ D c 0 ; (7.62)