Supercells - Severe Convective Storms and Tornadoes: Observations and Dynamics - page 168

Taking the dot product of the equation in (4.11) with the differential position
vector defined by (4.12), we find that

p 0

1

Dv = Dt dr ¼ J ð

2 v

v Þ E dr þ½ð JT v Þ T v E dr ¼ 0 J dr þ Bdz ð 4 : 13 Þ

E E E

Integrating (4.13) along a streamline (v), it follows that the Lamb term ½ð JT v Þ T v
is zero, because ½ð JT v Þ T v E v ¼ 0, and so
ð ð

Þ dx þ Ð @=@

2 2 2

1 1 1

@=@ x ð

2 j v j

y ð

2 j v j

Þ dy þ @=@ z ð

2 j v j

Þ dz

ð ð Bdz ð 4

p 0 p 0 p 0 ¼ 0 ð@ =@ xdx þ@ =@ ydy þ@ =@ zdz Þþ : 14 Þ

Since the LHS of (4.14) may be expressed as ð d ð

2

1

2 j v j

Þ , it follows that

ð d ð

Þþ 0 ð dp 0

ð Bdz ¼ 0

2

1

2 j v j

ð 4 : 15 Þ

Integrating (4.15) from an initial location (denoted by the subscript ''i '') to a final
position (denoted by the subscript ''f ''), we find that

ð Bdz ¼ 0

2 2

þ 0 p 0 f 0 p 0 i

1 1

2 j v f j 2 j v i j

ð 4 : 16 Þ

Now, suppose an air parcel moving at constant altitude, at mid-levels, in a storm-
relative reference frame catches up with a convective storm and air flows around
the updraft, but forks to the left side ( Figure 4.3 ). Outside the updraft of the
storm B ¼ 0 and far upstream of the storm p 0 i ¼ 0. Following an argument Kerry
Emanuel reproduced in his text, suppose that the air speeds up as it flows around
the edge of updraft such that

2 2

j v f j 2 j v i j ð 4 : 17 Þ

Figure 4.3. Idealized illustration of air at mid-levels catching up with and flowing around an
updraft inside which lower values of westerly momentum have been advected upward. Air
begins at I and is decelerated as it approaches the updraft; it then speeds up as it flows around
the left side of the updraft and ends up at F.