Stokes Constants

We have seen that the differential equation

$$w'' + h^2\,q(\hat{z})\,w = 0,$$ (1173)

where $'\equiv d/d\hat{z}$ , possesses approximate WKB solutions of the form

$$(a, \hat{z}) = q^{-1/4}(\hat{z}) \,\exp\left(\,{\rm i}\,h\int_a^{\hat{z}} q^{\,1/2}(\hat{z})\,d\hat{z}'\right) \left[1+{\cal O}\left(\frac{1}{h}\right)\right],$$ (1174)

$$(\hat{z}, a) = q^{-1/4}(\hat{z}) \,\exp\left(-{\rm i}\,h\int_a^{\hat{z}} q^{\,1/2}(\hat{z})\,d\hat{z}'\right)\left[1+{\cal O}\left(\frac{1}{h}\right)\right].$$ (1175)

Here, we have adopted an arbitrary phase reference level $\hat{z}=a$ . The convenient notation $(a, \hat{z})$ is fairly self explanatory: $a$ and $\hat{z}$ refer to the lower and upper bounds of integration, respectively, inside the exponential. It follows that the other WKB solution can be written $(\hat{z},a)$ (because we can reverse the limits of integration inside the exponential to obtain minus the integral in $\hat{z}$ from $\hat{z}=a$ to $\hat{z}=\hat{z}$ ).

Up to now, we have thought of $\hat{z}$ as a real variable representing scaled height in the ionosphere. Let us now generalize our analysis somewhat, and think of $\hat{z}$ as a complex variable. There is nothing in our derivation of the WKB solutions that depends crucially on $\hat{z}$ being a real variable, so we expect these solutions to remain valid when $\hat{z}$ is reinterpreted as a complex variable. Incidentally, we must now interpret $q(\hat{z})$ as some well-behaved function of the complex variable. An approximate general solution of the differential equation (1175) in the complex $\hat{z}$ -plane can be written as as a linear superposition of the two WKB solutions (1176)-(1177).

The parameter $h$ is assumed to be much larger than unity. It is clear from Equations (1176)-(1177) that in some regions of the complex plane one of the WKB solutions is going to be exponentially larger than the other. In such regions, it is not mathematically consistent to retain the smaller WKB solution in the expression for the general solution, because the contribution of the smaller WKB solution is less than the intrinsic error associated with the larger solution. Adopting the terminology introduced in Section 8.13, the larger WKB solution is said to be dominant, and the smaller solution is said to be subdominant. Let us denote the WKB solution (1176) as $(a,\hat{z})_d$ in regions of the complex plane where it is dominant, and as $(a,\hat{z})_s$ in regions where it is subdominant. An analogous notation is adopted for the second WKB solution (1177).

Suppose that $q(\hat{z})$ possesses a simple zero at the point $\hat{z}=\hat{z}_0$ (chosen to be the origin, for the sake of convenience). It follows that in the immediate neighborhood of the origin we can write

$$q(\hat{z})= a_1\,\hat{z} + a_2\,\hat{z}^{\,2} + \cdots,$$ (1176)

where $a_1\neq 0$ . It is convenient to adopt the origin as the phase reference point (i.e., $a=0$ ), so the two WKB solutions become $(0,\hat{z})$ and $(\hat{z},0 )$ . We can define anti-Stokes lines in the complex $\hat{z}$ plane (see Section 8.13). These are lines that satisfy

$${\rm Re} \left( {\rm i}\! \int_0^{\hat{z}} q^{1/2}(\hat{z}') \,d\hat{z}' \right) = 0.$$ (1177)

As we cross an anti-Stokes line, a dominant WKB solution becomes subdominant, and vice versa. Thus, $(0,\hat{z})_d \leftrightarrow (0,\hat{z})_s$ and $(\hat{z},0)_d \leftrightarrow (\hat{z},0)_s$ . In the immediate vicinity of an anti-Stokes line the two WKB solutions have about the same magnitude, so it is mathematically consistent to include the contributions from both solutions in the expression for the general solution. In such a region, we can drop the subscripts $d$ and $s$ , because the WKB solutions are neither dominant nor subdominant, and write the WKB solutions simply as $(0,\hat{z})$ and $(\hat{z},0 )$ .

It is clear from Equations (1176)-(1177) that the WKB solutions are not single-valued functions of $\hat{z}$ , because they depend on $q^{1/2}(\hat{z})$ , which is a double-valued function. Thus, if we wish to write an approximate analytic solution to the differential equation (1175) then we cannot express this solution as the same linear combination of WKB solutions in all regions of the complex $\hat{z}$ -plane. This implies that there must exist certain lines in the complex $\hat{z}$ -plane across which the mix of WKB solutions in our expression for the general solution changes discontinuously. These lines are called Stokes lines (see Section 8.13), and satisfy

$${\rm Im} \left( {\rm i}\! \int_0^{\hat{z}} q^{1/2}(\hat{z}') \,d\hat{z}' \right) = 0.$$ (1178)

As we cross a Stokes line, the coefficient of the dominant WKB solution in our expression for the general solution must remain unchanged, but the coefficient of the subdominant solution is allowed to change discontinuously. Incidentally, this is perfectly consistent with the fact that the general solution is analytic: the jump in our expression for the general solution due to the jump in the coefficient of the subdominant WKB solution is less than the intrinsic error in this expression due to the intrinsic error in the dominant WKB solution. Once we appreciate that the coefficient of the subdominant solution can only change at a Stokes line, we can retain both WKB solutions in our expression for the general solution throughout the complex $\hat{z}$ -plane. In practice, we can only see a subdominant solution in the immediate vicinity of an anti-Stokes line, but if we were to evaluate the WKB solutions to higher accuracy [i.e., by retaining more terms in the asymptotic series in Equations (1176)-(1177)] then we could, in principle, follow a subdominant solution all the way to a neighboring Stokes line.

In the immediate vicinity of the origin

$$\int_0^{\hat{z}} q^{\,1/2}(\hat{z}) \,d\hat{z}' \simeq \frac{2\sqrt{a_1}}{3} \,\hat{z}^{\,3/2}.$$ (1179)

It follows from Equations (1179) and (1180) that three Stokes lines and three anti-Stokes lines radiate from a zero of $q(\hat{z})$ . The general arrangement of Stokes and anti-Stokes lines in the vicinity of a $q=0$ point is sketched in Figure 23. Note that a branch cut must also radiate from the $q=0$ point in order to uniquely specify the function $q^{\,1/2}(\hat{z})$ . Thus, in general, seven lines radiate from a zero of $q(\hat{z})$ , dividing the complex $\hat{z}$ plane into seven domains (numbered 1 through 7).

Figure: The arrangement of Stokes lines (dashed) and anti-Stokes lines (solid) around a simple zero of $q(\hat{z})$ . Also shown is the branch cut (wavy line). All of the lines radiate from the point $q=0$ .

Let us write our general solution as

$$w(\hat{z}, h) = A\,(0,\hat{z}) + B\,(\hat{z},0)$$ (1180)

on the anti-Stokes line between domains 1 and 7, where $A$ and $B$ are arbitrary constants. Suppose that the WKB solution $(0,\hat{z})$ is dominant in domain 7. Thus, in domain 7 the general solution takes the form

$$w(7) = A\,(0,\hat{z})_d + B\,(\hat{z},0)_s.$$ (1181)

Let us move into domain 1. In doing so, we cross an anti-Stokes line, so the dominant solution becomes subdominant, and vice versa. Thus, in domain 1, the general solution takes the form

$$w(1) = A\,(0,\hat{z})_s + B\,(\hat{z},0)_d.$$ (1182)

Let us now move into domain 2. In doing so, we cross a Stokes line, so the coefficient of the dominant solution, $B$ , must remain constant, but the coefficient of the subdominant solution, $A$ , is allowed to change. Suppose that the coefficient of the subdominant solution jumps by $t$ times the coefficient of the dominant solution, where $t$ is an undetermined constant. It follows that in domain 2 the general solution takes the form

$$w(2) =( A+t\,B)\,(0,\hat{z})_s + B\,(\hat{z},0)_d.$$ (1183)

Let us now move into domain 3. In doing so, we cross an anti-Stokes line, so the the dominant solution becomes subdominant, and vice versa. Thus, in domain 3, the general solution takes the form

$$w(3) =( A+t\,B)\,(0,\hat{z})_d + B\,(\hat{z},0)_s.$$ (1184)

Let us now move into domain 4. In doing so, we cross a Stokes line, so the coefficient of the dominant solution must remain constant, but the coefficient of the subdominant solution is allowed to change. Suppose that the coefficient of the subdominant solution jumps by $u$ times the coefficient of the dominant solution, where $u$ is an undetermined constant. It follows that in domain 4 the general solution takes the form

$$w(4) =( A+t\,B)\,(0,\hat{z})_d + (B+ u\,[A+t\,B])\,(\hat{z},0)_s.$$ (1185)

Let us now move into domain 5. In doing so, we cross an anti-Stokes line, so the the dominant solution becomes subdominant, and vice versa. Thus, in domain 5 the general solution takes the form

$$w(5) =( A+t\,B)\,(0,\hat{z})_s + (B+ u\,[A+t\,B])\,(\hat{z},0)_d.$$ (1186)

Let us now move into domain 6. In doing so, we cross the branch cut in an anti-clockwise direction. Thus, the argument of $\hat{z}$ decreases by $2\pi$ . It follows from Equation (1178) that $q^{\,1/2}\rightarrow -q^{\,1/2}$ and $q^{\,1/4}\rightarrow -{\rm i}\,q^{\,1/4}$ . The following rules for tracing the WKB solutions across the branch cut (in an anti-clockwise direction) ensure that the general solution is continuous across the cut [see Equations (1176)-(1177)]:

$$(0,\hat{z}) \rightarrow -{\rm i}\,(\hat{z}, 0),$$ (1187)

$$(\hat{z}, 0) \rightarrow -{\rm i}\,(0,\hat{z}).$$ (1188)

Note that the properties of dominancy and subdominancy are preserved when the branch cut is crossed. It follows that in domain 6 the general solution takes the form

$$w(6) =-{\rm i}\,( A+t\,B)\,(\hat{z},0)_s -{\rm i}\,(B+ u\,[A+t\,B])\,(0,\hat{z})_d.$$ (1189)

Let us, finally, move into domain 7. In doing so, we cross a Stokes line, so the coefficient of the dominant solution must remain constant, but the coefficient of the subdominant solution is allowed to change. Suppose that the coefficient of the subdominant solution jumps by $v$ times the coefficient of the dominant solution, where $v$ is an undetermined constant. It follows that in domain 7 the general solution takes the form

$$w(7) =-{\rm i}\,( A+t\,B +v\,\{B+ u\,[A+t\,B]\}) \,(\hat{z},0)_s -{\rm i}\,(B+ u\,[A+t\,B])\,(0,\hat{z})_d.$$ (1190)

Now, we expect our general solution to be an analytic function, so it follows that the solutions (1183) and (1192) must be identical. Thus, we can compare the coefficients of the two WKB solutions, $(\hat{z},0)_s$ and $(0,\hat{z})_d$ . Because $A$ and $B$ are arbitrary, we can also compare the coefficients of $A$ and $B$ . Comparing the coefficients of $A\,(0,\hat{z})_d$ , we find

$$1 = -{\rm i}\, u.$$ (1191)

Comparing the coefficients of $B\,(0,\hat{z})_d$ yields

$$0 = 1 + u \,t.$$ (1192)

Comparing the coefficients of $A\, (\hat{z},0)_s$ gives

$$0 = 1 + v\,u.$$ (1193)

Finally, comparing the coefficients of $B\, (\hat{z},0)_s$ yields

$$1 = -{\rm i}\,(t + v + v\,u\,t).$$ (1194)

Equations (1193)-(1196) imply that

$$t = u = v = {\rm i}.$$ (1195)

In other words, if we adopt the simple rule that every time we cross a Stokes line in an anti-clockwise direction the coefficient of the subdominant solution jumps by ${\rm i}$ times the coefficient of the dominant solution then this ensures that our expression for the general solution, (1182), behaves as an analytic function. Here, the constant ${\rm i}$ is usually called a Stokes constant. Note that if we cross a Stokes line in a clockwise direction then the coefficient of the subdominant solution has to jump by $-{\rm i}$ times the coefficient of the dominant solution in order to ensure that our general solution behaves as an analytic function.