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## Math 597F, Notes 8: unstable Orr-Sommerfeld solutions for stable profiles

(this post was also posted here on my new blog address:http://blog.toannguyen.org/ )

We now turn to the delicate case: Orr-Sommerfeld solutions for stable profiles to Rayleigh. The results reported here are in a joint work with E. Grenier and Y. Guo, directing some tedious details of the proof to our paper. We consider the Orr-Sommerfeld problem:

$\displaystyle Ray_\alpha(\phi) = \epsilon \Delta_\alpha ^2 \phi,$

with zero boundary conditions on ${\phi}$ and ${\phi'}$, in which ${Ray_\alpha(\cdot)}$ denotes the Rayleigh operator and ${\Delta_\alpha = \partial_z^2 - \alpha^2}$.

We assume that the boundary layer ${U}$ is stable to the Rayleigh operator, that is, there is no unstable eigenvalue with ${\Im c_0 >0}$ to the Rayleigh problem (equivalently, the spectrum of Euler lies precisely on the imaginary axis). In term of the Wronskian determinant, this asserts that ${E(\alpha,c) \not =0}$ for all ${c}$ with ${\Im c>0}$. The dispersion relation studied in the last section shows that there is no eigenvalue to the Orr-Sommerfeld problem, with ${\Im c(\epsilon)}$ being bounded away from zero in the limit ${\epsilon \rightarrow 0}$. Hence, in search for a possible unstable point spectrum of Navier-Stokes, we will construct solutions to Orr-Sommerfeld equations in the regime where all parameters ${\alpha, c}$ and ${\epsilon}$ are sufficiently small. In particular, ${\alpha \log \Im c \ll 1}$, in order to take the inverse of the ${Ray_\alpha}$ operator as studied in the last section. Throughout the notes, ${z_c}$ denotes a complex number so that ${U(z_c) = c}$, for each fixed complex number ${c}$.

Clearly, ${\epsilon \Delta_\alpha^2 Ray_\alpha^{-1}}$ is not a good iteration operator, since by a view of Rayleigh solutions, ${Ray_\alpha^{-1}(f)}$ typically has a singularity of the form ${(z-z_c)\log (z-z_c)}$ and therefore ${\epsilon \Delta_\alpha^2 Ray_\alpha^{-1}}$ has a singularity of order ${(z-z_c)^{-3}}$. To deal with the singularity, we need to examine the leading operator in Orr-Sommerfeld equations near the singular point ${z = z_c}$. Indeed, introducing the blow-up variable: ${Z = (z-z_c)/\delta}$ and search for the ansatz solution ${\phi = \phi_\mathrm{cr}(Z)}$, which is in the literature often referred as the critical layer. It follows that ${\phi_\mathrm{cr}}$ approximately solves

$\displaystyle \partial_Z^4 \phi_\mathrm{cr} \approx Z\partial_Z^2 \phi_\mathrm{cr}$

with the critical layer size ${\delta \approx \epsilon^{1/3}}$. That is, within the critical layer, ${\partial_z^2 \phi_\mathrm{cr}}$ solves the classical Airy equation. In the light of the toy model problem in the previous section, we introduce the following iteration operator:

$\displaystyle Iter: = \underbrace{Airy^{-1}}_{\mbox{critical layer}} \circ \quad \underbrace{\epsilon \Delta^2_\alpha}_{\mbox{error}} \quad \circ \quad \underbrace{Ray_\alpha^{-1}}_{\mbox{inviscid}}.$

It suffices to show that the Iter operator is contractive in a suitable function space. This would take care o the singularity. (in the proof, we in fact need a slightly more complicated iteration operator to take care of a possible linear growth in ${z}$ of the Rayleigh solutions at infinity). Here, the Airy operator is defined by

$\displaystyle Airy(\phi) := \epsilon \partial_z^4 \phi - (U - c + 2 \epsilon \alpha^2) \partial_z^2 \phi . \ \ \ \ \ (1)$

Several issues to overcome. We need to appropriately define ${Airy^{-1}}$ and ${Ray_\alpha^{-1}}$, and most of all, to study the smoothing effect of Airy operator near the critical layer. The inverse ${Ray_\alpha^{-1}}$ is studied at length in the previous lecture. Similarly, the inverse ${Airy^{-1}}$ will be constructed through its Green kernel.

8.1. Green kernel

We recall the basic construction of the Green kernel of a linear, variable-coefficient, second-order ODE operator:

$\displaystyle L \phi : = a(z) \partial_z^2\phi + b(z)\partial_z\phi + c(z) \phi = f(z),\qquad z\ge 0,$

with ${a(z)}$ never vanishing. For simplicity, we assume that there are two independent solutions ${\phi_1, \phi_2}$ to the homogenous equation ${L\phi =0}$, with ${\phi_1}$ being bounded and ${\phi_2}$ unbounded at infinity ${z = \infty}$. We search for a particular (bounded) solution to the non-homogenous equation in the integral form:

$\displaystyle \phi (z) = \int_0^\infty G(x,z) f(x) \; dx,\ \ \ \ \ (2)$

in which ${G(x,z)}$ is called the Green kernel of ${L}$, which by the equation satisfies ${L G(x,z) = \delta_x(z)}$, where ${\delta_x(z)}$ denotes the delta function centered at ${z=x}$. Comparing the singularity in the equation for ${G(x,z)}$, one asserts that ${G(x,z)}$ must be continuous and ${\partial_zG(x,z)}$ has a jump across ${z=x}$; precisely,

$\displaystyle [G(x,z)]_{\vert_{z=x}} : = \lim_{z \rightarrow x^+} G(x,z) - \lim_{z\rightarrow x^-} G(x,z) = 0, \qquad [a(z)\partial_z G(x,z)]_{\vert_{z=x}} = 1.$

We now construct a Green kernel ${G(x,z)}$ (which is non unique, unless we impose a boundary condition). As ${z \not =x}$, ${G(x,z)}$ solves the homogenous equation and so can be constructed as a linear combination of the other two independent solutions ${\phi_1,\phi_2}$. This yields

\displaystyle G(x,z) = \left \{ \begin{aligned} C_1(x) \phi_1(z) , \qquad & z>x\\ - C_2(x) \phi_2(z), \qquad & z

The jump conditions across ${z=x}$ yield the linear equations:

$\displaystyle \begin{pmatrix} \phi_1 & \phi_2 \\ \phi'_1 & \phi'_2 \end{pmatrix} \begin{pmatrix} C_1\\C_2\end{pmatrix}(x) = \begin{pmatrix} 0 \\ 1/a(x)\end{pmatrix} ,$

which gives ${C_1(x)}$ and ${C_2(x)}$. Hence, the Green kernel ${G(x,z)}$ can be defined as

\displaystyle G(x,z) = \frac{1}{a(x)W[\phi_1, \phi_2](x)} \left \{ \begin{aligned} \phi_2(x) \phi_1(z) , \qquad & z>x\\ \phi_1(x) \phi_2(z), \qquad & z

in which ${W[\phi_1, \phi_2]}$ denotes the Wronskian determinant, which can be computed through the Abel’s identity: ${W(x) = W(0) \mbox{exp}(-\int_0^x b(z)/a(z)\; dz)}$. In particular, when ${b\equiv 0}$, the Wronskian determinant is constant. Note that the Green kernel is not unique, as we did not take care of the boundary conditions. Certainly, this construction yields a particular solution in the integral form (2).

8.2. Airy operator

We now study the Airy operator, defined as in (1). Note that ${\partial_z^2\phi}$ does not exactly solve the classical Airy equation: ${\partial_z^2 u - z u =0}$. We shall make a change of variables and unknowns in order to go back to the classical Airy equation. This change is very classical in physical literature, and called the Langer’s transformation: ${(z,\phi) \mapsto (\eta,\Phi)}$, with ${\eta = \eta(z)}$ defined by

$\displaystyle \eta (z) = \Big[ \frac 32 \int_{z_c}^z \Big( \frac{U-c}{U'_c}\Big)^{1/2} \; dz \Big]^{2/3} \ \ \ \ \ (4)$

and ${\Phi = \Phi(\eta)}$ defined by the relation

$\displaystyle \partial_z^2 \phi (z) = \dot z ^{1/2} \Phi(\eta), \ \ \ \ \ (5)$

in which ${\dot z = \frac{d z( \eta)}{d \eta} }$ and ${z = z(\eta)}$ is the inverse of the map ${\eta = \eta(z)}$. By a view of the definition (4), we note that ${(U-c)\dot z^2 = U'_c \eta}$, with ${U'_c = U'(z_c)}$. The following lemma links the Airy operator (1) with the classical Airy equation.

Lemma 1 Let ${(z,\phi) \mapsto (\eta, \Phi)}$ be the Langer’s transformation defined as above. The function ${\Phi(\eta)}$ solves the classical Airy equation:

$\displaystyle \epsilon \partial^2_\eta \Phi - U_c' \eta \Phi = f(\eta)\ \ \ \ \ (6)$

if and only if the function ${\phi = \phi(z)}$ solves

$\displaystyle Airy ( \phi) = \dot z ^{-3/2} f(\eta(z))+ \epsilon [ \partial_z^2 \dot z^{1/2} \dot z^{-1/2} - 2\alpha^2 ]\partial_z^2 \phi (z) .\ \ \ \ \ (7)$

Proof: The lemma follows from direct calculations. $\Box$

8.2.1. The classical Airy. Thanks to the Langer’s transformation, we first solve the classical Airy equation (6) for ${\Phi}$. Let us denote

$\displaystyle \delta = \Bigl( { \epsilon\over U_c'} \Bigr)^{1/3} = e^{-i \pi / 6} (\alpha R U_c')^{-1/3}$

to be the critical layer size, and introduce the notation ${Z = \delta^{-1} \eta }$. Then ${\Psi(Z) = \Phi(\eta)}$ solves the truly classical Airy: ${\Psi'' - Z \Psi =U_c' \delta f(\delta Z)}$. We use the following classical lemma:

Lemma 2 The classical Airy equation ${\Psi'' - z \Psi =0}$ has two independent solutions ${Ai(z)}$ and ${Ci(z)}$ so that the Wronskian determinant of ${Ai}$ and ${Ci}$ equals to one. In addition, ${Ai(e^{i \pi /6} x)}$ and ${Ci(e^{i \pi /6} x)}$ converge to ${0}$ as ${x\rightarrow \pm \infty}$ (${x}$ being real), respectively. Furthermore, there hold asymptotic bounds:

$\displaystyle \Bigl| Ai(k, e^{i \pi / 6} x) \Bigr| \le {C| x |^{-k/2-1/4} } e^{-\sqrt{2 | x|} x / 3}, \qquad k\in \mathbb{Z}, \quad x\in \mathbb{R},$

and

$\displaystyle \Bigl| Ci(k, e^{i \pi / 6} x) \Bigr| \le {C| x |^{-k/2-1/4} } e^{\sqrt{2 | x|} x / 3}, \qquad k\in \mathbb{Z},\quad x\in \mathbb{R},$

in which ${Ai(0,z) = Ai(z)}$, ${Ai(k,z) = \partial_z^{-k} Ai(z)}$ for ${k\le 0}$, and ${Ai(k,z)}$ is the ${k^{th}}$ primitives of ${Ai(z)}$ for ${k\ge 0}$ and is defined by the inductive path integrals

$\displaystyle Ai(k, z ) = \int_\infty^z Ai(k-1, w) \; dw$

so that the integration path is contained in the sector with ${|\arg(z)| < \pi/3}$. The Airy functions ${Ci(k,z)}$ for ${k\not =0}$ are defined similarly.

Thus, we can now introduce the Green kernel of the classical Airy equation (6):

$\displaystyle G_{a}(X,Z) = \delta \epsilon^{-1} \left\{ \begin{array}{rrr} Ai(X)Ci(Z), \qquad &\mbox{if}\qquad \xi > \eta,\\ Ai (Z) Ci(X) , \qquad &\mbox{if}\qquad \xi < \eta, \end{array}\right.$

in which ${X = \delta^{-1} \xi, Z = \delta^{-1}\eta}$. By definition, we have

$\displaystyle \epsilon\partial_\eta^2 G_{a}(X,Z) - U_c' \eta G_{a}(X,Z) = \delta_\xi(\eta). \ \ \ \ \ (8)$

It follows directly that

Lemma 3 There hold pointwise estimates:

\displaystyle \begin{aligned} |\partial_Z^\ell \partial_X^k G_\mathrm{a}(X,Z) | &\le C |\delta|^{-2}(1+|Z|)^{(k+\ell-1)/2} e^ {- {\sqrt{2} \over 3} \sqrt{|Z|}|X-Z| } \end{aligned}

for all ${k, \ell\ge 0}$.

8.2.2. An approximate Green kernel for Airy operator. Let us take ${\xi = \eta(x)}$ and ${\eta = \eta(z)}$ where ${\eta(\cdot)}$ is the Langer’s transformation and denote ${\dot x = 1/\eta'(x)}$ and ${\dot z = 1/ \eta'(z)}$. By a view of (5), we define the function ${G(x,z)}$ so that

$\displaystyle \partial_z^2G(x,z) =\dot x ^{3/2} \dot z^{1/2} G_{a}(\delta^{-1}\eta(x),\delta^{-1}\eta(z)), \ \ \ \ \ (9)$

in which the factor ${\dot x ^{3/2}}$ was added simply to normalize the jump of ${G(x,z)}$. It then follows from Lemma 1 together with ${\delta_{\eta(x)} (\eta(z)) = \delta_x(z)}$ that

$\displaystyle Airy( G(x,z) )= \delta_x(z) + \epsilon [ \partial_z^2 \dot z^{1/2} \dot z^{-1/2} - 2\alpha^2]\partial_z^2G(x,z) . \ \ \ \ \ (10)$

That is, ${G(x,z)}$ is indeed an approximate Green function of the Airy operator, defined as in (1), up to a small error term of order ${\epsilon \partial_z^2 G = \mathcal{O}(\delta)}$. It remains to solve (9) for ${G(x,z)}$, retaining the jump conditions on ${G(x,z)}$ across ${x=z}$. In view of primitive Airy functions, let us denote

$\displaystyle \widetilde Ci(1,z) =\delta^{-1} \int_0^z \dot y^{1/2} Ci(\delta^{-1}\eta(y))\; dy, \qquad \widetilde Ci(2,z) = \delta^{-1}\int_0^z \widetilde Ci(1,y)\; dy$

and

$\displaystyle \widetilde Ai(1,z) = \delta^{-1}\int_\infty^z \dot y^{1/2} Ai(\delta^{-1}\eta(y))\; dy, \qquad \widetilde Ai(2,z) =\delta^{-1} \int_\infty^z \widetilde Ai(1,y)\; dy.$

Thus, together with our convention that the Green function ${G(x,z)}$ should vanish as ${z}$ goes to ${+\infty}$ for each fixed ${x}$, we are led to introduce

\displaystyle G(x,z) = i \delta^3 \pi \epsilon^{-1} \dot x^{3/2} \left\{ \begin{aligned} \Big[ Ai(\delta^{-1}\eta(x)) \widetilde Ci(2,z) + \delta^{-1} a_1 (x) (z-x) + a_2(x) \Big] , &\quad \mbox{if }x>z,\\ Ci(\delta^{-1}\eta(x)) \widetilde Ai(2,z) , &\quad \mbox{if } x

in which ${a_1(x), a_2(x)}$ are chosen so that the jump conditions (see below) hold. Clearly, by definition, ${G(x,z)}$ solves (9), and hence (10). Here the jump conditions on the Green function read:

\displaystyle \begin{aligned} ~[G(x,z)]_{\vert_{x=z}} = [\partial_z G(x,z)]_{\vert_{x=z}} = [\partial_z^2 G(x,z)]_{\vert_{x=z}} =0 \end{aligned}\ \ \ \ \ (11)

and

\displaystyle \begin{aligned} ~[\epsilon \partial_z^3G(x,z)]_{\vert_{x=z}} = 1. \end{aligned}\ \ \ \ \ (12)

We note that from (9) and the jump conditions on ${G_a(X,Z)}$ across ${X=Z}$, the above jump conditions of ${\partial_z^2 G}$ and ${\partial_z^3 G}$ follow easily. In order for the jump conditions on ${G(x,z)}$ and ${\partial_zG(x,z)}$, we take

\displaystyle \begin{aligned} a_1 (x) &= Ci(\delta^{-1}\eta(x)) \widetilde Ai(1,x) - Ai(\delta^{-1}\eta(x)) \widetilde Ci(1,x) ,\\ a_2(x) &= Ci(\delta^{-1}\eta(x))\widetilde Ai(2,x) - Ai(\delta^{-1}\eta(x)) \widetilde Ci(2,x) . \end{aligned} \ \ \ \ \ (13)

This yields an approximate Green kernel for the Airy operator. Direct, but much tedious, calculations give sufficient point-wise estimates on the Green kernel, yielding the following convolution estimates; see this paper for details of the proof.

Lemma 4 Let ${G(x,z)}$ be the approximate Green kernel constructed as above. There holds

$\displaystyle \Big\|\int_{0}^{\infty} \partial^k_z G(x,\cdot ) f(x) dx \Big \|_{{\eta'}}\le \frac {C|\delta|^{-k}}{\eta - \eta'}\|f\|_{\eta} , \qquad k\ge 0,$

for arbitrary ${0\le \eta'<\eta}$, and for ${f \in X_\eta}$, denoting the function space that consists of decaying functions at an exponential rate of ${e^{-\eta z}}$.

8.2.3. Inverse for Airy operator. We study the inhomogeneous Airy equation: ${Airy(\phi) = f}$, for some source ${f(z)}$. Thanks to the above construction of an approximate Green kernel, we introduce an approximate inverse of ${Airy}$ operator:

$\displaystyle AirySolver(f) := \int_{0}^{\infty} G(x,z) f(x) dx . \ \ \ \ \ (14)$

Then, there holds

$\displaystyle Airy(AirySolver(f)) = f + AiryErr(f) \ \ \ \ \ (15)$

where the error operator ${AiryErr(\cdot)}$ is introduced due to the approximation of the Green kernel and thus is defined as

$\displaystyle AiryErr(f) : = \epsilon\int_{0}^{\infty} [ \partial_z^2 \dot z^{1/2} \dot z^{-1/2} - 2\alpha^2]\partial_z^2G(x,z) f(x) dx .$

From the convolution estimates, Lemma 4, there hold

$\displaystyle \| AirySolver(f)\|_{{\eta'}} \le C_\eta \| f\|_{\eta} , \qquad \| AiryErr(f) \|_{\eta} \le C|\delta| \|f \|_{\eta},$

for all ${f\in X_\eta}$ and ${\eta' <\eta}$. Recalling that ${\delta \rightarrow 0}$, and so the error is indeed small of order ${\mathcal{O}(\delta)}$. It is worth noting that unlike ${AirySolver(\cdot)}$, the ${AiryErr(\cdot)}$ maps from ${X_\eta}$ into itself. The reason is that the slight loss of an exponential decay in ${AirySolver(\cdot)}$ is due precisely to the linear growth in ${z}$ in the approximate Green function ${G(x,z)}$, whereas this linear growth vanishes in the calculation for ${\partial_z^2 G(x,z)}$. We recover the same rate of decay for ${AiryErr(f)}$ as that of ${f}$.

We now construct an inverse for Airy operator by iteration. Let us start with a fixed ${f \in X_\eta}$. Define

\displaystyle \begin{aligned} \phi_n &= - AirySolver(E_{n-1}) \\ E_n &= - AiryErr(E_{n-1}) \end{aligned} \ \ \ \ \ (16)

for all ${n \ge 1}$, with ${E_0 = f}$. It follows by induction that

$\displaystyle Airy (S_n) = f + E_n,\qquad S_n = \sum_{k=1}^n \phi_k ,$

for all ${n\ge 1}$. By induction, ${ \| E_n\|_\eta \le C \delta \|E_{n-1}\|_\eta \le (C\delta)^n \| f\|_\eta}$ and hence ${E_n \rightarrow 0}$ in ${X_\eta}$ as ${n \rightarrow \infty}$, for sufficiently small ${\delta}$. In addition, ${ \| \phi_n \|_{\eta'} \le C \| E_{n-1}\|_\eta \le C (C\delta)^{n-1} }$, which shows that ${\phi_n}$ converges to zero in ${X_{\eta'}}$ for arbitrary fixed ${\eta' <\eta}$ as ${n \rightarrow \infty}$. Furthermore the series

$\displaystyle S_n \rightarrow S_\infty$

in ${X_{\eta'}}$ as ${n \rightarrow \infty}$, for some ${S_\infty \in X_{\eta'}}$. We then denote ${AirySolver_\infty(f) = S_\infty}$, for each ${f \in X_\eta}$. In addition, we have ${ Airy (S_\infty) = f,}$ that is, ${AirySolver_\infty(f) }$ is the exact inverse for the Airy operator. Summarizing, we have proved the following theorem.

Lemma 5 Let ${\eta'<\eta}$ be positive numbers. Assume that ${\delta}$ is sufficiently small. There exists an exact solver ${AirySolver_\infty(\cdot)}$, which is a bounded operator from ${X_\eta}$ to ${X_{\eta'}}$, so that

$\displaystyle Airy(AirySolver_\infty (f)) = f.$

8.3. Singularities and contraction of Iter operator

In this section, we study the smoothing effect of the modified Airy function. Precisely, let us consider the Airy equation with a singular source:

$\displaystyle Airy(\phi) = \epsilon \partial_z^4 f(z)\ \ \ \ \ (17)$

in which ${f \in Y_{4,\eta}}$, that is ${f(z)}$ and its derivatives decay exponentially at infinity and behaves as ${(z-z_c)\log(z-z_c)}$ near the critical layer ${z=z_c}$. The singular source ${\epsilon \partial_z^4f}$ arises as an error of the inviscid solution when solving the full viscous problem. The key for the contraction of the iteration operator lies in the following lemma:

Lemma 6 Assume that ${z_c,\delta \lesssim \alpha}$. Let ${AirySolver_\infty(\cdot)}$ be the exact Airy solver of the ${Airy(\cdot)}$ operator constructed as in Proposition 5 and let ${f\in Y_{4,\eta}}$. There holds the estimate:

\displaystyle \begin{aligned} \Big\| AirySolver_\infty( \epsilon \partial_x^4 f ) \Big\|_{X_{2,\eta'}} \le C_\eta\|f\|_{Y_{4,\eta}} \delta(1+|\log \delta|) (1+|z_c/\delta| ) \end{aligned}\ \ \ \ \ (18)

for arbitrary ${\eta' < \eta}$.

Proof: The rough idea is that the convolution can be computed as

$\displaystyle G \star \epsilon \partial_z^4 f = - \epsilon \partial_z^3 G \star \partial_z f ,$

in which ${\epsilon \partial_z^3 G}$ is bounded and is localized near the critical layer of the size of order ${\delta}$. This indicates the bound by ${\delta \log \delta}$ as stated in the estimate (18). The factor ${1 + |z_c/\delta|}$ is precisely due to the linear growth in ${z}$ in the Green kernel ${G(x,z)}$. We refer to the paper, Section 5, for details of the proof. $\Box$

Having provided the estimates on ${Ray_\alpha^{-1} }$ and ${Airy^{-1}}$, and the convolution estimates, we can obtain the contraction of the ${Iter}$ operator, defined by

$\displaystyle Iter: = \underbrace{Airy^{-1}}_{\mbox{critical layer}} \circ \quad \underbrace{\epsilon \Delta^2_\alpha}_{\mbox{error}} \quad \circ \quad \underbrace{Ray_\alpha^{-1}}_{\mbox{inviscid}}.$

Precisely, we can prove the following lemma:

Lemma 7 For ${g \in X_{2,\eta}}$, the ${Iter(\cdot)}$ operator is a well-defined map from ${X_{2,\eta}}$ to ${X_{2,\eta}}$. Furthermore, there holds

$\displaystyle \| Iter(g)\|_{X_{2,\eta}} \le C \delta(1+|\log \delta|) (1+|z_c/\delta| ) \|g\|_{X_{2,\eta}},\ \ \ \ \ (19)$

for some universal constant ${C}$.

8.4. Slow modes

In this paragraph we explicitly compute the boundary contribution of the first terms in the expansion of the slow Orr-Sommerfeld modes, which are obtained from the Rayleigh solutions:

\displaystyle \begin{aligned} \phi_s(z;c) &= \phi_{Ray}(z;c) + Airy^{-1} (\epsilon \Delta_\alpha \phi_{Ray})(z;c) + \cdots \end{aligned}\ \ \ \ \ (20)

in which the second term is obtained by the interation via the Iter operator, plus higher order terms. We recall that the Rayleigh solution, again obtained via a perturbative analysis, is of the form:

$\displaystyle \phi_{Ray} (z;c)= e^{-\alpha z} (U-c + \mathcal{O}(\alpha)).$

It is crucial to note that the possible ${z\log z}$ singularity in the Rayleigh solution arises only at the order of ${\alpha}$. That is, we apply the Airy smoothing operator, Lemma 6, precisely to the ${\mathcal{O}(\alpha)}$ term, yielding

$\displaystyle \| Airy^{-1} (\epsilon \Delta_\alpha \phi_{Ray})\|_\eta \le C \epsilon + C\alpha \delta(1+|\log \delta|) (1+|z_c/\delta| ).$

This yields at once the following lemma:

Lemma 8 Let ${\phi_s}$ be the slow mode constructed above. For small ${z_c, \alpha, \delta}$, such that ${\delta \lesssim \alpha}$ and ${z_c\approx \alpha}$, there hold

\displaystyle \begin{aligned} \frac{\phi_s(0;c)}{\partial_z\phi_s(0;c)} &= \frac{1}{U'_0}\Big[ U_0 - c + \alpha \frac{(U_+-U_0)^2}{U'_0} + \mathcal{O}(\alpha^2\log \alpha) \Big] . \end{aligned} \ \ \ \ \ (21)

8.5. Fast modes

Similarly, the fast modes are constructed as a perturbation from the second primitive Airy solutions:

$\displaystyle \phi_{f,0}(z) : = \gamma_0 Ai(2,\delta^{-1}\eta(z)) , \qquad \gamma_0 := Ai(2,\delta^{-1}\eta(0))^{-1}.$

Here, ${\gamma_0}$ is to normalize the possible blow-up value of ${Ai(2,\cdot)}$ on the boundary ${z=0}$, since ${\delta^{-1} \eta(0) \approx e^{i 7\pi /6} |z_c/\delta|}$ could be arbitrarily large. By construction, there holds the following expansion of the fast mode ${\phi_f}$ on the boundary ${z=0}$:

$\displaystyle \phi_f(0) = \phi_{f,0}(0) + \mathcal{O}(\delta), \qquad \phi'_f(0) = \phi'_{f,0}(0) + \mathcal{O}(1).$

By definition, we have ${ \phi_{f,0}(0) = 1}$ and

$\displaystyle \phi'_{f,0}(0) = \delta^{-1} {Ai(1,\delta^{-1} \eta(0)) \over Ai(2,\delta^{-1} \eta(0)) } .$

In the study of the linear dispersion relation, we are interested in the ratio ${\phi_f/\phi'_f}$. From the above estimates on ${\phi_f(0)}$ and ${\phi'_{f}(0)}$, it follows at once that

$\displaystyle {\phi_{f}(0) \over \phi'_{f}(0)} = \frac{\delta C_{Ai}(\delta^{-1} \eta(0))}{1+\mathcal{O}( \delta) C_{Ai}(\delta^{-1} \eta(0)) } ( 1 + \mathcal{O}(\delta)), \qquad C_{Ai} (Y):= {Ai(2,Y) \over Ai(1,Y) } .$

As will be calculated below, ${\delta C_{Ai}(\delta^{-1} \eta(0)) \approx \delta (1+|\eta(0)/\delta|)^{-1/2} \ll 1}$. Hence, the above ratio is estimated by

$\displaystyle {\phi_{f}(0) \over \phi'_{f}(0)} =\delta C_{Ai}(\delta^{-1} \eta(0)) ( 1 + \mathcal{O}(\delta)) . \ \ \ \ \ (22)$

Here, we recall that ${\delta = e^{-i \pi / 6} (\alpha R U_c')^{-1/3}}$, and ${\eta(0) = - z_c + \mathcal{O}(z_c^2)}$. Therefore, we are interested in the ratio ${C_{Ai}(Y)}$ for complex ${Y = - e^{i \pi /6}y}$, for ${y}$ being in a small neighborhood of ${ \mathbb{R}^+}$. Without loss of generality, in what follows, we consider ${y \in \mathbb{R}^+}$. Directly from the asymptotic behavior of the Airy functions, we obtain the following lemma:

Lemma 9 Let ${C_{Ai}(\cdot)}$ be defined as above. Then, ${C_{Ai}(\cdot)}$ is uniformly bounded on the ray ${Y = e^{7i\pi/6} y}$ for ${y \in \mathbb{R}^+}$. In addition, there holds

$\displaystyle C_{Ai}(- e^{i \pi /6} y) = - e^{ 5i \pi / 12} y^{-1/2} (1+\mathcal{O}(y^{-3/2}))$

for all large ${y\in \mathbb{R}^+}$. At ${y = 0}$, we have ${ C_{Ai} (0) = - 3^{1/3} \Gamma(4/3).}$

This yields at once the following estimate on the ratio (22):

Lemma 10 As long as ${z_c/\delta}$ is sufficiently large, there holds

$\displaystyle {\phi_{f}(0) \over \phi'_{f}(0)} = - e^{\pi i/4} |\delta| |z_c/\delta|^{-1/2} (1+\mathcal{O}(|z_c/\delta|^{-3/2})) \ \ \ \ \ (23)$

In particular, the imaginary part of ${\phi_f / \phi'_f}$ becomes negative when ${z_c/\delta}$ is large (the ratio has a positive imaginary part when ${z_c/\delta}$ is small).

8.6. Linear dispersion relation

Any linear combination of the slow and fast modes is an exact solution to the Orr-Sommerfeld equation. The zero boundary conditions yield the dispersion relation:

$\displaystyle \frac{\phi_{s}(0; \alpha,\epsilon,c)}{ \phi'_{s}(0; \alpha,\epsilon,c)} = \frac{\phi_{f}(0; \alpha,\epsilon,c)}{\phi'_{f}(0; \alpha,\epsilon,c) }. \ \ \ \ \ (24)$

We shall show that for some ranges of ${(\alpha,\epsilon)}$, the dispersion relation yields the existence of unstable eigenvalues ${c}$. By Lemmas 8 and 10, the linear dispersion relation (24) simply becomes

\displaystyle \begin{aligned} \Big[ U_0 - c + \frac{\alpha (U_+-U_0)^2 }{U'_0} + \mathcal{O}(\alpha^2\log \alpha ) \Big] = - e^{\pi i/4} |\delta| |z_c/\delta|^{-1/2} (1+\mathcal{O}(|z_c/\delta|^{-3/2})). \end{aligned}\ \ \ \ \ (25)

We are interested in the region where ${z_c/\delta}$ is large. The dispersion relation yields

$\displaystyle | U_0 - c| \le C \alpha + C \delta (1+|z_c/\delta|)^{-1/2}.\ \ \ \ \ (26)$

Hence as ${\alpha, \epsilon, \delta \rightarrow 0}$, the eigenvalue ${c}$ converges to ${U_0}$ and so ${z_c \approx \alpha}$, from the equation ${U(z_c) = c}$. The existence of a unique ${c = c(\alpha,\epsilon)}$ near ${c_0 = U_0}$ so that the linear dispersion (25) holds, when ${\alpha, \epsilon}$ are sufficiently small, follows easily from the Implicit Function Theorem.

8.6.1 Lower stability branch: ${\alpha_\mathrm{low} \approx R^{-1/4}}$

Let us consider the case ${\alpha = A R^{-1/4}}$, for some constant ${A}$. We recall that ${\delta \approx (\alpha R)^{-1/3} = A^{-1/3} R^{-1/4}}$. That is, ${\alpha \approx \delta}$ for fixed constant ${A}$. In addition, since ${z_c\approx \alpha}$, we have

$\displaystyle z_c/\delta \quad \approx\quad A^{4/3}.\ \ \ \ \ (27)$

Thus, we are in the case that the critical layer goes up to the boundary with ${z_c/\delta}$ staying bounded in the limit ${\alpha,\epsilon \rightarrow 0}$. We obtain the following lemma.

Lemma 11 Let ${\alpha = A R^{-1/4}}$. For ${R}$ sufficiently large, there exists a critical constant ${A_{c}}$ so that the eigenvalue ${c = c(\alpha,\epsilon)}$ has its imaginary part changing from negative (stability) to positive (instability) as ${A}$ increases past ${A = A_c}$. In particular,

$\displaystyle \Im c \quad\approx \quad A^{-1} R^{-1/4}.$

Proof: The imaginary part of the dispersion relation (25) yields

$\displaystyle (-1 + \mathcal{O}(\alpha)) \Im c + \mathcal{O}(\alpha^2 \log \alpha) = \mathcal{O}(\delta (1+|z_c/\delta|)^{-1/2}).\ \ \ \ \ (28)$

which gives ${\Im c = \mathcal{O}(\delta (1+|z_c/\delta|)^{-1/2})}$ and so ${\Im c \approx A^{-1} R^{-1/4}}$. Next, also from Lemma 10, the right-hand side is positive when ${z_c/\delta}$ is small, and becomes negative when ${z_c/\delta \rightarrow \infty}$. Consequently, together with (27), there must be a critical number ${A_c}$ so that for all ${A > A_c}$, the right-hand side is positive, yielding the lemma as claimed. $\Box$

8.6.2. Intermediate zone: ${R^{-1/4} \ll \alpha \ll R^{-1/6}}$

Let us now turn to the intermediate case when

$\displaystyle \alpha = A R^{-\beta}$

with ${1/10 < \beta < 1/4}$. In this case ${\delta \approx \alpha^{-1/3} R^{-1/3} \approx A^{-1/3} R^{\beta/3 - 1/3}}$ and hence ${\delta \ll \alpha}$. That is, the critical layer is away from the boundary: ${\delta \ll z_c}$. We have the following lemma.

Lemma 12 Let ${\alpha = A R^{-\beta}}$ with ${1/6<\beta<1/4}$. For arbitrary fixed positive ${A}$, the eigenvalue ${c = c(\alpha,\epsilon)}$ always has positive imaginary part (instability) with

$\displaystyle \Im c \quad \approx\quad A^{-1 }R^{\beta-1/2}.$

Proof: As mentioned above, ${z_c/\delta}$ is unbounded in this case. Since ${z_c \approx \alpha}$, we indeed have

$\displaystyle z_c/\delta \quad \approx\quad A^{4/3} R^{(1-4\beta)/3} \rightarrow \infty,$

as ${R \rightarrow \infty}$ since ${\beta <1/4}$. By Lemma 10, the right hand side of the dispersion relation reads

$\displaystyle \Im\Big( {\phi_{f}(0) \over \phi'_{f}(0)} \Big) = \mathcal{O}(\delta (1+|z_c/\delta|)^{-1/2}) \approx A^{-1} R^{\beta -1/2}, \ \ \ \ \ (29)$

which is positive, since ${z_c/\delta \rightarrow \infty}$. It is crucial to note that in this case ${ \alpha^2 \log \alpha \approx R^{-2\beta} \log R,}$ which can be neglected in the dispersion relation (28) as compared to the size of the imaginary part of ${\phi_f/\phi'_f}$. This yields the lemma. $\Box$

8.6.3 Upper stability branch: ${\alpha_\mathrm{up} \approx R^{-1/6}}$

The upper branch of marginal stability is more delicate to handle. This is the case when the term of order ${\alpha^2 \log\alpha}$ on the left hand side of the dispersion relation has the same order as the right-hand side, and hence it is not clear whether ${\Im c}$ remains positive. We expect that the Rayleigh solutions will dominate when ${\alpha \gg \alpha_{\mathrm{up}}}$, and the imaginary part of ${c}$ will change from positive (instability) to negative (stability).