Schlieren measurement of axisymmetric internal wave amplitudes

Abstract

"Synthetic schlieren", which has been used to measure the amplitude of two-dimensional internal wave beams generated from an oscillating cylinder, is adapted to analyze axisymmetric internal waves generated by an oscillating sphere. This nonintrusive technique uses elementary inverse tomographic methods to measure the amplitude of the conical-structured wave beams everywhere in space and time. We compare the results with in situ probe measurements, and we examine the structure of the wave beams generated by a sphere oscillating at different amplitudes.

Scaling analysis

Here we give a detailed explanation of how the approximate solution to Eq. (6), given by Eq. (7), has been obtained through scaling and perturbation analysis. Equation (6) is simplified by dropping terms that are revealed to be small by a scaling analysis. In the x-direction, we expect variations on scales L _x , which is on the order of the sphere radius, on the order 1–10 cm in our experiments. In the z-direction, the length scale L _z is set to the vertical distance traveled by a ray as it progresses from the front to the back of the tank. In typical experiments, L _z ≃1 cm, N ₀ ²~1 s⁻² and γ~10⁻⁴ s²/cm. We transform to nondimensional variables (denoted by tildes) with the substitutions $\left( {x,z} \right) = \left( {L_x \tilde x,L_z \tilde z} \right),\;\alpha _0 = n_0 \tilde n$ and $\alpha _1 = - \gamma n_0 N_0 ^2 \widetilde{N^2 }$. Thus Eq. (6) becomes

$$ \matrix{ {{{L_z } \over {L_x^2 }}\tilde z''} \hfill & { + \gamma ^2 N_0^4 L_z {{\widetilde{N^4 }} \over {\tilde n^2 }}\tilde z - {{L_z^3 } \over {L_x^2 }}\gamma ^2 N_0^4 {{\widetilde{N^2 }\widetilde{N^2 }^\prime } \over {\left( {\tilde n'} \right)^2 }}\tilde z''\tilde z'\tilde z} \hfill \cr {} \hfill & { - {{L_z^2 } \over {L_x^2 }}\gamma N_0^2 {{\widetilde{N^2 }} \over {\tilde n}}\tilde z''\tilde z' = - \gamma N_0^2 {{\widetilde{N^2 }} \over {\tilde n}}.} \hfill \cr } $$

(17)

Substituting typical characteristic scales and supposing first that L _x ~1 cm, we obtain

$$ \matrix{ {\tilde z''} \hfill & { + 10^{ - 8} {{\widetilde{N^4 }} \over {\tilde n^2 }}\tilde z - 10^{ - 8} {{\widetilde{N^2 }\widetilde{N^2 }^\prime } \over {\left( {\tilde n'} \right)^2 }}\tilde z''\tilde z'\tilde z} \hfill \cr {} \hfill & { - 10^{ - 4} {{\widetilde{N^2 }} \over {\tilde n}}\tilde z''\tilde z' = - 10^{ - 4} {{\widetilde{N^2 }} \over {\tilde n}}.} \hfill \cr } $$

(18)

The leading order equation is

$$ {{{\rm{d}}^2 \tilde z^{(0)} } \over {{\rm{d}}\tilde x^2 }} = 0, $$

(19)

with solution

$$ \tilde z^{\left( 0 \right)} = z_i + \tilde x\cot \phi _i , $$

(20)

where z _i refers to the initial height of the ray and ϕ _i refers to its angle as it enters the stratified medium.

At next order, ε=10⁻⁴, perturbation theory with $z \simeq \tilde z^{\left( 0 \right)} + \varepsilon \tilde z^{\left( 1 \right)}$ gives the equation

$$ {{{\rm{d}}^{\rm{2}} \tilde z^{\left( 1 \right)} } \over {{\rm{d\tilde x}}^{\rm{2}} }} + {{\tilde N^2 } \over {\tilde n}}{{{\rm{d}}^{\rm{2}} \tilde z^{\left( 0 \right)} } \over {{\rm{d}}\tilde x^{\rm{2}} }}{{{\rm{d}}\tilde z^{\left( 0 \right)} } \over {{\rm{d}}\tilde x}} = - {{\tilde N^2 } \over {\tilde n}}. $$

(21)

Using Eq. (19), the solution is found:

$$ \tilde z^{\left( 1 \right)} = - \int_0^{\tilde x} {\int_0^{\tilde {\tilde x}} {{{\tilde N^2 \left( {\tilde {\tilde {\tilde x}}} \right)} \over {\tilde n\left( {\tilde {\tilde {\tilde x}}} \right)}}{\rm{d}}\tilde{ \tilde {\tilde x}}{\rm{d}}\tilde {\tilde x}} } . $$

(22)

Repeating this procedure but supposing L _x ≃10 cm gives the equation

$$ \matrix{ {\tilde z''} \hfill & { + 10^{ - 6} {{\widetilde{N^4 }} \over {\tilde n^2 }}\tilde z - 10^{ - 8} {{\widetilde{N^2 }\widetilde{N^2 }^\prime } \over {\left( {\tilde n'} \right)^2 }}\tilde z''\tilde z'\tilde z} \hfill \cr {} \hfill & { - 10^{ - 4} {{\widetilde{N^2 }} \over {\tilde n}}\tilde z''\tilde z' = - 10^{ - 2} {{\widetilde{N^2 }} \over {\tilde n}}.} \hfill \cr } $$

(23)

The leading order equation is again given by Eq. (19). At next order, ε=10⁻², the corresponding equation for $\tilde z^{(1)}$ is identical to Eq. (21) after using Eq. (19).

Therefore, for waves with horizontal scales ranging between 1 and 10 cm, we find

$$ z \simeq z_i + x\cot \phi _i - \gamma n_0 \int_0^x {\int_0^{\hat x} {{{N^2 \left( {\hat {\hat x}} \right)} \over {n\left( {\hat {\hat x}} \right)}}{\rm{d}}\hat {\hat x}{\rm{d}}\hat x} } . $$

(24)

Finally, we note that variations in the refractive index occur on the order Δn/n~10⁻⁵, whereas typical changes in the buoyancy frequency due to waves occur on the order ΔN ²/N ₀ ²~10⁻¹ to 10⁻². Thus, we can treat n(x) as approximately constant in the denominator in Eq. (24) to give Eq. (7).