Gravity wave

In fluid dynamics, gravity waves are waves generated in a fluid medium or at the interface between two media (e.g., the atmosphere and the ocean) which has the restoring force of gravity or buoyancy.

When a fluid element is displaced on an interface or internally to a region with a different density, gravity tries to restore the parcel toward equilibrium resulting in an oscillation about the equilibrium state or wave orbit. Gravity waves on an air–sea interface are called surface gravity waves or surface waves while internal gravity waves are called internal waves. Wind-generated waves on the water surface are examples of gravity waves, and tsunamis and ocean tides are others.

Wind-generated gravity waves on the free surface of the Earth's ponds, lakes, seas and oceans have a period of between 0.3 and 30 seconds (3 Hz to 0.03 Hz). Shorter waves are also affected by surface tension and are called gravity–capillary waves and (if hardly influenced by gravity) capillary waves. Alternatively, so-called infragravity waves, which are due to subharmonic nonlinear wave interaction with the wind waves, have periods longer than the accompanying wind-generated waves.

Atmosphere dynamics on Earth

This process plays a key role in controlling the dynamics of the middle atmosphere.

The clouds in gravity waves can look like Altostratus undulatus clouds, and are sometimes confused with them, but the formation mechanism is different.

Quantitative description

The phase speed $\backslash scriptstyle\; c$ of a linear gravity wave with wavenumber $\backslash scriptstyle\; k$ is given by the formula

$c=\backslash sqrt\{\backslash frac\{g\}\{k\}\},$

where g is the acceleration due to gravity. When surface tension is important, this is modified to

$c=\backslash sqrt\{\backslash frac\{g\}\{k\}+\backslash frac\{\backslash sigma\; k\}\{\backslash rho\}\},$

where σ is the surface tension coefficient, ρ is the density, and k is the wavenumber (spatial frequency) of the disturbance.

The gravity wave represents a perturbation around a stationary state, in which there is no velocity. Thus, the perturbation introduced to the system is described by a velocity field of infinitesimally small amplitude, $\backslash scriptstyle\; (u\text{'}(x,z,t),w\text{'}(x,z,t))$. Because the fluid is assumed incompressible, this velocity field has the streamfunction representation

:$\backslash textbf\{u\}\text{'}=(u\text{'}(x,z,t),w\text{'}(x,z,t))=(\backslash psi\_z,-\backslash psi\_x),\backslash ,$

where the subscripts indicate partial derivatives. In this derivation it suffices to work in two dimensions $\backslash scriptstyle\; \backslash left(x,z\backslash right)$, where gravity points in the negative z-direction. Next, in an initially stationary incompressible fluid, there is no vorticity, and the fluid stays irrotational, hence $\backslash scriptstyle\backslash nabla\backslash times\backslash textbf\{u\}\text{'}=0\backslash ,$. In the streamfunction representation, $\backslash scriptstyle\backslash nabla^2\backslash psi=0.\backslash ,$ Next, because of the translational invariance of the system in the x-direction, it is possible to make the ansatz

:$\backslash psi\backslash left(x,z,t\backslash right)=e^\{ik\backslash left(x-ct\backslash right)\}\backslash Psi\backslash left(z\backslash right),\backslash ,$

where k is a spatial wavenumber. Thus, the problem reduces to solving the equation

:$\backslash left(D^2-k^2\backslash right)\backslash Psi=0,\backslash ,\backslash ,\backslash ,\backslash \; D=\backslash frac\{d\}\{dz\}.$

We work in a sea of infinite depth, so the boundary condition is at $\backslash scriptstyle\; z=-\backslash infty$. The undisturbed surface is at $\backslash scriptstyle\; z=0$, and the disturbed or wavy surface is at $\backslash scriptstyle\; z=\backslash eta$, where $\backslash scriptstyle\backslash eta$ is small in magnitude. If no fluid is to leak out of the bottom, we must have the condition

:$u=D\backslash Psi=0,\backslash ,\backslash ,\backslash text\{on\}\backslash ,z=-\backslash infty$.

Hence, $\backslash scriptstyle\backslash Psi=Ae^\{k\; z\}$ on $\backslash scriptstyle\; z\backslash in\backslash left(-\backslash infty,\backslash eta\backslash right)$, where A and the wave speed c are constants to be determined from conditions at the interface.

The free-surface condition: At the free surface $\backslash scriptstyle\; z=\backslash eta\backslash left(x,t\backslash right)\backslash ,$, the kinematic condition holds:

:$\backslash frac\{\backslash partial\backslash eta\}\{\backslash partial\; t\}+u\text{'}\backslash frac\{\backslash partial\backslash eta\}\{\backslash partial\; x\}=w\text{'}\backslash left(\backslash eta\backslash right).\backslash ,$

Linearizing, this is simply

:$\backslash frac\{\backslash partial\backslash eta\}\{\backslash partial\; t\}=w\text{'}\backslash left(0\backslash right),\backslash ,$

where the velocity $\backslash scriptstyle\; w\text{'}\backslash left(\backslash eta\backslash right)\backslash ,$ is linearized on to the surface $\backslash scriptstyle\; z=0\backslash ,$. Using the normal-mode and streamfunction representations, this condition is $\backslash scriptstyle\; c\; \backslash eta=\backslash Psi\backslash ,$, the second interfacial condition.

Pressure relation across the interface: For the case with surface tension, the pressure difference over the interface at $\backslash scriptstyle\; z=\backslash eta$ is given by the Young–Laplace equation:

:$p\backslash left(z=\backslash eta\backslash right)=-\backslash sigma\backslash kappa,\backslash ,$

where σ is the surface tension and κ is the curvature of the interface, which in a linear approximation is

:$\backslash kappa=\backslash nabla^2\backslash eta=\backslash eta\_\{xx\}.\backslash ,$

Thus,

:$p\backslash left(z=\backslash eta\backslash right)=-\backslash sigma\backslash eta\_\{xx\}.\backslash ,$

However, this condition refers to the total pressure (base+perturbed), thus

:$\backslash left[P\backslash left(\backslash eta\backslash right)+p\text{'}\backslash left(0\backslash right)\backslash right]=-\backslash sigma\backslash eta\_\{xx\}.$

(As usual, The perturbed quantities can be linearized onto the surface z=0.) Using hydrostatic balance, in the form $\backslash scriptstyle\; P=-\backslash rho\; g\; z+\backslash text\{Const.\},$

this becomes

:$p=g\backslash eta\backslash rho-\backslash sigma\backslash eta\_\{xx\},\backslash qquad\backslash text\{on\; \}z=0.\backslash ,$

The perturbed pressures are evaluated in terms of streamfunctions, using the horizontal momentum equation of the linearised Euler equations for the perturbations,

:$\backslash frac\{\backslash partial\; u\text{'}\}\{\backslash partial\; t\}\; =\; -\; \backslash frac\{1\}\{\backslash rho\}\backslash frac\{\backslash partial\; p\text{'}\}\{\backslash partial\; x\}\backslash ,$

to yield $\backslash scriptstyle\; p\text{'}=\backslash rho\; c\; D\backslash Psi$.

Putting this last equation and the jump condition together,

:$c\backslash rho\; D\backslash Psi=g\backslash eta\backslash rho-\backslash sigma\backslash eta\_\{xx\}.\backslash ,$

Substituting the second interfacial condition $\backslash scriptstyle\; c\backslash eta=\backslash Psi\backslash ,$ and using the normal-mode representation, this relation becomes $\backslash scriptstyle\; c^2\backslash rho\; D\backslash Psi=g\backslash Psi\backslash rho+\backslash sigma\; k^2\backslash Psi$.

Using the solution $\backslash scriptstyle\; \backslash Psi=e^\{k\; z\}$, this gives

$c=\backslash sqrt\{\backslash frac\{g\}\{k\}+\backslash frac\{\backslash sigma\; k\}\{\backslash rho\}\}.$

Since $\backslash scriptstyle\; c=\backslash omega/k$ is the phase speed in terms of the angular frequency $\backslash scriptstyle\backslash omega$ and the wavenumber, the gravity wave angular frequency can be expressed as

$\backslash omega=\backslash sqrt\{gk\}.$

The group velocity of a wave (that is, the speed at which a wave packet travels) is given by

$c\_g=\backslash frac\{d\backslash omega\}\{dk\},$

and thus for a gravity wave,

$c\_g=\backslash frac\{1\}\{2\}\backslash sqrt\{\backslash frac\{g\}\{k\}\}=\backslash frac\{1\}\{2\}c.$

The group velocity is one half the phase velocity. A wave in which the group and phase velocities differ is called dispersive.

The generation of waves by wind

In the work of Phillips, the ocean surface is imagined to be initially flat (glassy), and a turbulent wind blows over the surface. When a flow is turbulent, one observes a randomly fluctuating velocity field superimposed on a mean flow (contrast with a laminar flow, in which the fluid motion is ordered and smooth). The fluctuating velocity field gives rise to fluctuating stresses (both tangential and normal) that act on the air-water interface. The normal stress, or fluctuating pressure acts as a forcing term (much like pushing a swing introduces a forcing term). If the frequency and wavenumber $\backslash scriptstyle\backslash left(\backslash omega,k\backslash right)$ of this forcing term match a mode of vibration of the capillary-gravity wave (as derived above), then there is a resonance, and the wave grows in amplitude. As with other resonance effects, the amplitude of this wave grows linearly with time.

The air-water interface is now endowed with a surface roughness due to the capillary-gravity waves, and a second phase of wave growth takes place. A wave established on the surface either spontaneously as described above, or in laboratory conditions, interacts with the turbulent mean flow in a manner described by Miles. This is the so-called critical-layer mechanism. A critical layer forms at a height where the wave speed c equals the mean turbulent flow U. As the flow is turbulent, its mean profile is logarithmic, and its second derivative is thus negative. This is precisely the condition for the mean flow to impart its energy to the interface through the critical layer. This supply of energy to the interface is destabilizing and causes the amplitude of the wave on the interface to grow in time. As in other examples of linear instability, the growth rate of the disturbance in this phase is exponential in time.

This Miles–Phillips Mechanism process can continue until an equilibrium is reached, or until the wind stops transferring energy to the waves (i.e., blowing them along) or when they run out of ocean distance, also known as fetch length.

See also

Notes