u x u t Internal Waves

Transcription

1 Internal Waves We now examine internal waves for the ase in whih there are two distint layers and in whih the lower layer is at rest. This is an approximation of the ase in whih the upper layer is muh thinner than the deeper layer but this makes the dynamis muh simpler. The linearied equations of motion (for small perturbations) are as before but now for the upper layer only: v + fv = g fu = g + [ v + ] = 0 where is the total thikness of the upper layer. The upper surfae h is free to move vertially as is the bottom of the upper layer. Reall before that in this ase there an be no pressure gradients in the lower layer (layer ) if it is to remain at rest. If perturbations to the internal layer interfae are small and its depth is otherwise at a onstant 0 then the vanishing of pressure gradients at any depth in the lower layer = - 0 requires the following: P ( x y x y P 0 ) = ρ g( + h) + ρ g( ( = 0 h h ( ρ ρ ) / ρ = h ρ / ρ 0 + h h )) and Upward displaements of the free surfae must be aompanied by downward displaements of the interfae to prevent pressure gradients in the lower layer. We have defined the displaements of the interfae so that a downward displaement of the interfae is equivalent to a positive value of h. The reason for this will be apparent shortly. Sine the density differene is going to be a small ompared to the density of the surfae layer ( ρ/ρ<<) hanges in the free surfae are going to be small ompared to those of the interfae. Thus in the third equation above we an ignore variations in the surfae height ompared to hanges in the interfae height. When we now substitute this approximation for h and into the above equations we obtain:

2 v fv + fu + 0 ρ g' g. ρ = g' = g' [ v + ] = 0 where Note that these are exatly the equations for a homogeneous fluid of uniform density with two exeptions: the water depth now beomes the mean depth of the upper layer ( 0 ) and the gravity term beomes a redued gravity (g g ). Thus all of the previous types of surfae waves an be immediately taken over for internal waves with these two differenes in mind. The small differenes make some substantial hanges in the properties of these waves. Consider two quantities whih were important for all of the surfae waves: the shallow water phase speed and the Rossby Radius of deformation a. These will take on different numerial values. We need to first estimate some basi quantities suh as the density ratio and the mean interfae depth. For this we will use two stations from either side of the Gulf Stream (they were used in one of the homework problems).

3 In the above figure two hypothetial ases for dividing up the water olumn into two layers have been maded. In the first the station (#65) is south of the Gulf Stream in a thik layer of subtropial mode water (Eighteen Degree Water). We have used a surfae density appropriate for what might be found in winter and a pynoline depth of 750m. The stations were oupied in September when the seasonal pynoline is fully developed. In the seond we used a shallower pynoline (00m) but a larger density ratio (x0-3 ). The phase speed and radius of deformation are given by: a = g' = 0 / f = g' 0 / f We see that the phase speed has been redued from 00 m/s to -3 m/s and the radius of deformation is redued from 000km to 0-30km (for a value of f of 0-4 s -.) While this hange in two of the key parameters governing wave motion is two orders of magnitude there is no qualitative differene in the types of waves or their dispersion relations. We will briefly review this now. Internal gravity/inertial waves: σ = f + κ where κ ( k l )& κ = ( k + l ) and = g' Kelvin waves: ( h v) = ( h( x) v( x)) e σ = l = g' where i( ly σt) h h = 0 giving Rossby waves:

4 u v h ( uˆ vˆ hˆ) e βa k σ = + κ a a = g' / f i( kx+ ly σt ) where κ = k = ( uˆ vˆ hˆ)[os( kx + ly σt) + i sin( kx + ly σt)] + l where A smaller radius of deformation a means among other things that barolini sales are muh smaller than barotropi sales: the adjustment problem in whih a olumn of upper layer fluid is released and allowed to adjust to the rotation of the earth will yield a muh narrower geostrophi urrent and a smaller sale for the slumping of the interfae in the upper layer. Whether Poinaré (or internal/inertial gravity) waves or Rossby waves are long or short ompared to the radius of deformation will be greatly affeted: most of these that were short for barotropi motion will be long for barolin waves. The frequeny gap between the lowest frequeny internal wave and the highest frequeny internal Rossby wave will be muh larger beause the lower limit for the former is the same as before (still the inertial frequeny) but the upper limit for the latter is muh smaller: internal Rossby waves have muh longer periods than their barotropi ounterparts. For example we estimated earlier that the minimum period (maximum frequeny) of a bartopi Rossby wave would be 3.6 days. For the barolini ase just onsidered this would beome 360 days. In other words we would expet no propagating barolini Rossby waves to have a period shorter than about a year! Barolini motions an have large amplitude displaements of the pynoline: of 00m. These an be seen at the oean surfae although redued in amplitude by a fator of ρ/ρ. The satellite altimeter has provided the best view of these barolini waves beause though the amplitudes be small (0 m) the periods are long and thus easily samples by the 0 day repeat orbit of the Topex-Poseiden altimeter. In a paper by Chelton and Shlax (Siene 996). They showed a spae-time series of altimeter variability at seleted latitudes in the North Paifi Oean whih we will now disuss. In the diagram at the right altimeter data have been filtered to look at signals having periods between 0.5 and yrs. For the three years of data displayed surfae highs (depressions of the pynoline) and lows an be seen to be

5 propagating to the west most learly in the western part of the Paifi. These have phase speed of order 0 m/s and are onsistent with speeds of the lowest mode barolini Rossby wave (similar to our approximation above). While the phase speeds seem slightly faster than expeted and this has been a fous of muh debate in the literature the evidene for the existene of these waves and their importane to long- term variability in the oean is Image removed due to undisputed. Also evident is the opyright onerns. evidene that barolini waves propagate faster towards the equator. This an be understood in terms of the variation of the radius of deformation with latitude: in the tropis where f is smaller the radius of deformation is larger than in mid-latitudes. Sine long Rossby waves propagate have a phase speed that is proportional to the square of the radius of deformation ( βa p = ) as one approahes the equator waves will take a shorter time to ross a basin than at higher latitudes. At N above it will take about 4 years for a wave to ross the domain plotted whereas it will take approximately 0 years at not that long and we have to extrapolate to 39N. Of ourse the time series is see this. Internal waves are responsible for muh of the high frequeny variability in open oean observations. This an be dedued by the relationship between observed horiontal and vertial veloities and how they depend upon frequeny. In shallow water suh as Massahusetts Bay internal waves in the seasonal thermoline (pynoline) are generated by tidal flow over Stellwagen Bank and an produe a visible signature at the oean surfae (due to surfae onvergenes and divergenes and their effet on surfae waves) and substantial mixing at the oean bottom. In fat they an be a major soure of nutrient supply to the upper oean as these steep waves mix and oasionally break. A good desription of these an be seen in a paper by aury Brisoe and Orr (Nature 979) whih will be disussed in lass.