AOS 103 Week 9 Discussion Internal Waves 1) What type of conditions are necessary for internal wave propagation? ! ! 2) What is the medium on which internal waves propagate?! ! 3) What types of processes can generate internal waves?! ! 4) What is the restoring force that is involved in internal wave propagation (sketch and write an equation for the sketch)! ! 5) For 2 density layers of equal depth:! 1) What are the internal wave phase and group speeds?! 2) Are these internal waves dispersive?! 3) What can we say about the relation of phase and group speed with depth?! 4) How do the long and short waves behave for this regime?! ! ! 1) What type of conditions are necessary for internal wave propagation? Internal waves (generally) need stratification to propagate. That is, there needs to be a gravitationally stable “layering” of density with depth (light over heavy). ! 2) What is the medium on which internal waves propagate? Internal waves propagate along a density (i.e., temperature and salinity) interface. They can be visualized by looking for undulations of isopycnals, isotherms or isohalines Internal Waves (Time,Depth) View (Distance,Depth) View CTD measurements of the water column capture large variations in the depths of isopycnals/isotherms over time. !

Often associated with high acoustic Depth backscatter intensity, which indicates the presence of biological Temperature material. Time ! Wave amplitudes reach hundreds of meters. ! Typical wave speeds Acoustic Backscatter Intensity are around 1 m/s. ! 3) What types of processes can generate internal waves?! Generally, internal waves are generated by flows over variable topography.! ! On the continental shelf this can commonly be due to tidal flow, but can also be due to currents flowing in a stratified regime that somehow cause a displacement of a density surface (and trigger the buoyant restoring force)

Internal Wave Movie

Estuarine inlets (like the Columbia River inlet shown on the left) are usually very active internal wave sites because of the strong stratification and tidal flow ebbing and flooding back and forth through the inlet ! Ocean4) What Stratification is the restoring force that is involved in internal wave propagation (sketch and write an equation for the sketch) Suppose we displace a parcel of water from the bottom layer into the top layer. If ρThe > restoring ρ then force in which is the buoyantdirection restoring force: it is analogous2 1 to gravity (for surface waves) and is essentially does the parcel experience thea netdownward force? force [Here that you fluid should parcels feel in a stratified think of ρ as a potential densitymedium. –The we buoyant are ignoring restoring force the is a function of gravity increase in density due to compression.]and the density difference between 2 layers ! In the upper layer p(z) = - ρ g z A. Up 1 B. Down F = Fpressure + Fgravity =(⇢1 ⇢2) gV C. Left Fpressure = + ρ1gV

D. Right Fgravity = - ρ2gV Reduced g (⇢2 ⇢1) g0 = Gravity ⇢2 The net force is F = F + F = (ρ -ρ ) pressure gravity 1 2 1/2 gV < 0, so the parcel is ! =(gk tanh(kH)) pushed down. ! ! 1/2 ! Cp = =(g/k tanh(kH)) ! k < 2H

g 1/2 C p ⇠ k ⇣ ⌘ 2⇡ k = > 20H

C gH p ⇠ 1/2 ! = Cpk =(pgH) k

@! C = =(gH)1/2 g @k

2 2 @! 0 (k l ) Cg = = @k (k2 + l2)2 g 1/2 ! = C k = k =(gk)1/2 p k ⇣ ⌘

@! g 1/2 C = = g @k 4k ⇣ ⌘

⇢1 < ⇢2

! 5) For 2 density layers of equal depth:! 1) What are the internal wave phase and group speeds?! 2) Are these internal waves dispersive?! 3) What can we say about the relation of phase and group speed with depth?! 4) How do the long and short waves behave for this regime?!

g H H C = 0 1 2 p H + H The waves are not dispersive because the group r 1 2 speed quals the phase speed.! H ! H = H = 1 2 2 Phase and group speed (for a constant reduced gravity), will increase with an increase in depth 1 C = g H (H).! p 2 0 ! p The long and short waves travel at the same k speeds because the phase and group speeds are ! = C k = g H p 2 0 independent of wavenumber(i.e., wavelength) @! 1p C = = g H g @k 2 0 p F = F + F =(⇢ ⇢ ) gV pressure gravity 1 2

g (⇢2 ⇢1) g0 = ⇢2

! =(gk tanh(kH))1/2 ! C = =(g/k tanh(kH))1/2 p k

< 2H

g 1/2 C p ⇠ k ⇣ ⌘ 2⇡ k = > 20H

C gH p ⇠ 1/2 ! = Cpk =(pgH) k

@! C = =(gH)1/2 g @k

2 2 @! 0 (k l ) Cg = = @k (k2 + l2)2 The'Con9nental'Slope'

• The'ocean'depth'decreases'rapidly,'some9mes'by'thousands'of' meters,'across'rela9vely'narrow'con$nental)slopes.'

• Remember' that' the' perspec9ve' is' oJen' exaggerated:' typical' gradients'range'from'1)in)100'to'1)in)10.' CoastalSlope Currents Ocean 1) Where would• a potentialThe' slopes' vorticity are'(PV) conserving punctuated' slope by' submarine) canyons,' typically' Where would a PV conserving slope current flow in Santa current flow in Santa Monica Bay?! Monica Bay? formed'by'submarine)landslides.' ! ! ! ! ! ! ! ! The'Con9nental'Slope'

• The'ocean'depth'decreases'rapidly,'some9mes'by'thousands'of' meters,'across'rela9vely'narrow'con$nental)slopes.'

• Remember' that' the' perspec9ve' is' oJen' exaggerated:' typical' gradients'range'from'1)in)100'to'1)in)10.' Slope Currents@v @u 1)• WhereThe' would slopes'ˆ a potential are' punctuated' vorticity (PV) conserving by' submarine) slope canyons,' typically' ⇣Where=( would currenta PVv conserving) flowk in Santa= slope Monica current Bay?flow in! Santa Monica Bay? formed'by'submarine)landslides.' ! r⇥ · @x @y A slope current would flow on the slope. The reason it flows on ! the slope is the water parcels in ! @v @u the current want to conserve ⇣relative! = = ⇣ their potential vorticity (PV) and ! @x @y do not want to change their ! depth (H), so the current stays ! on a line of equal depth.! ! ⇣ ! =2⌦ sin()=f planetary On the slope (indicated on the right by the locations where isobaths get very close to each ⇣absolute = ⇣planetary + ⇣relative = f other),+ ⇣ the depth (H) can change with very little lateral displacement (i.e., if you move up the slope a bit you get very shallow very quick). So the (H) f + ⇣ term in PV can change by a lot q = even for a small movement on- H or offshore ! Coastal Ocean 2) What do a vertical profiles of temperature and density look like on the continental shelf (where are the mixed layers, is there a thermocline)?! ! The'Con9nental'Shelf'Continental Shelf ! ! ! Blue lines represent ! isopycnals (lines of ! constant density)! ! ! !

Jim'McWilliams'

• The'Whatcon$nental)shelf does a vertical'depth'ranges'from'tens'to'hundreds'of' profile of density (or meters.'temperature) look like in the continental ' shelf? • Temperature/density' exhibit' top' and' bogom)mixed)layers,' with'a'thermocline/pycnocline)in'between.' Where are there mixed layers here?

Is there a thermocline? ! 2) What do a vertical profiles of temperature and density look like on the continental shelf (where are the mixed layers, is there a thermocline)?! ! ! ! ! Surface mixed layer ! Thermocline/Pycnocline ! ! Botttom mixed layer ! !

The shelf is unique in the sense that it has BOTH surface AND bottom mixed layer.! ! The presence of a bottom mixed layer is due to the relatively shallow depths (~10-100m). It is possible for substantial currents to exist at these depths and be in contact with the bottom, which creates bottom stresses which can drive vertical mixing ! ! Coastal Ocean 3) How do waves (shallow-water surface waves) approaching the beach at an angle behave (and why)?! ! 4) Do you expect stronger wave activity in headlands or bays?! ! 5) What is the fundamental cause of wave breaking?! ! 6) Using the relation for phase speed of shallow-water surface gravity waves explain why these waves break when they get to shore (assuming their period remains constant)! ! 7) Bonus: if you were standing at the shoreline and looking along the beach (not at the water), how could you tell where there is a rip current based on the sand?! ! ! ! ! ! @! c = g @k

cp = cg c = c p 6 g ! 3) How do waves(shallow-water surface waves) approaching the beach at an angle behave! = (and! (why)?k) ! As waves approach a beach at an angle they refract: that is, they bend parallel to the shore.! ! This is because the phase speed of 1the/2 waves closest to shore is slower than the phaseg speed of the wave further from shore (becausecp of= Clicker'Ques9on'small depth (H) nearshore relative to offshore) k This'figure'shows'wave'crests'approaching'the'beach.'What'do'⇣ ⌘ 1/2 you'expect'to'happen'to'the'waves'as'they'approach'the'beach?'cp =(gH) ' ' ' ' Shallow' Cp = gH ' =>'Cp'small' ' ' Deep' ' =>'Cp'larger' ' A. They'will'turn'to'lie'parallel'to'the'shore.' B. They'will'turn'to'lie'perpendicular'to'the'shore.' C. They'will'keep'propaga9ng'at'the'same'angle.' ! 4) Do you expect stronger wave activity in headlands or bays? We expect stronger wave activity at headlands.! ! Wave'Refrac9on' This expectation is rooted in refraction.! When'waves'approach'the'shore'and'start'to'“feel”'the'bogom,'! or' when'Headlands they' stick experience' out relative changes' to the rest in' current' of the coast speed,' and they' are' refracted,'or'“bent”'such'that'crests'arrive'parallel'to'the'shore.'thus are subject to a convergence of wave energy ! 5) What is the fundamental cause of wave breaking?

The fundamental cause of wave breaking is an increase in steepness of the waves. When waves become too steep, they break. CCp == gHgH ! p 6) Using the relation for phase speed of shallow-waterp surface gravity waves explain why these !waves!== CbreakCppkk when=p= theygHkgHk get to shore (assuming their period remainsC constant)p = gH p C = pgH ! =! C=constantppk =p gHk The phase speed of shallow-water surface ! =constant waves is only variable with depth (H) ! = Cpk =pp gHk ! =constantp !HH=constant00 !! As waves approach shore, the depth decreases: H 0 H! 0 k! If frequency is constant, the wavenumber must k " increase with a decrease in depth: " kk "" Steepness will increase if wavenumber ak increases: ak " akak" Once the wave becomes too steep it will break (and all" the" wave equations/assumptions we have been working withg H are H thrown out the window) C = 0 1 2 p gg00HH11HH2 2 Cp = gH0H1 1+HH2 2 CCpp == r H1 + H2 rHH1++HH2 rr 1 HH2 H1H1==HH22 = H2H2 HH1 == HH2 == 1 12 2 C =1 g H2 C p= 12 g0 H Cp = 12 g H0 C p = 2 pg0 H p pk 0 ! = Cpk2p= g0H kk2 ! = Cpk =p p g0H ! = Cpk@!= k21 g0H Cg = =2 g0H ! = Cpk@k= 2 pg0H @!@! 121pp CCgg = == g0gH0H F = Fpressure@!@+@kkFgravity212p=(⇢1 ⇢2) gV C = = g H g @k 2pp 0 FF==FFpressure ++FFgravityg (⇢2 =(⇢=(1)⇢1⇢ ⇢2⇢) gV) gV pressure g0 = gravityp 1 2 ⇢2 F = Fpressure + Fgravity =(⇢1 ⇢2) gV g (⇢2 ⇢1) 1/2 !g0=(=gkgtanh((⇢2kH⇢))1) g0!= ⇢2 C = =(g/k ⇢tanh(2 kH))1/2 p k g (⇢2 ⇢1) ! g=(0 =gk tanh(kH))1/2 ⇢2 1/2 ! =(! gktanh(< 2HkH)) C = =(g/k tanh(kH))1/2 p !k 1/2 1/2 Cp !==(=(gk tanh(g/k gtanh(kH1/2 ))kH)) k Cp ! <⇠ 2Hk 1/2 Cp = =(g/k⇣tanh(⌘ kH)) k < 2H g 1/2 Cp ⇠< k2gH 1/2 C ⇣ ⌘ p ⇠ k ⇣g ⌘1/2 C p ⇠ k ⇣ ⌘ ! 7) Bonus: if you were standing at the shoreline and looking along the beach (not at the water), how could you tell where there is a rip current based on the sand?! Rip currents occur where there is a convergence of mass transport by waves on the shoreline and they flow seaward in at these places via deep channels. The rip currents can “carve out” these deep channels into the sand in the swash zone that would be visible from the shoreline. This channel can either be visible through the high-tide line or an actual “valley” in the sand on the shoreline that is the channel itself