Equatorial Ocean Dynamics and the Undercurrent

CHAPTER 7 Equatorial ocean dynamics and the undercurrent 7.1. Linearization about a state of rest Particularly useful approximate solutions of the primitive equations can be obtained by linearizing them about a state of rest and assuming that the bac! round "mean) vertical density structure is horizontally but not vertically uniform. In order to simplify the presentation &e shall assume incompressibility and hydrostatic equilibrium which &e discussed in earlier lectures. It is possible to drop the first assumption without too much trouble for dealin with the atmosphere but &e consider the ocean application here. The momentum equations for small perturbations about the state of rest are ∂u fv= 1 ∂p ∂t − − ρ0 ∂x "7$($(# ∂v +fu= 1 ∂p ∂t − ρ0 ∂y gρ= ∂p − ∂z The Coriolis parameterf is 2Ω sinϕ whereΩ is the angular velocity of the Earth andρ 0 =ρ 0(z) is the assumed mean density profile. As before the equation of continuity in the incompressible case is ∂w ∂u ∂v "7$($)# + + = 0 ∂z ∂x ∂y If &e assume that there are no sources of heat and salinity for the present then the dρ incompressible version of mass conservation implies that dt = 0$ *hen linearized this becomes ∂ρ ∂ρ "7$($+# +w 0 = 0 ∂t ∂z 7.2. Separation of Variables and the vertical Sturm-Liouville equations These five equations have the five unkno&ns(u, v, w, p, ρ) and eneral solutions a re possible. Those of interest to us are separable in the vertical direction$ *e choose the following separation for the five variables −→u= u(z) −→U(x,y,t) p= p(z)η(x, y, t) ρ=�ρ(z)d(x,y,t) w=� h(z) w(x, y, t) � where for reasons that will become clearer� � later &e assume that u and h have the dimensions of length whereas p and ρ have their ordinary units "implying, of � 22 � � � 7.2. SEPARATION OF VARIABLES AND THE VERTICAL STURM-LIOUVILLE EQUATIONS23 course, that the horizontal dependent components are dimensionless)$ The vertical equations can no& be deduced up to a separation constant from the linearized set above. *e obtain the four equations for the vertical part of the solutions: ρ0 u= p/g g ρ= p z "7$)$(# hz� =� u/H 0 �ρ=(� ρ 0)z h � � where the factorg, the acceleration due t o ravity, has been included in the first � � equation for dimensional consistency andH 0 is the separation constant which has dimension of length. Combining the first and third equations and the second and fourth reduces this to gH h = p/ρ "7$)$)# 0 z 0 zp =g(ρ 0)z h � � and this is easily condensed further to � � 1 N 2 "7$)$+# (ρ0hz)z + 2 h=0 ρ0 ce 1 1/2 � � whereN gρ0− (ρ0)z is termed the Brunt-Vaisala frequency and measures the stability≡ of− the bac! round stratification. 0ote that is al&ays real if the density profile is stable.� Parcels� of fluid displaced within a density stratification will tend to oscillate at this frequency due to buoyancy effects. The separation constant has 2 been “renamed”c e gH 0 wherec e is referred to as shallo& &ater speed for reasons ≡ 2 that will become clearer later$ The reason for assuming thatc e is positive is because 1 the operatorH= − 2 ∂ ρ ∂ may easily be sho&n to be positive providing that ρ0N z 0 z 2 the stratification is stable and hence has the positive ei envaluesc e− $ 5or the ocean the lo&er boundary condition is that the vertical velocity vanishes and so "7$)$6# h(0) = 0 At the ocean surface the hydrostatic� relation shows that the pressure near the unperturbed surfaceL is p ρ (L)gz+p ≈ 0 0 wherez is the perturbation ve rtical displacement andp 0 is constant "exercise)$ 7ifferentiating this with respect to time ives us the upper boundary condition "7$)$8# ρ0(L)gh(L) = p(L) Equations "7$)$+#, "7$)$6# and "7$)$8# to ether with the assumption thatN>0 � � form a Sturm-Liouville ei ensystem. The mathematical literature on such systems 2 is extensive and the ei envaluesc e can be shown to be discrete, bounded above, and, as already noted, positive. To a hi h degree of accuracy in the ocean 1 1 "7$)$;# − 2 ∂zρ0∂z = − 2 ∂zz ρ0N N and so &e can see that the ei envalues and ei envectors are determined by the vertical profile of the Brunt-Vaisala frequency$ The ei envectors in this system are called the normal, vertical or baroclinic/barotropic modes. The applications of 7.3. AN ILLUSTRATIVE SOLUTION TO THE VERTICAL EQUATIONS 24 these modes are very extensive particularly in ocean dynamics. The mode with the reatest ei envalue and the simplest "one si ned) vertical structure for its ei envector is the so-called barotropic mode which has a shallo& &ater speed of around 1 200 300ms − . The other modes are called the baroclinic modes and have more complex− vertical structures. 5or the observed stratifications they have much smaller shallo& &ater speeds "the first baroclinic mode has a typical shallo& &ater speed 1 of around3ms − #$ Because the system here is Sturm-Liouville, the vertical modes satisfy ortho onality conditions. These can be obtained by simple manipulation of the ei envector equations from equation "7$)$+#- d dhn dhm 1 1 2 ρ0hm ρ 0hn = 2 2 ρ0N hmhn dz � dz − dz � cm − cn � � � � � � � � If this is integrated from the bottom of the ocean to the top &e obtain with the use of the boundary conditions and equations "7$)$)#- L 2 "7$)$7# ρ0N hm(z)hn(z)dz+ρ 0ghm(L)hn(L)=0 if m=n ˆ0 � 5rom this, the equations� "7$)$)#� and the bound� ary� conditions &e can also obtain by integration by parts that L 1 "7$)$<# ρ0− mp (z) np(z)dz=0 if m=n ˆ0 � 7.3. An illustrative� solution� to the vertical equations To illustrate this system &e consider the simplest example in whichN 2 =K 2 = const. To further simplify matters &e will assume that equation "7$)$;# holds. The solutions then of equation "7$)$+# are obviously sinusoids and cosines how- ever boundary condition "7$)$6# means that only the former are appropriate. *e write a eneral solution then as h(z)=h 0 sin λz subject to the constraint following� from the ei envector equation that 2 2 2 "7$+$(# cn =K /λ The second boundary condition "7$)$8# &hen combined with the 'rst equation from "7$)$)# then leads to 2 "7$+$)#c n cos(λL)=g sin(λL) and then combinin "7$+$(# with "7$+$)# leads to K2 "7$+$+# tan(λL)= λg which clearly has an infinite set of solutions which are the intersections of the K2L hyperbola (λL)g with the multiple branch function tan(λL) "see 5i ure 7$+$(#$ 3 0o& in eneral in the oceanK 10 − andL 4000m which means the first solution occurs in a region for which∼ tan(λL) ∼ λL and other solutions oc cur for � 7.3. AN ILLUSTRATIVE SOLUTION TO THE VERTICAL EQUATIONS 25 Figure 7.3.1. Constant stratification solutions tan(λL) 0 "exercise)$ Thus the first "barotropic) ei envector and ei envalue, which &e� label with a zero subscript, have approximately K 0 = √gL c0 = √gL 1 5or the parameters chosen above this shows thatc 0 200ms− . Also note that 1 � 0 !! L− which shows that the ei enmode has an approximately uniform struc- nπ ture in the vertical$ 5or the other solutions λL 0 implies that n L and hence that � � KL 1.3 1 c = ms− n nπ � n 0ote the1/n dependency of the spec trum for hi her order modes. This !ind of rela- 2 tion is approximately true for real profiles ofN . 0ote also that then �th baroclinic ei enmodes are sinusoids withn nodes and have a vert ical velocity at the surface of approximately zero$ These properties hold for the “real4 modes and the latter is sometimes referred to as the ri id lid approximation "it obviously cannot hold for the barotropic mode)$ 5or real profiles the 'rst baroclinic mode is somewhat faster and many of the hi her order modes have most of their structure near the surface "see 5i ure 7$+$)#$ It is also &orth noting that because the baroclinic modes have most of their vertical motion in the interior of the ocean they are sometimes called internal &aves while the barotropic mode is sometimes called the exterior mode. 7.5. WIND FORCED EQUATIONS 26 Figure 7.3.2. Realistic vertical structure functions for the ocean 7.4. he shallo! !ater equations Corresponding to the four equations in the vertical "7$)$(# there are a set of equations overning the horizontal flo& ∂U f#= g ∂η ∂t − − ∂x ∂V fU= g ∂η "7$6$(# ∂t − − ∂y w= H ( ∂U + ∂V ) − 0 ∂x ∂y w= ∂d = ∂η − ∂t ∂t � The third and fourth equations� here can be combined and a new variableh gη introduced. The resulting equations ≡ U f#= h t − − x "7$6$)# #t fU= h y 2 − − ht +c n(Ux +# y) = 0 are commonly referred to as the shallo& &ater equations. 7.5. #ind forced equations The wind forced linearized momentum equations may be written as ∂u fv= 1 ∂p + 1 ∂X ∂t − − ρ0 ∂x ρ0 ∂z ∂v +fu= 1 ∂p + 1 ∂Y ∂t − ρ0 ∂y ρ0 ∂z 7.6. HORIZONTAL SOLUTIONS ON THE EQUATORIALβ-PLANE 27 where($,%) is the wind stress vec tor$ 5or the ocean let us assume that the atmospheric stress is deposited into the ocean in an infinitesimal layer of thic!ness dz near the surface.

Load more