-ry i s the surface slope obtained from a barotropic model. Then using equation 3.15b for the pressure at a point in the 36 lower layer, 1 ' 3.19a. Defining$ - ^ 0-^, the above becomes, after rearranging, 3. 20. Eguation 3.20 i s the fourth eguation reguired to complete the problem. The pressure gradient term in the integrated upper layer eguation 3.18. becomes then (ignoring variations in p when not differentiated) t f 3.21. Now to derive the momentum eguation for the entire water column. A l l terms except the pressure gradient term are quite straightforward. The pressure term i s : ± If d*- + ( 5 - L ^ ? = f ° 3.22a Substituting p z p o - ^ # and doing a bit of rearranging, we get the above expression to egual: ^ ( D + nr\ ) t\ x + ^ C D + * 1 ) £ c5 ( / * ( - £ ) ) x I 3.22b. 37 Note that i f we consider the terms proportional to D + r*\ , and require them to equal pbojCr^, we get p -as in eguation 3.20. The momentum eguation for the entire water column i s then (Oo "I 3. 23. The l e f t hand side represents the momentum terms in the barotropic t i d a l equation. The f i r s t terra on the riqht hand side represents the correction made to g•( 0 * x because i t contains the pressure gradient required to maintain the river flow. The pressure gradient associated with the river flow i s (#/p»X <^/*""^ * * a n d this i s subtracted from r\x , in eguation 3.23, to give <^£-r^, the gradient calculated in a model with no river effects. The second term on the right represents a correction to the barotropic pressure because the upper layer i s somewhat lighter. The other terms, proportional to - j£ have l i t t l e effect on the barotropic motion, whose driving terms, the le f t hand side of 3.23, are proportional to D. Thus the presence of the river flow has negligible effects on a barotropic model, and the surface slopes and velocities obtained from that model 38 may be used as forcing for the upper layer model. To summarize, we will rewrite the eguations for the upper layer in terms of a variable z which is zero at the bottom o f the upper layer and increases upwards, and also w i l l include the second horizontal dimension, and the Coriolis force. Note that the thickness of the upper layer i s h=/^-.£. CQJTIJJOITY * A r r -T" J ° * + f- / vdx = \AJf - W „ 3.24. SALT A I jsd* + i. f usdz + A. fvsdz So % - 5 \AJ„ . 0 3.2 5. DIRECTED MOMENTUM h 4 ^ V, 2J o o pa 3. 26 X-BISECTED MOMENTUM A ~ " ~~ A A- ^ 3.27. 39 There are several things which have yet to be specified in this model. 1. We have to provide the profiles of u, v, and s, so that, for example, we can relate J ^ " " ^ ^ to J u . 2. wp,w,0, u, s, and r^«-f must be specified in terms of flow properties such as the density and velocity differences between the upper and lower layer, and the thickness of the upper layer. ry # u0# and v G must be obtained from a barotropic tidal model (Crean, 1977), which is the solution of eguations of the type 3.8, 3.23, with the right hand sides set to zero. Items 1 and 2 above are related to properties of the flow, being related to the turbulent structure present, and can only be parameterized in terms of the large scale properties of the flow. Item 3 can be considered as external driving of the flow. However, the extent of the forcing produced by u Q and v 0 depends on the type of turbulent interactions specified in items 1 and 2. Having gone through considerable algebra to get to the above eguations, we should check that they agree with the physics we want to model. First, consider the continuity and salt equations, as in the control volume approach described earlier, the guantity of water in the plume changes, ^ V^r, because of horizontal divergences, in particular the river flow, and by fluxes relative to the boundary, Wp and wn. Similarly, the salt content can change by internal rearrangement, or by influxes of salt, w ps 0, and effluxes, w^ s. In the momentum eguation, we see that the change in no momentum of a column of flu i d i s approximately given by: Or J buoyant spreading j t>/^^ I/^OJS/I7') * f r i c t i o n a l interaction^ * gain or loss of water and i t s associated momentum, wfUt>-\aJ„ ia , + forcing by the barotropic t i d a l slopes, aj S r ^ . It i s d i f f i c u l t to estimate the relative importance of these terms, since the plume i s spatially and temporally variable. However, Table 3 presents very coarse estimates of the order of magnitude of the terms in the momentum equation..The f i r s t part of the table l i s t s the scaling parameters, and the second part l i s t s the sizes of the various terms in the momentum eguation, for the region near the river mouth, and for the far fi e l d . For the salinity and continuity eguations, we notice that the ratio of the advective terms to the source terms i s uh/(wL), which i s also given in Table 3. Wp i s estimated from the numerical model of Chapter 5, where wp was calculated according to the formula wP = 0.0001u. Except for the action of winds and possibly horizontal eddy viscosity and diffusivity, the eguations derived in this chapter appear to have a l l the necessary terms to describe the plume. BOONDaHY CONDITIONS In solving any differential equation system, one has to specify the appropriate boundary conditions. The actual boundary conditions used will be discussed in chapter 5, but I would like to discuss here the theoretical boundary condition requirements. Consider a simplification of the above equations, in x-t space 41 only, (we define a new variable, =s„-s, the salinity defect.) £A t 2 ( uA) - o. dt 3* 3x These equations can be thought of as homogeneous eguations the behaviour of whose solutions dictate the behaviour of the eguations with forcing and dissipation, as long as the forcing and dissipation do not contain derivatives of the same or higher order. Writing g=gk, these may be put in matrix form A \ I k 1 " o 0 { 1* o o O (A or H^+fiHy=0. The eigenvalues TV # and l e f t eigenvectors, £• , of A are: Multiplying the matrix form of the differential eguations by i J , we get: I; l-i: Since ,4 tj - J: ~}\ S c ' i , there results 3 Thus, in the direction = <9 , the d t characteristic form of the differential equations. Explicitly, 42 the characteristic equations are : ^ dt J The reason for putting the equations i n characteristic form i s that the required boundary conditions at open boundaries become more obvious. The basic requirement i s that one should prescribe as many boundary conditions as there are characteristics pointing into the region under consideration. Thus, one must prescribe salinity on an open boundary i f i t s characteristic i s directed into the region. Since the characteristic speed i s u, the flow velocity, one must prescribe s on an inflow, and must devise a way for i t to be determined at an outflow boundary by flow conditions in the interior of the computational region. If |uj i s greater than c, and u is an inflow, then both u+c and u-c point into the computational region, and two independent pieces of information about u and c, in addition to s discussed above, must be specified. If |u| i s less than than c, then one of u+c, u-c points into the region, and one points out, so either u or c, or a relation between them, must be specified. If u=c, a very complicated situation arises, in which a boundary in x-t space becomes a characteristic. This i s a situation which becomes very tricky in problems with a time-dependent boundary condition, and requires recourse to further aspects of the physical system. 43 CHAPTER 4 AIDS TO INTDITION ABOUT THE PLUME Before presenting the numerical model in the next chapter, i t i s worthwhile to look at some simple models of plume dynamics. These models do not include a l l terms in the eguations of Chapter 3, so can't be expected to describe the plume adequately, but they are useful.in clarifying various aspects of the plume's behaviour. The f i r s t sub-model, the compressible flow analogy, shows that near the river mouth, the predictions of a frictionless model are at variance with observations, leading one to conclude that fri c t i o n and entrainment are important features of the plume near the river mouth. The second model, a time-independent model of surfacing isopycnals, illustrates the roles of depletion and entrainment in causing the isopycnals to rise as one proceeds downstream in the plume. The third model, conditions at a strong frontal discontinuity, discusses the motion of the strongly contrasted colour fronts frequently found in the Strait, and suggests that these fronts induce considerable vertical circulation. The fourth model, a kinematic wave approach to frontal motion, is intended to i l l u s t r a t e , in a very simple manner, the way in which fronts arise in a time-dependent situation, due to the tidally varying river flow. The f i f t h model , mixing and fluxes across an interface, shows how an upper layer model, as developed in this thesis, i s compatible with a diffusive {eddy diffusivity) model of the vertical salinity distribution. The sixth section, analogy with turbulent jets, is an attempt to motivate the use of entrainment and 44 depletion by showing how they arise in a more accessible system, a turbulent plane jet. i i . COHPfiESSIBLE FLOW ANALOGY There i s an exact analogy between the frictionless flow of a compressible f l u i d and the frictionless flow of a liquid with a free surface. The method of solution of the equations derived below was developed for compressible flow (e.g. Shapiro and Edelman, 1947), and later adopted for use in hydraulic engineering (Ippen,1951). Using this method, one i s able to predict the velocity and thickness of a f l u i d discharged from a channel into an unbounded region. The solution for water flowing oyer a solid surface (as developed in Rouse et a l , 1951) i s identical to the solution for lighter fluid flowing over heavier fl u i d (as required in a plume theory), i f q, the acceleration of qravity, i s replaced by g*=g , as shown in Chapter 3., This then i s a buoyancy spreading model, representing a balance between the convective accelerations and the spreading tendency of the pressure gradient. The equations of continuity and momentum conservation for a steady-state, frictionless plume are: u a „' * v IA Y v- g 'h x = o , 4m 2 u i A + V * ^ ' ^ y = °- 4.3. Fundamental to this method is the requirement that the vorticity of the flow be zero, 3 ' l / * - O . 4. 4. This approximation i s assumed to be valid in a region around the river mouth. 45 The method of solution i s as follows. From 4.2, 4.3, and 4.4 one obtains the Bernoulli eguation, Z Z 4.5. where c7'=• h The continuity eguation may be put in the form ( is*--C*) ry, da. + C • *• - c * ) d is = O , where ^ = ^ ^ +. / ^ i ^ z . ^ i 1 4.6. U2- - tz du , (pi + ^ U ) <_/ d i/ = ( d-¥ + m l y ) d x Thus, in the direction /dv. - m, du and dv satisfy the ordinary differential eguation 4.6. Since there are two types of m, depending on the sign of the square root, there are two eguations, in addition to the Bernoulli equation, from which to obtain u, v, and c 2. However, a l l this depends on m being real, that i s on u 2+v 2 being greater than c 2. Thus; this model applies to supercritical flow only. There i s a fair amount of evidence (Wright and Coleman, 1971; Garvine, 1977) that the flow at a river mouth is internally c r i t i c a l or supercritical, i.e. that (u 2+v 2)>c 2. Thus, one can use eguation 4.6 to obtain a solution around the river mouth, as long as u2+v2 remains greater than or egual to c 2. Figure 56a illustrates the orientation of the two characteristics, C+ and C- (defined by / j x ^ or m_ ) , and a streamline, with respect to an x-y coordinate system. 46 Defining u=gcos0, v=gsinO, c=gsin/w, and ~X-+= <9 ± / * , (Fig. 56a) one can eventually manipulate eguation 4.6 into the form 9 ± P(m) =constant 4.7. on ^/dx-Wi: - , and One interesting thing about the above solution i s that i t i s possible to integrate analytically the differential eguations along the characteristics. The problem i s set up as follows. An opening in a solid wall i s assumed, through which water is flowing with v=0, u=c. One then f i l l s up the computational region with the two families of characteristics, and uses tabulated values of to obtain values of u and v at the intersection of characteristics. (Fig. 56b) . Rouse et al, (1951), worked out by hand the solution to this problem, with F = u/c = 1, 2, 4, (Fig 57). The results, for the values of F examined , are a l l quite similar - as one proceeds outward from the river mouth, the upper layer thins and spreads. Since the downstream thinninq of the layer constitutes a pressure gradient, the flow accelerates. In contrast to this model we observe, (Cordes, 1977), that the flow outward from the river mouth slows down, rather than speeds up. The inadequacy of the model in this respect points out the importance of retarding forces at the river mouth. The retarding forces could come from three sources: 1) an adverse pressure gradient in the barotropic tide, caused by the large geostrophic slope during the ebb cycle of the tide in the Strait; 2) entrainment of water with zero momentum in the downstream direction of the plume; 3) f r i c t i o n a l interaction between the upper layer and lower layer. To determine how important these terms are, we f i r s t estimate ^' h x from the analytical solution. Fig. 57. For F2=1, 3'h- - x ( ^ ; J ' where L^/<>^t^j~i is measured from the diagram, h „ i s the i n i t i a l depth of flow, and b i ) 2 _ i s the halfwidth of the river mouth. The factor Ch>/bv ) converts from the non-dimensional units of the analytic solution to units appropriate to the Fraser River. Taking the width of the river to be 600 meters, and g* to be 10m/sec*x.01=0.1m/sec2, and h 0 to be 8 meters, we can evaluate c>hf$% for the region along the centreline where h changes from 1 to 0.3, a distance of 2.5 halfwidths: 9 V3-x") ^ f 7 = O.l/Z-S. Thus, g*h„~7.5x10-* m/secz. in the region where h changes from 0.3 to 0.1, the pressure gradient , calculated in the same manner, i s 2.0x10-* m/sec. From the geostrophic relation, fv=g £>< , the crosschannel pressure gradient for a current of 1 m/sec (an upper limit) i s about 1x10-* m/sec*. This slope results in the water level being higher on the western side of the Strait of Georgia during an ebb, which constitutes an adverse pressure gradient to the river flow in the vicinity of the river mouth. Thus i t appears that the cross channel barotropic t i d a l slope is the same order of magnitude as the buoyant spreading pressure gradient. 48 One can estimate the relative effect of vertical entrainment as follows. If the contribution of entrainment to the continuity eguation i s written as dt 3 and one considers the vertically integrated momentum equation, then the average momentum equation is Jjt _ 1 & ~ MS u . Dt h In Thus, entrainment acts as linear f r i c t i o n , with f r i c t i o n coefficient w / k . . He wi l l assume w=Eu, and E=2x10_* (Keulegan,1966}. This order of magnitude for E was verified by both Cordes (1977) and de Lange Boom (1976) for the Fraser Biver plume. An estimate of u from the Bernoulli equation i s At h/ho=0.6, u ^ 1.2 m/sec. This i s also a reasonable value for the measured speed near the river mouth. Hith this value for u, and h=.6x8m, the retarding force due to entrainment, wu/h, i s 0.4x10-* m/sec*. This value is one f u l l order of magnitude less than $'hx near the river mouth and somewhat closer to g*h v at points further downstream. It appears that entrainment has a significant but not dominant effect on the plume near the river mouth. There i s an indirect effect also. As entrainment proceeds, g', proportional to the density difference, decreases, and h has a tendency to increase. The actual pressure gradient driving u is (j/zk) Cs'* • D u e t o entrainment, since g' decreases, and h has a tendency to increase (opposed by i t s L\9 buoyant spreading tendency) i t is d i f f i c u l t to predict this pressure gradient without a more detailed model, such as that discussed in Chapter 5. The third possible retarding force is f r i c t i o n . If one assumes quadratic f r i c t i o n , and equates hg*h^ . to Ku|u|, then to be important, Ku2/h must be close to 7,5x10-* m/sec2. Assuming Ku2/h=7.5x10-* m/sec2, we get K~2x10~3. This value of K i s similar to the value of drag coefficient used in many calculations. For instance, the drag coefficient for wind over water is about 1.5x10-3, and the drag coeffiecient for bottom fr i c t i o n in a t i d a l channel i s about 2-4x10-3, and the interfacial drag coefficient in a laboratory scale flow i s of order 10 - 3 (Lofguist, 1960). Thus i t appears that f r i c t i o n plays an essential role in the plume dynamics. Is., h lI«IrINDEPENDENT MODEL OF SURFACING ISOPYCNALS As demonstrated in Chapter 2, figures 15, 27a, and 27b in particular, the surfacing of isopycnals i s a dominant feature of the plume in regions away from the river mouth. The model discussed here is an attempt to explain this phenomenon in terms of entrainment and depletion. This model applies to the region from around station c to station j . Fig. 27a. Here the plume i s thought to be more or less uniform across the Strait, and advected back and forth by the tide, with a small mean velocity to carry river water out of the Strait. The tidal excursion in this area i s about 10 km, so the plume i s advected back and forth by the tide a rather large distance. One could imagine performing an average over a few t i d a l cycles and obtaining a set of data describing a stationary plume. One also needs to 50 perform averages across the Strait (and hence across the plume), or else assume lateral uniformity. The plume i s fed water at i t s upstream end, and this water leaks out from the forward or leading end by means of the depletion mechanism discussed in Chapter 3. The equations for this model of the plume, a simplified form of 3.24, 3.25, and 3.26 are: + VUn - \AJp = O -4.7. . 2 (Us) + \AJ„ 5 - \AJP So = O • 4.8. h h 4.9. , where T(j0-s) - j( 9°'f) i s an equation of state, 0 i s a transport, (vertically integrated velocity), and K' i s a coefficient of linear f r i c t i o n . A schematic drawing of the model is shown in Figure 58a. The use of linear f r i c t i o n i s not unrealistic in that the t i d a l average of square law f r i c t i o n i s linear in the residual flow (Gruen and Groves, 1966). To make the solution of this model very easy, assume w^ and Wpare constants, and that w n>Wp, which i s valid near the outer edge of the plume., The continuity equation, 4.7, has the immediate solution UL -- Ua ~ C v u n - M J P ) * Y where D e i s the transport at x=0, and where x=0 is taken at the upstream boundary of the region of applicability of this model (Fig. 58a) . Defining <^ =w^ ,-Wp, and I^Uo/^, we have 0=«<{L-x). Note that 0=0 at x=L, so the length of the plume i s L., The salt ahd continuity eguation may be combined to give: 51 UL S * +\AS?(S-S*)^ O. 4. jo. Define s-s0= 2T,<0; y=L-x. Then, the above equation becomes - o with solution =Ay ; the salinity defect i s proportional to the distance from the leading edqe, y, to the power vpA<. Imposing the boundary condition that at x= 0 s=si , we get A= (s^ -s 0)L In terms of the variables y and , the momentum eguation becomes <2y k K With .2_=Ay , we get (3 Trying a solution h=Byr gives A 6 y ^ £ * ^ ; + Z£ C w„ + /< J y Equating, powers of y: 0 = (2 - ^ ^L)/j. &nd finding a value of B to make the l e f t hand side equal to zero: - A ( z ^ 4 w?/^J Because i t is a non-dimensional number, and perhaps the most important one for the plume, the internal Proude number is of some interest. It i s given by: 3 = 52 a constant.. To summarize the solution U -- a C L - * ) 4. 1 1. 4. 12. In = 4.13. (S.-SJ l - ^p U C f/3 - 'h**^ And the velocity, u, i s given by 4.14., Thus, the transport, 0, decreases in the downstream direction, as plume water becomes sal t i e r and is redefined as lower layer water; the average salinity of the plume increases downstream; and the thickness of the layer decreases downstream. It i s reassuring to note that the f l u i d velocity, u, also decreases downstream, going to zero at x=L. The data for July 3, 1975 (Fig. 27a) seem a good choice for comparison with this model, in that the salinity section for that day shows the surfacing of isopycnals very clearly. Referring to Fig. 27a, consider station d as x=0. L i s then 23 km, i f the plume boundary i s defined as the 25 0/oO contour. With 53 s Q = 25 °/oo, the average salinity at station d, slr is 18.6 */oi> , from an integration of the actual salinity profile, fe need an estimate for U0. Assume that on average, half the river discharge leaves through the northern channels and half through the southern channels. Assuming that the river discharge is 8000m3/sec, and that by the time river water reaches station d i t has entrained an egual volume of salt water then the transport to the south i s 8000m3/sec, flowing along the axis of the Strait. At station d, the Strait i s about 16000m wide. Assuming uniform discharge across the Strait, 0o= (8000/16000)=0.5 m2/sec. Thus <=< =0 „ A = . 5/ (23x 105) =2 . 2x10-s m/sec. We can get an estimate of vp/*. by f i t t i n g the salinity change from station d to station j . At station j , the average salinity in the upper layer i s 23"/"° and x=11km;,We obtain, from eguation 4.12, vp /U-1.8. With =2.2x10~s m/sec, we find wp= 4x10-5 m/sec, and w„= 6x10~s m/sec. This fixes h (*•) 3 ha ( LJU)°'z/3using y - o}((J»y)/(S>>-s) =.83, {average of salinity profiles for stations d, j, 1) we calculate K*, the linear f r i c t i o n coefficient, to be 5.3x10~3 m/sec* Eguating K'u and Cu|u| at x=0, we get an eguivalent C=2.6x10—2. This i s about 10 times the usual value of a drag coefficient, 3x10~3. However, K* i s really C U r m 5 , (Groen and Groves, 1966), and the rms value of velocity is probably quite a bit higher than the t i d a l mean used here. The computed and observed distribution of h is shown in Figure 58b. The f i t uses the two h points at the upstream and downstream ends of the plume, so there is only one point left to check the f i t . To show the sensitivity of the f i t , the curve h= ti „ (^-[r)3 i s plotted as a dashed line. The model is substantially correct in that i t predicts the rising of isopycnals (modelled as a thinning of the layer and an increase in average salinity of the layer), but i t would be extremely unlikely that the entrainment and depletion processes can be modelled very well by constant values of wn and w,o. The model would be improved by obtaining data for f i t t i n g that were truly cross-channel and t i d a l l y averaged; and by obtaining, empirically, better formulae for w^ and Vp. 2 j l CONDITIONS AT A STRONG I1QNTAL DISCONTINUITY As discussed in Chapter 2, particularly with reference to the data of July 2, 1975, Fig. 31, there i s often a distinct colour front bounding the plume. The purpose of this section i s to c l a r i f y the role these fronts play in determining the motion of the water behind them. One would like to have a model of a front suitable for use with a larger scale model of the entire plume, and for these purposes a detailed model of the circulation at the front i s not reguired, but rather relations between frontal velocity and the fluxes of mass, momentum and salt into the frontal region. r With reference to the dye experiment described in Chapter 2, on July 3,1975, a "tank tread" model of frontal circulation suggests i t s e l f . Thus, i f one considers a military tank moving over the ground at speed V, to an observer on the ground the upper tread i s moving at 27, the tank at V, and the bottom tread is stationary.„This is similar to the velocity profiles measured in the plume, eg. Fig. HO, where the bottom part of the plume is "attached" to the lower water, and the upper part i s moving at a significantly different speed. Continuing with the tank 55 tread analogy, and changing the co-ordinate system to that of an observer sitting on the tank, the top tread i s moving at V, the bottom at -V, and the tank i t s e l f appears stationary. Similarly, an observer travelling with the front sees water at the surface of the plume coming toward him (as the dye on the surface ran toward the front, July 2), and sees deeper water travelling away from him, on the under side of the plume (like the dye which was later found at depth, behind the front). Also, to an observer on the tank, the ground on which the tank is travelling i s coming toward him, and then passing under him; similarly, to an observer at the front, the dark blue water appears to travel toward him, and then flow under the plume. A model to describe this motion can be developed from a control volume approach. Consider a cross-section of the front, and draw a control volume around i t . Fig. ., 59a. , Assuming the front moves ater, the control volume also moves atG". Taking z=0 at the base of the plume, at some z=h' (Fig. 59a), the water speed u (relative to a stationary co-ordinate system) equals the front speed CT . Above this level, water i s flowing into the control volume, and below this level, water is flowing out. The speed of water in the control volume or frontal co-ordinate system i s u-cT. Conservation of mass requires J (u - cr) J ? + Q --. IC cr- u \ dz ^ or 56 A c^A - j u c/z - Q = O. 0 4.15a. Q represents a rate of entrainment specifically due to frontal processes and i s entirely different from vp discussed previously. wp- and wn have been ignored because the control volume length i s assumed very small, of order tens of meters, relative to the entire plume. Similarly, conservation of salt reguires: A h' j Ct*-cr)sdz + & 5> * J $(tA-cr)dz = O 5 or A h '& ( sd* - f us dz ~ O s0 = o . 4.16a. Conservation of momentum reguires: [ (u -cr)*dz- + f (jj-crfj z - O'er + A p - K = O , A' or, using the continuity eguation, A ' ' / ul ds. ~ & J u d z + A p - l< ~- O . 4.17a. Here, K i s the excess momentum reaction at the front, due mainly to form drag. as derived in Chapter 3, the net pressure force is 1/2gh 2^£ , i f P i s independent of depth in the layer. It i s possible to obtain the pressure force f a i r l y simply for a 57 variablep (z). Consider a plume of thickness h, with i t s base at z=- (h-Ah) (Fig. 59b). For zero pressure gradient at depth, (the assumed condition in the water beneath the plume), we reguire h \ , k or £>k -Cp»-(p) k where p = J_ f J^ The net pressure force, divided by the average density i i 4 p P9 dz ' d ? - Z 4. 18. For the case p = 7^ = a constant, this becomes i ^f-c -£1 - dk- H , p 2 L { y» J 2, p as in Chapter 3. If p - ^s 2- /j^ * a linear profile, then the net pressure force i s '//^ (p0-ps) /p0 , correct to f i r s t order in If we had used the average density in the formula 'l^g^. &fIp , we would have J/^ t p0-pj) [ p* ) significantly different from / / 2 . C fo'fs^)/p> • Now, to apply the eguations to some fie l d data. On July 23, 1975,. we measured a current profile and a salinity profile at about 200 meters from the front, in the plume. (Fig, 39 and Fig. 58 40). The current profile was actually a speed profile, so the zero crossing and direction were only inferred. Referring the velocities to 17m, the deepest measured, the resulting front speed is that relative to tid a l l y moving deep water. Assuming a plume depth of 6m, p e =1.017 , ps = 1.003, and assuming a linear density profile, p - ps ^o.pjj ' t f ee pressure term 4,18, turns out to be . 42m2s-2. The integrals of u and s are: k j. udz = 1.25m2/s 6 k J u2dz = 0.72 mVs 2 k J sdz = 100 m ppt o A f usdz = 12.9 ppt m2/s. 0 With s =23.4, we get, by solving eguations 4.15a, 4.16a, 4.17a, ^ =.41m/s Q= 1.21 m2/s K = 0.63 m3/s 2; recalling that CT i s the front speed, Q is the extra frontal entrainment, and K is the extra form drag associated with propogation of the front. Note that the front speed i s considerably larger than the average speed, 1.25/6 =• .21m/s. Also note that Q^Judz. That i s , a front extends i t s e l f , or propagates, by mixing egual parts of plume water and salt water. Also, one could write K=1/2CD h. 59 where C p i s a drag coefficient, which turns out to be 1.2 in this case. Since values of C 0 for flow over blunt objects such as cylinders are about 1, i t appears that K may be identified with the drag exerted by the salt water as i t flows under the blunt leading edge of the plume. It is informative to re-derive eqns. 4.15a, 4.16a, and 4.17a from the differential form of the equations. Chapter 3 . Thus, in one horizontal dimension, suppose that for x~~6 equal to zero, since there is no plume ahead of the front; and finally take ^ ^ o . There results: * 4.15b. A A ir f 5 dz - f u sd? - Q $b ~ o. J J 4.16b. O A 60 - y w * -9*% ' k " °-Notice that the quantities involving w^ , wA, and Ru|u| disappear, since they are proportional to<£, which approached zero. The justification for this procedure is that as one proceeds from Sand Heads to the front, properties change only gradually, and are appropriately related to each other by partial differential equations.,Then, at the front, the fields of thickness, velocity and salinity change drastically in a very short distance, of the order of a small boat length, and can only be related to each other in terms of weak solutions. (A weak solution i s a set of relations between the changes of various properties across a discontinuity, tfhitham, 1 9 7 4 ) . We can compare (approximately) the speed of the front to the calculated speed, 0.41m/s. Referring to Fig.38a, a l l three drogues were inserted at the front, and the shallow drogue, S, stayed with the front. Assuming the deep drogue travelled with the bottom water (not exactly true), the relative speed i s about .33 m/s. This i s 8055 of the calculated speed, but the d r i f t time of the drogues, 1.5 and 2 hours, is not very small compared to the tidal period, (nor was the current profile measurement time), so great accuracy can't be expected.. Summary of frontal circulation The following description of a front emerges from the equations derived above. The front i s a mixing region of water which i s pushed outward by the momentum flux and the pressure gradient, and retarded by the mixing of ambient water and the 61 form drag i t experiences from the ambient water. The intense mixing at the front can be visualized as follows. Relative to the depth at which u=0"i guite fresh water above this level flows into the mixing region at the front. Rater below this level, flowing away from the front, is quite salty, and could only have picked up this salt by an intense , churninq mixing at the front. ta. ISINEMATIC WAVE APPROACH TO FBONTAL MOTION The following model i s intended to demonstrate, as simply as possible, the way in which fronts, described in Chapter 2, develop in a time-dependent plume model. We consider a one dimensional model..The continuity eguation, without entrainment, dt 1.22... We further suppose that a l l the dynamics governing the plume can be characterized by h ' 4.23. where F. i s a constant.. ,F is similar to a Froude number, but has the dimensions of an" acceleration to simplify subsequent mathematical expressions. Although the assumption that F is constant may not be a good approximation for the plume, for the present purposes the simplicity of the resulting mathematical analysis more than compensates for the physical inadeguacy of that assumption. The approach used here i s called the kinematic wave method because a l l the dynamic interactions are summarized in a simple rule, equation 4.23, and we look at the motion prescribed by the continuity eguation 4.22 (Whitham, 1974) . 62 Substituting for u in the continuity eguation we find: 4.24. The lines (3/2) u are the characteristics. At x=0, representing the river mouth, assume the velocity versus time graph i s a series of triangles, representing t i d a l modulation of river flow (plotted along the time axis, Fig. 61). Thus, on x=G: u= (2/3)t; 0~~