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For this set of experiments, ψt and (ut, vt) are plotted in Fig. 1a, ψg and (ug, vg)
$ \equiv $ ($ - {\partial _y}{\psi _{\rm{g}}},\; - {\partial _x}{\psi _{\rm{g}}}$ ) are plotted in Fig. 1b, the vorticity ζt$ \equiv $ $ {\nabla ^2} $ ψt is plotted in Fig. 1c, and the geostrophic vorticity ζg$ \equiv $ $ {\nabla ^2} $ ψg is plotted in Fig. 1d. Figure 1c shows that the absolute vorticity, defined by f + ζt, is positive everywhere so the nonlinearly balanced wavering jet flow is inertially stable over the entire domain (see the proof in Appendix C of Xu, 1994). Figure 1c also shows that the geostrophic vorticity ζg is larger than −f/2 (= −f0/2) everywhere, so the NBE is elliptic over the entire domain and its associated boundary value problem in Eq. (1) is well posed.Figure 1. (a) ψt plotted by color contours every 4.0 in the unit of 106 m2 s−1 and (ut, vt) plotted by black arrows over domain D
$ \equiv $ [−L ≤ x ≤ L, −L ≤ y ≤ L] with L = 2000 km for the first set of experiments. (b) As in (a) but for ψg and (ug, vg) with ψg$ \equiv $ ϕ/f and ϕ computed from ψt by setting f = f0 = 10−4 s−1 as described in section 3.3. (c) Vorticity${\zeta _{\rm{t}}} \equiv {\nabla ^2}{\psi _{\rm{t}}}$ plotted by color contours every 0.1 in the unit of 10−4 s−1 over domain D. (d) As in (c) but for geostrophic vorticity${\zeta _{\rm{g}}} \equiv {\nabla ^2}{\psi _{\rm{g}}} $ . The wavering jet axis is along the green contour of ψt = 0 in (a) with its ridge at x = 0 and two troughs at x = ±L on the west and east boundaries of domain D.The relative error of ψk with respect to ψt can be denoted and defined by
where || || denotes the RMS of discretized field of the variable inside || || computed over all the grid points (including the boundary points) of domain D. The accuracy of the solution ψk obtained during the iterative process in each experiment can be evaluated by the RMS of ε(ψk), denoted and defined by
where || || is defined in Eq. (18). The accuracy to which the NBE is satisfied by ψk can be measured by E[N(ψk)] defined in Eq. (14).
Table 1 lists the values of E(ψk) and E[N(ψk)] for the initial guess ψ0 (= ψg) in row 1 and the optimally truncated solutions ψK from the four experiments in rows 2−5. As shown in row 2 versus row 1 of Table 1, M1a reaches the optimal truncation at k = K = 6 where E[N(ψk)] is reduced (from 0.120 at k = 0) to its minimum [= 2.411×10−3 < E(
$ {\nabla ^2}$ ϕ) = 3.25×10−3—the NBE discretization error defined in Eq. (17)] with E(ψk) reduced (from 2.43×10−2 at k = 0) to 4.87×10−4. Figure 2a shows that E(ψk) reaches its minimum (= 4.79×10−4) at k = 10. This minimum is slightly below E(ψK) = 4.87×10−4 but undetectable in real-case applications.E(ψk) E[N(ψk)] k ψ0 2.43×10−2 0.120 k = 0 M1a 4.87×10−4 2.41×10−3 k = K = 6 M1b 1.68×10−3 1.81×10−2 k = K = 38493 M2a 4.55×10−3 3.55×10−2 k = K = 19 M2b 2.69×10−3 2.66×10−2 k = K = 26 Table 1. Values of E(ψk) and E[N(ψk)] listed in row 1 for the initial guess ψ0 (= ψg) with k = 0 and in rows 2−5 for ψK from the four iterative procedures in the first set of experiments (with Ro = 0.1). Here, E(ψk) is defined in Eq. (19), E[N(ψk)] is defined in Eq. (14), k is the iteration number, and ψK is the optimally truncated solution at k = K.
Figure 2. (a) E[N(ψk)] and E(ψk) from M1a in the first set of experiments plotted by red and blue curves, respectively, as functions of k over the range of 1 ≤ k ≤ 20. (b) As in (a) but from M1b plotted over the range of 1 ≤ k ≤ 4×104. (c) As in (a) but from M2a plotted over the range of 1 ≤ k ≤ 60. (d) As in (c) but from M2b. In each panel, the ordinate of E[N(ψk)] is on the left side labeled in red and the ordinate of E(ψk) is on the right side labeled in blue.
E(ψk) E[N(ψk)] k ψ0 4.86×10−2 0.243 k = 0 M1a 1.24×10−3 5.23×10−3 k =K = 13 M1b 5.14×10−3 2.20×10−2 k = K = 48057 M2a 6.31×10−3 4.17×10−2 k = K = 26 M2b 3.96×10−3 2.94×10−2 k = K = 35 Table 2. As in Table 1 but for the second set of experiments (with Ro = 0.2).
E(ψk) E[N(ψk)] k ψ0 9.72×10−2 0.57 k = 0 M1a 8.20×10−2 0.13 k = K = 2 M1b 8.31×10−2 0.15 k = K = 10325 M2a 8.25×10−2 0.11 k = K = 26 M2b 8.26×10−2 0.10 k = K = 29 Table 3. As in Table 1 but for the third set of experiments (with Ro = 0.4 and x0 = 0).
On the contrary, as shown in row 3 of Table 1 and Fig. 2b, M1b reaches the optimal truncation very slowly at k = K = 38493 where E[N(ψk)] is reduced to its global minimum (= 1.81×10−2) with E(ψk) reduced to 1.68×10−3. Here, E[N(ψk)] has three extremely shallow and small local minima (at k = 32408, 38490 and 38497) not visible in Fig. 2b. These local minima are detected and passed by setting m = 3 in Eq. (14) for M1b. Clearly M1b is less accurate and much less efficient than M1a.
Figure 2c (or 2d) shows that M2a (or M2b) reaches the optimal truncation at k = K = 19 (or 26) where E[N(ψk)] is reduced to its global minimum [= 3.55×10−2 (or 2.66×10−2)] with E(ψk) reduced to 4.55×10−3 (or 2.69×10−3), and E(ψk) decreases continuously toward its minimum [= 2.45×10−3 (or 1.62×10−3)] as k increases beyond K. Thus, M2a and M2b are less efficient and much less accurate than M1a for Ro = 0.1.
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For this set of experiments, ψt and (ut, vt) have the same patterns as those in Fig. 1a, and ψg and (ug, vg) are similar to those in Fig. 1b, but the contour intervals of ψt and ψg are reduced by 50% as L is reduced from 2000 to 1000 km with Ro increased to 0.2, so the wavering jet flow is on the sub-synoptic scale. In this case, the nonlinearly balanced jet flow is still inertially stable over the entire domain since ζt > −f everywhere as shown in Fig. 3a, but ζg < −f/2 in the two small yellow colored areas as shown in Fig. 3b, so the NBE becomes hyperbolic locally in this small area and the boundary value problem in Eq. (1) is not fully well posed.
Figure 3. (a) ζt plotted by color contours every 0.25 in the unit of 10−4 s−1 in domain D with L = 1000 km and Ro = 0.2 for the second set of experiments. (b) As in (a) but for ζg. As shown in (b), ζg < −f/2 (= −f0/2) in the two small yellow colored areas where the NBE becomes locally hyperbolic.
In this case, as shown in row 2 versus row 1 of Table 2, M1a reaches the optimal truncation at k = K = 13 where E[N(ψk)] is reduced (from 0.243 at k = 0) to its minimum [= 5.23×10−3 close to E(
$ {\nabla ^2}$ ϕ) = 4.33×10−3] with E(ψk) reduced (from 4.86×10−2 at k = 0) to 1.24×10−3. The rapid descending processes of E(ψk) and E[N(ψk)] (not shown) are similar to those in Fig. 2a for M1a in the first set of experiments.As shown in row 3 of Table 2, M1b takes K = 48057 iterations to reach the optimal truncation and the values of E[N(ψk)] and E(ψk) at k = K are about four times larger than those from M1a. The extremely slow descending processes of E(ψk) and E[N(ψk)] (not shown) are similar to those in Fig. 2b for M1b in the first set of experiments. As shown in row 4 (or 5) of Table 2, M2a (or M2b) reaches the optimal truncation at k = K = 26 (or 35) and the values of E[N(ψK)] and E(ψK) are more than (or about) 4 times of those from M1a. Thus, M1a is still more accurate and much more efficient than M1b and is more efficient and much more accurate than M2a and M2b for Ro = 0.2, although the boundary value problem in Eq. (1) in this case is not fully (but nearly) well posed.
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For this set of experiments, ψt and (ut, vt) have the same patterns as those in Fig. 1a but the contour interval of ψt is reduced 4 times as L is reduced from 2000 to 500 km with Ro increased to 0.4, so the wavering jet flow is on the meso-α scale. Figure 4a shows the fields of ψg and (ug, vg) for the nonlinearly balanced jet flow. This nonlinearly balanced jet flow is inertially unstable in the yellow colored area south of the ridge of wavering jet axis in the middle of domain D where ζt < −f as shown in Fig. 4c. Figure 4d shows that ζg < −f/2 in the long and broad yellow colored area along and around the wavering jet, so the NBE is hyperbolic in this area and the boundary value problem in Eq. (1) becomes seriously ill-posed.
Figure 4. (a) ψg plotted by color contours every 1.0 in the unit of 106 m2 s−1 and (ug, vg) plotted by black arrows over domain D with L = 500 km and Ro = 0.4 for the third set of experiments. (b) As in (a) but for ε(ψ0) = ε(ψg) plotted by color contours every 5.0 in the unit of 10−2. (c) As in (a) but for ζt plotted by color contours every 0.5 in the unit of 10−4 s−1 in domain D. (d) As in (c) but for ζg. As shown in (c), ζt < −f in the yellow colored area south of the ridge of wavering jet axis where the jet flow becomes inertially unstable. As shown in (c), ζg < −f/2 (= −f0/2) in the long and broad yellow colored area (along and around the wavering jet) where the NBE becomes hyperbolic.
In this case, as shown in row 2 of Table 3 and Fig. 5a, M1a reaches the optimal truncation at k = K = 2 where E[N(ψk)] is decreased (from 0.57 at k = 0) to its minimum (= 0.13), while E(ψk) decreases from 9.72×10−2 at k = 0 to 8.20×10−2 at k = K = 2 and then to its minimum (= 7.38×10−2) at k = 6. As k increases beyond 6, M1a diverges. Its optimally truncated solution ψK is merely slightly more accurate than the initial guess ψ0. As shown in row 3 of Table 3 and Fig. 5b, M1b reaches the optimal truncation at k = K = 10325 where E[N(ψk)] is decreased to its global minimum (= 0.15), while E(ψk) decreases to 8.31×10−2 at k = K and then to its minimum (= 7.68×10−2) at k = 23515. Thus, M1b is still less accurate and much efficient than M1a.
Figure 5. (a) E[N(ψk)] and E(ψk) from M1a in the third set of experiments plotted by red and blue curves, respectively, as functions of k over the range of 1 ≤ k ≤ 8, (b) As in (a) but from M1b plotted over the range of 1 ≤ k ≤ 3×104. (c) As in (a) but from M2a plotted over the range of 1 ≤ k ≤ 60. (d) As in (c) but from M2b. In each panel, the ordinates of E[N(ψk)] and E(ψk) are placed and labeled as in Fig. 2.
Figure 5c (or 5d) shows that M2a (or M2b) reaches the optimal truncation at k = K = 26 (or 29) where E[N(ψk)] is reduced to its minimum [= 0.11 (or 0.10)], while E(ψk) is reduced to its minimum [= 8.24×10−2 (or 8.24×10−2)] at k = 25 (or 26) and then increases slightly to 8.25×10−2 (or 8.26×10−2) at k = K = 26 (or 29). As shown in row 4 (or 5) versus row 2 of Table 3, E(ψK) from M2a (or M2b) is larger than that from M1a, so M2a (or M2b) is still less accurate than M1a in this case.
Figure 6a (or 6b) shows that ε(ψK) from M1a (or M1b) peaks positively and negatively in the middle of domain D as ε(ψ0) does in Fig. 4b but with slightly reduced amplitudes. Figure 6c (or 6d) shows that ε(ψK) from M2a (or M2b) has a broad negative peak south of the ridge of wavering jet axis similar to that of ε(ψ0) in Fig. 4b but with a slightly enhanced amplitude. In this case, M1a is still slightly more accurate than the other three iterative procedures, but it cannot effectively reduce the solution error in the central part of the domain where not only is NBE hyperbolic (with ζg < −f/2 as shown in Fig. 4d), but also the jet flow is strongly anti-cyclonically curved and subject to inertial instability (with ζt < −f as shown in Fig. 4c).
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For this set of experiments, ψt and (ut, vt) are plotted in Fig. 7a. These fields represent the same nonlinearly balanced wavering westerly jet flow as that in the third set of experiments except that the wave fields are shifted by a half-wavelength so the jet flow is curved cyclonically in the middle of domain D. In this case, ψg and (ug, vg) are nearly the same as the half-wavelength shifted fields (not shown) from Fig. 4a but with small differences, mainly along and around the trough and ridge lines due to the boundary condition, ϕ
$ \equiv $ fψg = fψt, used here along the two trough lines (instead of the two ridge lines in Fig. 4a) for solving ϕ from${\nabla ^2} $ ϕ = N(ψt). Figure 7c shows that the jet flow becomes inertially unstable in the two yellow colored areas (where ζt < −f) around the west and east boundaries of domain D. Figure 7d shows that the NBE becomes hyperbolic in the long and broad yellow colored area (where ζg < −f/2) that is nearly the same as the yellow colored area in Fig. 4d but half-wavelength shifted, so the area of ζg < −f (that is, the area of ζ0 + f < 0 in which the initial guess field is inertially unstable) in Fig. 4d is moved with the ridge line to the west and east boundaries in Fig. 7d. As the area of ζg < −f and area of ζt < −f are moved away from the domain center to the domain boundaries where ψt is known and given by ϕ/f, the NBE becomes less difficult to solve in this fourth set of experiments than in the third set.Figure 7. (a) As in Fig. 4a but for ψt and (ut, vt) in the fourth set of experiments with L = 500 km and x0 = L (instead of x0 = 0). (b) As in (a) but for ε(ψ0) = ε(ψg) plotted by color contours every 6.0 in the unit of 10−2. (c) As in (a) but for ζt plotted by color contours every 0.5 in the unit of 10−4 s−1 in domain D. (d) As in (c) but for ζg.
In this case, as shown in row 2 of Table 4 and Fig. 8a, M1a reaches the optimal truncation at k = K = 7 where E[N(ψk)] is decreased (from 0.76 at k = 0) to its minimum (= 3.81×10−2), while E(ψk) decreases from 9.71×10−2 at k = 0 to 2.29×10−2 at k = K = 7 and then to its flat minimum (= 2.25×10−2) at k = 12, so ψK is significantly more accurate than ψ0 and slightly less accurate than ψk at k = 12 (which is undetectable in real-case applications). As shown in row 3 of Table 4 and Fig. 8b, M1b reaches the optimal truncation at k = K = 31830 where E[N(ψk)] is decreased to its global minimum (= 4.54×10−2), while E(ψk) decreases to 2.37×10−2 at k = K and then to its minimum (= 2.21×10−2) at k = 57 586. Thus, M1b is still much less efficient and less accurate than M1a.
E(ψk) E[N(ψk)] k ψ0 9.71×10−2 0.76 k = 0 M1a 2.29×10−2 3.81×10−2 k = K = 7 M1b 2.37×10−2 4.54×10−2 k=K= 31830 M2a 3.03×10−2 5.42×10−2 k = K = 27 M2b 2.64×10−2 4.66×10−2 k = K = 32 Table 4. As in Table 1 but for the fourth set of experiments (with Ro = 0.4 and x0 = L).
Figure 8. (a) E[N(ψk)] and E(ψk) from M1a in the fourth set of experiments plotted by red and blue curves, respectively, as functions of k over the range of 1 ≤ k ≤ 24, (b) As in (a) but from M1b plotted over the range of 1 ≤ k ≤ 6×104. (c) As in (a) but from M2a plotted over the range of 0 ≤ k ≤ 60. (d) As in (c) but from M2b. In each panel, the ordinates of E[N(ψk)] and E(ψk) are placed and labeled as in Fig. 2.
Figure 8c (or 8d) shows that M2a (or M2b) reaches the optimal truncation at k = K = 27 (or 32) where E[N(ψk)] is reduced to its minimum [= 5.42×10−2 (or 4.66×10−2)], E(ψk) reduces to 3.03×10−2 (or 2.64×10−2) at k = K and then to its minimum [= 2.72×10−2 (or 2.43×10−2)] at k = 36 (or 44), so M2a (or M2b) is still less efficient and less accurate than M1a in this case.
Figure 7b shows that ε(ψ0) has a broad positive (or negative) peak south (or north) of the trough of wavering jet axis in the middle of domain D. These broad peaks are mostly reduced by M1a as shown by ε(ψK) in Fig. 9a but slightly less reduced by M1b as shown in Fig. 9b and less reduced by M2a (or M2b) as shown in Fig. 9c (or 9d). However, the small secondary negative peak of ε(ψg) near the west or east boundary in Fig. 7b is reduced only about 30% by M1a (or M1b) as shown by ε(ψK) in Fig. 9a (or 9b) and even less reduced by M2a (or M2b) as shown in Fig. 9c (or 9d). Thus, all the four iterative procedures have difficulties in reducing the errors of their optimally truncated solutions near the west and east boundaries where not only is the NBE hyperbolic (with ζg < −f/2 as shown in Fig. 7d), but also the jet flow is subject to inertial instability (with ζt < −f as shown in Fig. 7c). Nevertheless, since the area of ζt < −f is moved with the ridge of wavering jet axis to the domain boundaries in Fig. 7c, all of the four iterative procedures perform significantly better in this set of experiments than in the previous third set, as shown in Fig. 9 and Table 4 versus Fig. 6 and Table 3. In this case, M1a is still most accurate and M1b is still least efficient among the four iterative procedures.
E(ψk) | E[N(ψk)] | k | |
ψ0 | 2.43×10−2 | 0.120 | k = 0 |
M1a | 4.87×10−4 | 2.41×10−3 | k = K = 6 |
M1b | 1.68×10−3 | 1.81×10−2 | k = K = 38493 |
M2a | 4.55×10−3 | 3.55×10−2 | k = K = 19 |
M2b | 2.69×10−3 | 2.66×10−2 | k = K = 26 |