# Multiple Equilibria in a Land–Atmosphere Coupled System

• Corresponding author: Jianping HUANG, hjp@lzu.edu.cn
• Funds:

Supported by the National Science Foundation of China (41521004 and 41705047), Strategic Priority Research Program of Chinese Academy of Sciences (XDA2006010301), Foundation of Key Laboratory for Semi-Arid Climate Change of the Ministry of Education in Lanzhou University from the Fundamental Research Funds for the Central Universities (lzujbky-2017-bt04), and China 111 Project (B 13045)

• doi: 10.1007/s13351-018-8012-y
• Many low-order modeling studies indicate that there may be multiple equilibria in the atmosphere induced by thermal and topographic forcings. However, most work uses uncoupled atmospheric model and just focuses on the multiple equilibria with distinct wave amplitude, i.e., the high- and low-index equilibria. Here, a low-order coupled land–atmosphere model is used to study the multiple equilibria with both distinct wave phase and wave amplitude. The model combines a two-layer quasi-geostrophic channel model and an energy balance model. Highly truncated spectral expansions are used and the results show that there may be two stable equilibria with distinct wave phase relative to the topography: one (the other) has a lower layer streamfunction that is nearly in (out of) phase with the topography, i.e., the lower layer ridges (troughs) are over the mountains, called ridge-type (trough-type) equilibria. The wave phase of equilibrium state depends on the direction of lower layer zonal wind and horizontal scale of the topography. The multiple wave phase equilibria associated with ridge- and trough-types originate from the orographic instability of the Hadley circulation, which is a pitch-fork bifurcation. Compared with the uncoupled model, the land–atmosphere coupled system produces more stable atmospheric flow and more ridge-type equilibrium states, particularly, these effects are primarily attributed to the longwave radiation fluxes. The upper layer streamfunctions of both ridge- and trough-type equilibria are also characterized by either a high- or low-index flow pattern. However, the multiple wave phase equilibria associated with ridge- and trough-types are more prominent than multiple wave amplitude equilibria associated with high- and low-index types in this study.
• Fig. 1.  The second one (left panels) and third one (right panels) of the three equilibrium states for m = 3.7 at Cg = 50 W m–2. They belong to “High 2” and “High 1” equilibria, respectively. The streamfunction fields of the (a, e) upper and (b, f) lower layers, respectively. The temperature fields of (c, g) the atmosphere and (d, h) the land, respectively. The contour intervals are (a, e) 2.0 × 107 m2 s–1, (b) 2.0 × 105 m2 s–1, (f) 3.0 × 105 m2 s–1, and (c, d, g, h) 10 K. The background dotted lines show the topographic heights in the model, with negative regions shaded.

Fig. 2.  As in Fig. 1, but for the third one of the three equilibrium states for m = 3.7 at Cg = 55 W m–2 (left panels) and for the third one of the five equilibrium states for m = 6 at Cg = 30 W m–2 (right panels). They belong to “Low 1” and “Low 2” equilibria, respectively. The “Low 2” equilibria has nondimensional solutions with $({\psi _1}, \;{\psi _2}, \;{\psi _3}, \;{\theta _1}, \;{\theta _2}, \;{\theta _3}, \;{T_{{\rm g}, 1}}, \;{T_{{\rm g}, 2}}, \;{T_{{\rm g}, 3}})$ = (0.0216, –0.0079, –0.0051, 0.0311, –0.0098, –0.0043, 0.0771, –0.0175, –0.0077). The contour intervals are (a) 2.0 × 107 m2 s–1, (e) 1.0 × 107 m2 s–1, (b, f) 1.0 × 106 m2 s–1, (c, d) 10 K, and (g, h) 4 K.

Fig. 3.  The equilibrium bifurcation associated with the change in meridional differential solar heating parameter Cg for m = 3.7 (left panels) and m = 6 (right panels), respectively. The ordinate shows the nondimensional equilibrium values of (a, d) the zonal component $\psi _1^1$ and the wave components (b, e) $\psi _2^1$ and (c, f) $\psi _3^1$, respectively. Different branches of equilibrium solutions have different colors. The crosses denote unstable equilibria, the circles denote stable high-index equilibria, and the asterisks denote stable low-index equilibria.

Fig. 5.  (a) Stability curves of the Hadley circulation in the coupled land–atmosphere model (Case 1). The blue solid lines enclose the region of orographic instability. The red solid (black dashed) lines and the top x-axis and the right y-axis enclose the region of baroclinic instability in the presence (absence) of topography. Comparison of the regions of (b) orographic instability, (c) baroclinic instability in the presence of topography, and (d) baroclinic instability in the absence of topography for the four experiments (Cases 1–4).

Fig. 4.  A traveling wave solution in the absence of topography for m = 3.7 at Cg = 50 W m–2, with $({\psi _1}, \;{\psi _2}, \;{\psi _3}, \;{\theta _1}, \;{\theta _2}, \;{\theta _3}, \;{T_{{\rm g}, 1}}, \;{T_{{\rm g}, 2}}, \;{T_{{\rm g}, 3}})$= (0.0534, 0.0214, 0.0016, 0.0534, 0.0125, –0.0008, 0.1311, 0.0014, 0.0048) at this moment. (a, b) The streamfunction fields of the upper and lower layers, respectively. The contour intervals are (a) 2.0 × 107 m2 s–1 and (b) 2.0 × 106 m2 s–1. (c) Temporal evolution for the nondimensional equilibrium values of the zonal component $\psi _1^1$ (solid line) and $\psi _1^3$ (dashed line). (d) Temporal evolution for the nondimensional equilibrium values of the wave component $\psi _3^1$ (solid line) and $\psi _3^3$ (dashed line).

Fig. 6.  As in Fig. 3, but for Case 3 (without heat flux) with (a) m = 3.7 and (b) m = 6, (c) for Case 2 (without coupling) with m = 3.7, and (d) for Case 4 (without longwave radiation) with m = 3.7. Each ordinate shows the nondimensional equilibrium solution of the wave component $\psi _3^1$.

Fig. 7.  As in Fig. 1, but for the only stable equilibrium state for m = 3.7 at Q = 50 W m–2 in Case 2 (left panels) and for the third one of the three equilibrium states for m = 3.7 at Cg = 50 W m–2 in Case 4 (right panels). The former has nondimensional solutions with $({\psi _1}, \;{\psi _2}, \;{\psi _3}, \;{\theta _1}, \;{\theta _2}, \;{\theta _3})$ = (0.0656, –0.0679, 0.0259, 0.0613, –0.0616, 0.0253). The latter has nondimensional solutions with $({\psi _1}, \;{\psi _2}, \;{\psi _3}, \;{\theta _1}, \;{\theta _2}, \;{\theta _3}, \;{T_{{\rm g}, 1}}, \;{T_{{\rm g}, 2}}, \;{T_{{\rm g}, 3}})$ = (0.0662, –0.0863, 0.0278, 0.0615, –0.0776, 0.0272, 0.1762, –0.1551, 0.0544). They both belong to “Low 1” equilibria. The contour intervals are (a, e) 4.0 × 107 m2 s–1, (b, f) 2.0 × 106 m2 s–1, and (c, g, h) 10 K. Note that there is no land temperature field in Case 2 (left panel).

Fig. 8.  As in Fig. 1, but for the second one (left panels) and third one (right panels) of the three equilibrium states for m = 3.7 at Cg = 50 W m–2 in Case 3. They have nondimensional solutions with $({\psi _1}, \;{\psi _2}, \;{\psi _3}, \;{\theta _1}, \;{\theta _2}, \;{\theta _3}, \;{T_{{\rm g}, 1}}, \;{T_{{\rm g}, 2}}, \;{T_{{\rm g}, 3}})$ = (0.0623, –0.0002, –0.0020, 0.0626, –0.0004, –0.0019, 0.1596, –0.0004, –0.0021) and (0.0628, –0.0006, 0.0052, 0.0621, –0.0003, 0.0052, 0.1591, –0.0003, 0.0056), respectively. They belong to “High 2” and “High 1” equilibria, respectively. The contour intervals are (a, e) 2.0 × 107 m2 s–1, (b, f) 1.0 × 105 m2 s–1, and (c, d, g, h) 10 K.

Fig. 9.  The heating fields of the “High 2” (left panels) and “High 1” (right panels) equilibrium states shown in Fig. 1, respectively. (a, e) The shortwave radiation, (b, f) the longwave radiation, (c, g) the heat flux, and (d, h) the net diabatic heating absorbed by the atmosphere. All of the contour intervals are 10 W m–2. The background dotted lines show the topographic heights in the model, with negative regions shaded.

Fig. 10.  (a)–(d) The net diabatic heating absorbed by the atmosphere for the equilibrium states shown in Figs. 7 and 8, respectively. The contour intervals are (a, b) 30 W m–2 and (c, d) 5 W m–2. The background dotted lines show the topographic heights in the model, with negative regions shaded.

Fig. 11.  As in Fig. 3, but for dimensional variables for m = 3.7. Only stable equilibrium states are shown here. The blue (red) branch represents trough-type (ridge-type) equilibria, and the black branch represents the Hadley equilibria.

Fig. 12.  As in Fig. 11, but for m = 6.

Fig. 13.  Phase diagrams of dimensional variables for m = 3.7 (left panels) and m = 6 (right panels), respectively. Each ordinate shows the variable AH, and the abscissa gives (a, e) ΔTa, (b, f) ΔTg, (c, g) ATa, and (d, h) ATg. The meaning of colors and symbols are same as that in Fig. 3.

###### 通讯作者: 陈斌, bchen63@163.com
• 1.

沈阳化工大学材料科学与工程学院 沈阳 110142

## Multiple Equilibria in a Land–Atmosphere Coupled System

###### Corresponding author: Jianping HUANG, hjp@lzu.edu.cn;
• Key Laboratory for Semi-Arid Climate Change of the Ministry of Education, College of Atmospheric Sciences, Lanzhou University, Lanzhou 730000
Funds: Supported by the National Science Foundation of China (41521004 and 41705047), Strategic Priority Research Program of Chinese Academy of Sciences (XDA2006010301), Foundation of Key Laboratory for Semi-Arid Climate Change of the Ministry of Education in Lanzhou University from the Fundamental Research Funds for the Central Universities (lzujbky-2017-bt04), and China 111 Project (B 13045)

Abstract: Many low-order modeling studies indicate that there may be multiple equilibria in the atmosphere induced by thermal and topographic forcings. However, most work uses uncoupled atmospheric model and just focuses on the multiple equilibria with distinct wave amplitude, i.e., the high- and low-index equilibria. Here, a low-order coupled land–atmosphere model is used to study the multiple equilibria with both distinct wave phase and wave amplitude. The model combines a two-layer quasi-geostrophic channel model and an energy balance model. Highly truncated spectral expansions are used and the results show that there may be two stable equilibria with distinct wave phase relative to the topography: one (the other) has a lower layer streamfunction that is nearly in (out of) phase with the topography, i.e., the lower layer ridges (troughs) are over the mountains, called ridge-type (trough-type) equilibria. The wave phase of equilibrium state depends on the direction of lower layer zonal wind and horizontal scale of the topography. The multiple wave phase equilibria associated with ridge- and trough-types originate from the orographic instability of the Hadley circulation, which is a pitch-fork bifurcation. Compared with the uncoupled model, the land–atmosphere coupled system produces more stable atmospheric flow and more ridge-type equilibrium states, particularly, these effects are primarily attributed to the longwave radiation fluxes. The upper layer streamfunctions of both ridge- and trough-type equilibria are also characterized by either a high- or low-index flow pattern. However, the multiple wave phase equilibria associated with ridge- and trough-types are more prominent than multiple wave amplitude equilibria associated with high- and low-index types in this study.

Reference (51)

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